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# square planar complexes

January 7, 2021

The Pt 0 complex [Pt(PPh 3)(Eind 2-BPEP)] with a pyridine-based PNP-pincer-type phosphaalkene ligand (Eind 2-BPEP) has a highly planar geometry around Pt with â(Pt)=358.6°.This coordination geometry is very uncommon for formal d 10 complexes, and the Pd and Ni homologues with the same ligands adopt distorted tetrahedral geometries. The complexes forming square planar geometry has the electron configuration ending in d8. For example, d 8 square-planar Pd(ii) and Pt(ii) complexes are invariably low spin, while d 3 âd 7 tetrahedral complexes are high spin. Xem thêm: Substitution of square planar complexes , Substitution of square planar complexes , Substitution of square planar complexes . Ligand substitution in square planar complexes â¢ Associative Mechanism at D4h Centers â¢ Factors Which Affect The Rate Of Substitution 5. Which of the following is/are correct about [Cu(NH3)4]SO4 (A) It is a square planar complex. Treatment of Fe 2 (Mes) 4 (Mes = 2,4,6-Me 3 C 6 H 2) with monodentate phosphine and phosphite ligands furnished square planar trans-P 2 Fe(Mes) 2 derivatives. Platinum gives 2 of the 8 (6 + 2(due to charge)) electrons it had to 4 chlorine atoms. The Trans Effect in Square-Planar Platinum(II) ComplexesâA Density Functional Study 2371 Journal of Computational Chemistry DOI 10.1002/jcc. Cite this: J. Do take this answer with a grain of salt, because I cannot access the paper Iâm referencing here on holidays (I could when Iâm back at work) but the supplementaries confirm what Iâm trying to prove. Square planar compounds for p-elements are much rarer, but $\ce{XeF4}$ adopt such structure. This means that most square planar complexes are low spin, strong field ligands. (2-) indicates addition of two more electrons to platinum. Noté /5: Achetez Square Planar Complexes de Cross: ISBN: 9780138379643 sur amazon.fr, des millions de livres livrés chez vous en 1 jour * Corresponding authors a Department of Chemistry and Chemical Biology, Baker Laboratory, Cornell University, Ithaca, NY, USA E-mail: pc92@cornell.edu Fax: 607-255-4137 Now, researchers in the US have thrown away the textbook by synthesising a square-planar Fe(ii) complex that is not only high spin, but has a different core (FeO 2 NCl) to the only other examples of this complex type, all of which feature an FeO 4 core. Download âMolecular Orbital Theory â Octahedral, Tetrahedral or Square Planar Complexesâ ATOICV1-7-2-Molecular-Orbital-Theory-Octahedral-Tetrahedral-or-Square-Planar-Complexes.pdf â Downloaded 38 times â 834 KB In structure 1, the polymeric chain is formed by the alternating cationic [Rh(L1)(CO) 2] + and the anionic [Rh(CO) 2 Cl 2]-units leading to a neutral pseudo linear 1D chain with metallophilic contacts. B. To understand CFT, one must understand the description of the lobes: d xy: lobes lie in-between the x and the y axes. The square planar geometry is prevalent for transition metal complexes with d 8 configuration. Square planar is favoured electronically by d8 complexes. We find that the square planar complexes have the greatest crystal field splitting energy compared to all the other complexes. Description of d-Orbitals. Catalysis and Inhibition of Ligand Substitution in Palladium(II) Square-Planar Complexes: Effects of DNA. The square planar Ag(I) stereochemistry is generally acknowledged as rare, with only â¼2% of all reported silver complexes possessing this stereochemistry. Trans effect 6. Square planar complexes e.g [Ni(CN) 4]2-, [PtCl 4] 2-Symmetry analysis ( Ï-only) âo. Bibliography 3. Square Planar Complexes. Tetrahedral complexes form a two-tiered crystal field diagram. Synthesis of 14-Ï nickel(II) complexes with disubstituted benzoN4 macrocycles and electrochemical studies of 14-Ï nickel(II) complexes with monosubstituted benzoN4 macrocycles. They find this configuration particularly favourable as they completely fill the "lower" (i.e. Squaring the circle : The novel dienamido pincer ligand N(CHCHPt Bu 2) 2 â affords the isolation of the unusual squareâplanar iridium(II) and iridium(III) amido complexes [IrCl{N(CHCHPt Bu 2) 2}] n (n =0 (1 ), +1 (2 )).In contrast, the corresponding iridium(I) complex of the redox series (n =â1) is surprisingly unstable.The diamagnetism of 2 is attributed to strong NâIr Ï donation. Square Planar Complexes Consider a CFT diagram of a tetragonal elongation taken to its extreme: tetragonal elongation removal of z ligands eg t2g b2g dxydxzdyz eg dz2 dx2-y2 dxzdyz dxy dz2 dx2-y2 a1g b1g b2g eg dxzdyz dxy dz2 dx2-y2 a1g b1g â1,sp Octahedral Square Planar Î> Î . The configuration of the electrons in tetrahedral complexes can be from d0 or d10. experiment in the cited studies is due to large cancellation of errors, this approach is fully acceptable in our case since we are interested only in the relative (not absolute) activation energies of the reactions of the same class). Square-Planar Complexes The crystal field theory can be extended to square-planar complexes, such as Pt(NH 3) 2 Cl 2. The geometry is prevalent for transition metal complexes with d 8 configuration. The most common coordination polyhedra are octahedral, square planar and tetrahedral. In square planar molecular geometry, a central atom is surrounded by constituent atoms, which form the corners of a square on the same plane. Polyhedron 2002 â¦ 12-6 Substitution Reactions of Square-Planar Complexes 12-9 Reactions of Coordinated Ligand 12-3 Kinetic Consequences of Reaction Pathways 12-2 Substitution Reactions 12-1 History and Principles Chapter 12 Coordination Chemistry IV: Reaction and Mechanisms âInorganic Chemistryâ Third Ed. Platinum has 6 valence electrons. Packing of square planar Rh complexes favor formation of one-dimensional chains. As the z-ligands move away, the ligands in the square plane move a little closer to the metal. 2H 2 O complex containing tetradentate mda ligand (mda stands malamido-N,Nâ²-diacetato ion) .In the past decade, we worked with linear tetradentate ligands and corresponding copper(II) complexes as well .Three square-pyramidal (SPy) copper(II) complexes containing linear tetradentate edda-type â¦ Matteo Cusumano, Maria Letizia Di Pietro, Antonino Giannetto, Maria Anna Messina, and ; Francesco Romano; View Author Information. asked Oct 12, 2019 in Co-ordinations compound by KumarManish ( 57.6k points) coordination compounds à¸à¸à¸°à¹à¸£ 4. à¸à¸²à¸£à¹à¸à¸£à¸µà¸¢à¸à¹à¸à¸µà¸¢à¸à¹à¸à¸à¹à¸à¸µà¸¢à¸à¸à¹à¸²à¸à¸à¸±à¸ - Square Planar à¹à¸à¸µà¸¢à¸à¸à¸±à¸ Tetrahedral Complexes à¹à¸à¸£à¸¹à¸à¹à¸à¸à¸à¸²à¸£à¸²à¸ 5. à¸ªà¸£à¸¸à¸ . In square planar molecular geometry, a central atom is surrounded by constituent atoms, which form the corners of a square on the same plane. This includes Rh(I), Ir(I), Pd(II), Pt(II), and Au(III). Contribution from the Dipartimento di Chimica Inorganica, Chimica Analitica e Chimica Fisica, University of Messina, 98166 Messina, Italy . For example, [Co(NH 3) 6] 3+ is octahedral, [Ni(Co) 4] is tetrahedral and [PtCl 4] 2â is square planar. Click hereðto get an answer to your question ï¸ For the square planar complex, [M(a)(b)(c)(d)] (Where M = central metal and a, b, c and d are monodentate ligands), the number of â¦ Efficient Stereoselective Synthesis and Optical Properties of Heteroleptic SquareâPlanar Platinum(II) Complexes with Bidentate Iminopyrrolyl Ligands Dr. Ryo Inoue Corresponding Author E 18 vs 16-electron rules â18 electronâ rule use all available s, p and d orbitals to accommodate electrons â16 electronâ rule which orbital is not being used? Homoleptic: Complexes in which a metal is bound to only one kind of donor â¦ Tá»« khóa liên quan. Molecular Orbital Theory â Octahedral, Tetrahedral or Square Planar Complexes The crystal field theory fails to explain many physical properties of the transition metal complexes because it does not consider the interaction between the metal and ligand orbitals. 1. d-Orbital Splitting in Square Planar Coordination. The Spectrochemical Series . In compound 2, the square planar [Rh(CO) 2 Cl 2]-anions form only dinuclear anion pairs instead of â¦ The geometric preferences of a family of four coordinate, iron(II) d 6 complexes of the general form L 2 FeX 2 have been systematically evaluated. Thank you for prompting me to look for square planar complexes that are not $\mathrm{d^8}$; I learnt some valuable stuff while researching the answer to this question! Square planar coordination can be imagined to result when two ligands on the z-axis of an octahedron are removed from the complex, leaving only the ligands in the x-y plane. Each chlorine atom requires 1 electron to complete its octet. Square Planar Complexes Theories of trans effect â¢ Electrostatic Polarisation Theory â¢ Î  Bonding Theory â¢ Molecular Orbital Theory for Ï & Ï trans effects 7. Substitution in Square Planar Metal Complexes Extensively studied for square planar Ni2+, Pd2+ and Pt2+ (d8) substitutions in water and non-polar solvents: M(A)2(T)(X) + Y â M(A)2(T)(Y) + X â¢ rates depend on [M(A)2(T)(X)] and [Y] â¢ âSâ¡ and âVâ¡ are generally large and negative â¢ sensitive to nature of Y What type of mechanism is it? Transitions metals such as Rh(I), Ir (II), etc has square planar â¦ Title: Substitution reactions of square planar complexes 1 Substitution reactions of square planar complexes 2 especially d8 Ni(II), Rh(I), Pd(II), Ir(I), Pt(II), Au(III) 3 ML3X ? 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# 3.2: The Derivative as a Function

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As we have seen, the derivative of a function at a given point gives us the rate of change or slope of the tangent line to the function at that point. If we differentiate a position function at a given time, we obtain the velocity at that time. It seems reasonable to conclude that knowing the derivative of the function at every point would produce valuable information about the behavior of the function. However, the process of finding the derivative at even a handful of values using the techniques of the preceding section would quickly become quite tedious. In this section we define the derivative function and learn a process for finding it.

## Derivative Functions

The derivative function gives the derivative of a function at each point in the domain of the original function for which the derivative is defined. We can formally define a derivative function as follows.

Definition: derivative function

Let $$f$$ be a function. The derivative function, denoted by f′, is the function whose domain consists of those values of $$x$$ such that the following limit exists:

$f′(x)=\lim_{h→0}\frac{f(x+h)−f(x)}{h}.$

A function $$f(x)$$ is said to be differentiable at a if $$f'(a)$$ exists. More generally, a function is said to be differentiable on $$S$$ if it is differentiable at every point in an open set $$S$$, and a differentiable function is one in which $$f′(x)$$ exists on its domain.

In the next few examples we use Equation to find the derivative of a function.

Example $$\PageIndex{1}$$: Finding the Derivative of a Square-Root Function

Find the derivative of $$f(x)=\sqrt{x}$$.

Solution

Start directly with the definition of the derivative function.

 $$f′(x)=\lim_{h→0}\frac{\sqrt{x+h}−\sqrt{x}}{h}$$ Substitute$$f(x+h)=\sqrt{x+h}$$ and $$f(x)=\sqrt{x}$$ into $$f′(x)=\lim_{h→0}\frac{f(x+h)−f(x)}{h}$$. $$=\lim_{h→0}\frac{\sqrt{x+h}−\sqrt{x}}{h}⋅\frac{\sqrt{x+h}+\sqrt{x}}{\sqrt{x+h}+\sqrt{x}}$$ Multiply numerator and denominator by $$\sqrt{x+h}+\sqrt{x}$$ without distributing in thedenominator. $$=\lim_{h→0}\frac{h}{h(\sqrt{x+h}+\sqrt{x}})$$ Multiply the numerators and simplify. $$=\lim_{h→0}\frac{1}{(\sqrt{x+h}+\sqrt{x})}$$ Cancel the $$h$$. $$=\frac{1}{2\sqrt{x}}$$ Evaluate the limit

Example $$\PageIndex{2}$$: Finding the Derivative of a Quadratic Function

Find the derivative of the function $$f(x)=x^2−2x$$.

Solution

Follow the same procedure here, but without having to multiply by the conjugate.

 $$f′(x)=\lim_{h→0}\frac{((x+h)^2−2(x+h))−(x^2−2x)}{h}$$ Substitute $$f(x+h)=(x+h)^2−2(x+h)$$ and $$f(x)=x^2−2x$$ into $$f′(x)=\lim_{h→0}\frac{f(x+h)−f(x)}{h}$$ $$\lim_{h→0}\frac{x^2+2xh+h^2−2x−2h−x^2+2x}{h}$$ Expand $$(x+h)^2−2(x+h)$$. $$=\lim_{h→0}\frac{2xh−2h+h^2}{h}$$ Simplify $$=\lim_{h→0}\frac{h(2x−2+h)}{h}$$ Factor out $$h$$ from the numerator $$=\lim_{h→0}(2x−2+h)$$ Cancel the common factor of $$h$$ $$=2x−2$$ Evaluate the limit

Exercise $$\PageIndex{1}$$

Find the derivative of $$f(x)=x^2$$.

Hint

Use Equation and follow the example.

$$f′(x)=2x$$

We use a variety of different notations to express the derivative of a function. In Example we showed that if $$f(x)=x^2−2x$$, then $$f′(x)=2x−2$$. If we had expressed this function in the form $$y=x^2−2x$$, we could have expressed the derivative as $$y′=2x−2$$ or $$\frac{dy}{dx}=2x−2$$. We could have conveyed the same information by writing $$\frac{d}{dx}(x^2−2x)=2x−2$$. Thus, for the function $$y=f(x)$$, each of the following notations represents the derivative of $$f(x)$$:

$$f′(x), \frac{dy}{dx},y′,\frac{d}{dx}(f(x))$$.

In place of $$f′(a)$$ we may also use $$\frac{dy}{dx}∣x=a$$ Use of the $$\frac{dy}{dx}$$ notation (called Leibniz notation) is quite common in engineering and physics. To understand this notation better, recall that the derivative of a function at a point is the limit of the slopes of secant lines as the secant lines approach the tangent line. The slopes of these secant lines are often expressed in the form $$\frac{Δy}{Δx}$$ where $$Δy$$ is the difference in the $$y$$ values corresponding to the difference in the $$x$$ values, which are expressed as $$Δx$$ (Figure). Thus the derivative, which can be thought of as the instantaneous rate of change of y with respect to $$x$$, is expressed as

$$\frac{dy}{dx}=\lim_{Δx→0}\frac{Δy}{Δx}$$.

Figure $$\PageIndex{1}$$: The derivative is expressed as $$\frac{dy}{dx}=\lim_{Δx→0}\frac{Δy}{Δx}$$.

## Graphing a Derivative

We have already discussed how to graph a function, so given the equation of a function or the equation of a derivative function, we could graph it. Given both, we would expect to see a correspondence between the graphs of these two functions, since $$f′(x)$$ gives the rate of change of a function $$f(x)$$ (or slope of the tangent line to $$f(x)$$).

In Example we found that for $$f(x)=\sqrt{x},f′(x)=1/2\sqrt{x}$$. If we graph these functions on the same axes, as in Figure, we can use the graphs to understand the relationship between these two functions. First, we notice that $$f(x)$$ is increasing over its entire domain, which means that the slopes of its tangent lines at all points are positive. Consequently, we expect $$f′(x)>0$$ for all values of x in its domain. Furthermore, as $$x$$ increases, the slopes of the tangent lines to $$f(x)$$ are decreasing and we expect to see a corresponding decrease in $$f′(x)$$. We also observe that $$f(0)$$ is undefined and that $$\lim_{x→0^+}f′(x)=+∞$$, corresponding to a vertical tangent to $$f(x)$$ at $$0$$.

Figure $$\PageIndex{2}$$: The derivative $$f′(x)$$ is positive everywhere because the function $$f(x)$$ is increasing.

In Example we found that for $$f(x)=x^2−2x,f′(x)=2x−2$$. The graphs of these functions are shown in Figure. Observe that $$f(x)$$ is decreasing for $$x<1$$. For these same values of $$x,f′(x)<0$$. For values of $$x>1$$, $$f(x)$$ is increasing and $$f′(x)>0$$. Also, $$f(x)$$ has a horizontal tangent at $$x=1$$ and $$f′(1)=0$$.

Figure $$\PageIndex{3}$$: The derivative $$f′(x)<0$$ where the function $$f(x)$$ is decreasing and $$f′(x)>0$$ where $$f(x)$$ is increasing. The derivative is zero where the function has a horizontal tangent

Example $$\PageIndex{3}$$: Sketching a Derivative Using a Function

Use the following graph of $$f(x)$$ to sketch a graph of $$f′(x)$$.

Solution

The solution is shown in the following graph. Observe that $$f(x)$$ is increasing and $$f′(x)>0$$ on $$(–2,3)$$. Also, $$f(x)$$ is decreasing and $$f′(x)<0$$ on $$(−∞,−2)$$ and on $$(3,+∞)$$. Also note that $$f(x)$$ has horizontal tangents at $$–2$$ and $$3$$, and $$f′(−2)=0$$ and $$f′(3)=0$$.

Exercise $$\PageIndex{2}$$

Sketch the graph of $$f(x)=x^2−4$$. On what interval is the graph of $$f′(x)$$ above the $$x$$-axis?

Hint

The graph of $$f′(x)$$ is positive where $$f(x)$$ is increasing.

(0,+∞)

## Derivatives and Continuity

Now that we can graph a derivative, let’s examine the behavior of the graphs. First, we consider the relationship between differentiability and continuity. We will see that if a function is differentiable at a point, it must be continuous there; however, a function that is continuous at a point need not be differentiable at that point. In fact, a function may be continuous at a point and fail to be differentiable at the point for one of several reasons.

Differentiability Implies Continuity

Let $$f(x)$$ be a function and $$a$$ be in its domain. If $$f(x)$$ is differentiable at $$a$$, then $$f$$ is continuous at $$a$$.

Proof

If $$f(x)$$ is differentiable at $$a$$, then $$f′(a)$$ exists and

$$f′(a)=\lim_{x→a}\frac{f(x)−f(a)}{x−a}$$.

We want to show that $$f(x)$$ is continuous at $$a$$ by showing that $$\lim_{x→a}f(x)=f(a).$$ Thus,

$$\lim_{x→a}f(x)=\lim_{x→a}(f(x)−f(a)+f(a))$$

$$=\lim_{x→a}(\frac{f(x)−f(a)}{x−a}⋅(x−a)+f(a))$$ Multiply and divide $$f(x)−f(a)$$ by $$x−a$$.

$$=(\lim_{x→a}\frac{f(x)−f(a)}{x−a})⋅(\lim_{x→a}(x−a))+\lim_{x→a}f(a)$$

$$=f'(a)⋅0+f(a)$$

$$=f(a).$$

Therefore, since $$f(a)$$ is defined and $$\lim_{x→a}f(x)=f(a)$$, we conclude that $$f$$ is continuous at $$a$$.

We have just proven that differentiability implies continuity, but now we consider whether continuity implies differentiability. To determine an answer to this question, we examine the function $$f(x)=|x|$$. This function is continuous everywhere; however, $$f′(0)$$ is undefined. This observation leads us to believe that continuity does not imply differentiability. Let’s explore further. For $$f(x)=|x|$$,

$$f′(0)=\lim_{x→0}\frac{f(x)−f(0)}{x−0}=\lim_{x→0}\frac{|x|−|0|}{x−0}=\lim_{x→0}\frac{|x|}{x}$$.

This limit does not exist because

$$\lim_{x→0^−}\frac{|x|}{x}=−1$$ and $$\lim_{x→0^+}\frac{|x|}{x}=1$$.

See Figure.

Figure $$\PageIndex{4}$$: The function $$f(x)=|x|$$ is continuous at $$0$$ but is not differentiable at $$0$$.

Let’s consider some additional situations in which a continuous function fails to be differentiable. Consider the function $$f(x)=\sqrt[3]{x}$$:

$$f′(0)=\lim_{x→0}\frac{\sqrt[3]{x}−0}{x−0}=\lim_{x→0}\frac{1}{\sqrt[3]{x^2}}=+∞$$.

Thus $$f′(0)$$ does not exist. A quick look at the graph of $$f(x)=\sqrt[3]{x}$$ clarifies the situation. The function has a vertical tangent line at $$0$$ (Figure).

Figure $$\PageIndex{5}$$: The function $$f(x)=\sqrt[3]{x}$$ has a vertical tangent at $$x=0$$. It is continuous at $$0$$ but is not differentiable at $$0$$.

The function $$f(x)=\begin{cases}xsin(\frac{1}{x}) & if x≠0\\0 if x=0\end{cases}$$ also has a derivative that exhibits interesting behavior at $$0$$. We see that

$$f′(0)=\lim_{x→0}\frac{xsin(1/x)−0}{x−0}=\lim_{x→0}sin(\frac{1}{x})$$.

This limit does not exist, essentially because the slopes of the secant lines continuously change direction as they approach zero (Figure).

Figure $$\PageIndex{6}$$: The function $$f(x)=\begin{cases}xsin(\frac{1}{x})&ifx≠0\\0 & ifx=0\end{cases}$$ is not differentiable at $$0$$.

In summary:

1. We observe that if a function is not continuous, it cannot be differentiable, since every differentiable function must be continuous. However, if a function is continuous, it may still fail to be differentiable.
2. We saw that $$f(x)=|x|$$ failed to be differentiable at $$0$$ because the limit of the slopes of the tangent lines on the left and right were not the same. Visually, this resulted in a sharp corner on the graph of the function at $$0.$$ From this we conclude that in order to be differentiable at a point, a function must be “smooth” at that point.
3. As we saw in the example of $$f(x)=\sqrt[3]{x}$$, a function fails to be differentiable at a point where there is a vertical tangent line.
4. As we saw with $$f(x)=\begin{cases}xsin(\frac{1}{x})&ifx≠0\\0 & ifx=0\end{cases}$$ a function may fail to be differentiable at a point in more complicated ways as well.

Example $$\PageIndex{4}$$: A Piecewise Function that is Continuous and Differentiable

A toy company wants to design a track for a toy car that starts out along a parabolic curve and then converts to a straight line (Figure). The function that describes the track is to have the form $$f(x)=\begin{cases}\frac{1}{10}x^2+bx+c & if x<−10\\−\frac{1}{4}x+\frac{5}{2} & if x≥−10\end{cases}$$ where $$x$$ and $$f(x)$$ are in inches. For the car to move smoothly along the track, the function $$f(x)$$ must be both continuous and differentiable at $$−10$$. Find values of b and c that make $$f(x)$$ both continuous and differentiable.

Figure $$\PageIndex{7}$$: For the car to move smoothly along the track, the function must be both continuous and differentiable.

Solution

For the function to be continuous at $$x=−10$$, $$\lim_{x→10^−}f(x)=f(−10)$$. Thus, since

$$\lim_{x→−10^−}f(x)=\frac{1}{10}(−10)^2−10b+c=10−10b+c$$

and $$f(−10)=5$$, we must have $$10−10b+c=5$$. Equivalently, we have $$c=10b−5$$.

For the function to be differentiable at $$−10$$,

$$f′(10)=\lim_{x→−10}\frac{f(x)−f(−10)}{x+10}$$

must exist. Since $$f(x)$$ is defined using different rules on the right and the left, we must evaluate this limit from the right and the left and then set them equal to each other:

$$\lim_{x→−10^−}\frac{f(x)−f(−10)}{x+10}=\lim_{x→−10^−}\frac{\frac{1}{10}x^2+bx+c−5}{x+10}$$

$$=\lim_{x→−10^−}\frac{\frac{1}{10}x2+bx+(10b−5)−5}{x+10}$$ Substitute $$c=10b−5$$.

$$=\lim_{x→−10^−}\frac{x^2−100+10bx+100b}{10(x+10)}$$

$$=\lim_{x→−10^−}\frac{(x+10)(x−10+10b)}{10(x+10)}$$ Factor by grouping

$$=b−2$$.

We also have

$$\lim_{x→−10^+}\frac{f(x)−f(−10)}{x+10}$$

$$=\lim_{x→−10^+}\frac{−\frac{1}{4}x+\frac{5}{2}−5}{x+10}$$

$$=\lim_{x→−10^+}\frac{−(x+10)}{4(x+10)}$$

$$=−\frac{1}{4}$$.

This gives us $$b−2=−\frac{1}{4}$$. Thus $$b=\frac{7}{4}$$ and $$c=10(\frac{7}{4})−5=\frac{25}{2}$$.

Exercise $$\PageIndex{3}$$

Find values of a and b that make $$f(x)=\begin{cases}ax+b & if x<3\\x^2 & if x≥3\end{cases}$$ both continuous and differentiable at $$3$$.

Hint

Use Example as a guide.

$$a=6$$ and $$b=−9$$

## Higher-Order Derivatives

The derivative of a function is itself a function, so we can find the derivative of a derivative. For example, the derivative of a position function is the rate of change of position, or velocity. The derivative of velocity is the rate of change of velocity, which is acceleration. The new function obtained by differentiating the derivative is called the second derivative. Furthermore, we can continue to take derivatives to obtain the third derivative, fourth derivative, and so on. Collectively, these are referred to as higher-order derivatives. The notation for the higher-order derivatives of $$y=f(x)$$ can be expressed in any of the following forms:

$$f''(x),f'''(x),f^{(4)}(x),…,f^{(n)}(x)$$

$$y''(x),y'''(x),y^{(4)}(x),…,y^{(n)}(x)$$

$$\frac{d^2y}{dx^2},\frac{d^3y}{dy^3},\frac{d^4y}{dy^4},…,\frac{d^ny}{dy^n}.$$

It is interesting to note that the notation for $$\frac{d^2y}{dx^2}$$ may be viewed as an attempt to express $$\frac{d}{dx}(\frac{dy}{dx})$$ more compactly. Analogously, $$\frac{d}{dx}(\frac{d}{dx}(\frac{dy}{dx}))=\frac{d}{dx}(\frac{d^2y}{dx^2})=\frac{d^3y}{dx^3}$$.

Example $$\PageIndex{5}$$: Finding a Second Derivative

For $$f(x)=2x^2−3x+1$$, find $$f''(x)$$.

Solution

First find $$f′(x)$$.

 $$f′(x)=\lim_{h→0}\frac{(2(x+h)^2−3(x+h)+1)−(2x^2−3x+1)}{h}$$ Substitute $$f(x)=2x^2−3x+1$$ and $$f(x+h)=2(x+h)^2−3(x+h)+1$$ into $$f′(x)=\lim_{h→0}\frac{f(x+h)−f(x)}{h}.$$ $$=\lim_{h→0}\frac{4xh+h^2−3h}{h}$$ Simplify the numerator. $$=\lim_{h→0}(4x+h−3)$$ Factor out the $$h$$ in the numerator and cancel with the $$h$$ in the denominator. $$=4x−3$$ Take the limit.

Next, find $$f''(x)$$ by taking the derivative of $$f′(x)=4x−3.$$

 $$f''(x)=\lim_{h→0}\frac{f′(x+h)−f′(x)}{h}$$ Use $$f′(x)=\lim_{h→0}\frac{f(x+h)−f(x)}{h}$$ with $$f ′(x)$$ in place of $$f(x).$$ $$=\lim_{h→0}\frac{(4(x+h)−3)−(4x−3)}{h}$$ Substitute $$f′(x+h)=4(x+h)−3$$ and $$f′(x)=4x−3.$$ $$=\lim_{h→0}4$$ Simplify. $$=4$$ Take the limit.

Exercise $$\PageIndex{4}$$

Find $$f''(x)$$ for $$f(x)=x^2$$.

Hint

We found $$f′(x)=2x$$ in a previous checkpoint. Use Equation to find the derivative of $$f′(x)$$

$$f''(x)=2$$

Example $$\PageIndex{6}$$: Finding Acceleration

The position of a particle along a coordinate axis at time $$t$$ (in seconds) is given by $$s(t)=3t^2−4t+1$$ (in meters). Find the function that describes its acceleration at time $$t$$.

Solution

Since $$v(t)=s′(t)$$ and $$a(t)=v′(t)=s''(t)$$, we begin by finding the derivative of $$s(t)$$:

$$s′(t)=\lim_{h→0}\frac{s(t+h)−s(t)}{h}$$

$$=\lim_{h→0}\frac{3(t+h)^2−4(t+h)+1−(3t^2−4t+1)}{h}$$

$$=6t−4.$$

Next,

$$s''(t)=\lim_{h→0}\frac{s′(t+h)−s′(t)}{h}$$

$$=\lim_{h→0}\frac{6(t+h)−4−(6t−4)}{h}$$

$$=6.$$

Thus, $$a=6 m/s^2$$.

Exercise $$\PageIndex{5}$$

For $$s(t)=t^3$$, find $$a(t).$$

Hint

Use Example as a guide.

$$a(t)=6t$$

## Key Concepts

• The derivative of a function $$f(x)$$ is the function whose value at $$x$$ is $$f′(x)$$.
• The graph of a derivative of a function $$f(x)$$ is related to the graph of $$f(x)$$. Where $$f(x)$$ has a tangent line with positive slope, $$f′(x)>0$$. Where $$f(x)$$ has a tangent line with negative slope, f′(x)<0. Where $$f(x)$$ has a horizontal tangent line, $$f′(x)=0.$$
• If a function is differentiable at a point, then it is continuous at that point. A function is not differentiable at a point if it is not continuous at the point, if it has a vertical tangent line at the point, or if the graph has a sharp corner or cusp.
• Higher-order derivatives are derivatives of derivatives, from the second derivative to the $$nth$$ derivative.

## Key Equations

• The derivative function

$$f′(x)=\lim_{h→0}\frac{f(x+h)−f(x)}{h}$$

## Glossary

derivative function
gives the derivative of a function at each point in the domain of the original function for which the derivative is defined
differentiable at a
a function for which $$f′(a)$$ exists is differentiable at $$a$$
differentiable on S
a function for which $$f′(x)$$ exists for each $$x$$ in the open set $$S$$ is differentiable on $$S$$
differentiable function
a function for which $$f′(x)$$ exists is a differentiable function
higher-order derivative
a derivative of a derivative, from the second derivative to the nth derivative, is called a higher-order derivative

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### 12th Standard Physics Study material & Free Online Practice Tests - View and download Sample Question Papers with Solutions for Class 12 Session 2019 - 2020 TN Stateboard

#### 12th Physics - Term 1 Model Question Paper - by Pavithra - Karaikkudi Sep 21, 2019 - View & Download

• 1)

An electric dipole is placed at an alignment angle of 300 with an electric fi eld of 2 × 105 NC-1. It experiences a torque equal to 8 N m. Th e charge on the dipole if the dipole length is 1 cm is

• 2)

A parallel plate capacitor stores a charge Q at a voltage V. Suppose the area of the parallel plate capacitor and the distance between the plates are each doubled then which is the quantity that will change?

• 3)

A carbon resistor of (47 ± 4.7 ) k Ω to be marked with rings of diff erent colours for its identifi cation. Th e colour code sequence will be

• 4)

A non-conducting charged ring of charge q, mass m and radius r is rotated with constant angular speed ω. Find the ratio of its magnetic moment with angular momentum is

• 5)

When the current changes from +2A to −2A in 0.05 s, an emf of 8 V is induced in a coil. Th e co-efficient of self-induction of the coil is

#### 12th Physics - Term 1 Five Mark Model Question Paper - by Sarojini - Tiruppur Sep 16, 2019 - View & Download

• 1)

When two objects are rubbed with each other, approximately a charge of 50 nC can be produced in each object. Calculate the number of electrons that must be transferred to produce this charge.

• 2)

Two electric bulbs marked 20 W – 220 V and 100 W – 220 V are connected in series to 440 V supply. Which bulb will be fused?

• 3)

What is the value of x when the Wheatstone’s network is balanced?
P = 500 Ω, Q = 800 Ω, R = x + 400, S = 1000 Ω

• 4)

A cell supplies a current of 0.9 A through a 2 Ω resistor and a current of 0.3 A through a 7 Ω resistor. Calculate the internal resistance of the cell.

• 5)

In a plane Electromagnetic wave the electric field oscillates sinusoidally at a frequency of 1.5 X 1010Hz & amplitude 36 Vm-1.
(i) What is the wavelength of a wave?
(ii) What the amplitude of the oscillating magnetic field?
(iii) Straight the average energy density of the electric field $\left( \overrightarrow { E } \right)$ , is equal to average energy density of the magnetic field $\left( \overrightarrow { B } \right)$

#### 12th Physics - Electromagnetic Waves Two Marks Questions - by Sarojini - Tiruppur Sep 16, 2019 - View & Download

• 1)

What is displacement current?

• 2)

What are electromagnetic waves?

• 3)

Write down the integral form of modified Ampere’s circuital law.

• 4)

Explain the concept of intensity of electromagnetic waves.

• 5)

What is meant by Fraunhofer lines?

#### 12th Physics - Electromagnetic Induction and Alternating Current Two Marks Questions - by Sarojini - Tiruppur Sep 16, 2019 - View & Download

• 1)

How is Eddy current produced? How do they flow in a conductor?

• 2)

Mention the ways of producing induced emf.

• 3)

Give the principle of AC generator.

• 4)

List out the advantages of stationary armature-rotating field system of AC generator.

• 5)

What do you mean by resonant frequency?

#### 12th Physics - Magnetism and Magnetic Effects of Electric Current Two Marks Questions - by Sarojini - Tiruppur Sep 16, 2019 - View & Download

• 1)

What is meant by magnetic induction?

• 2)

Define magnetic flux.

• 3)

What is magnetic susceptibility?

• 4)

State Biot-Savart’s law.

• 5)

What is meant by hysteresis?

#### 12th Physics - Current Electricity Two Marks Questions - by Sarojini - Tiruppur Sep 16, 2019 - View & Download

• 1)

What is superconductivity?

• 2)

What is electric power and electric energy?

• 3)

State Kirchhoff’s voltage rule.

• 4)

State the principle of potentiometer.

• 5)

What is Peltier effect?

#### 12th Physics Chapter 1 Electrostatics Two Marks Questions - by Sarojini - Tiruppur Sep 16, 2019 - View & Download

• 1)

What is meant by quantisation of charges?

• 2)

Write down Coulomb’s law in vector form and mention what each term represents.

• 3)

What is mean by ‘Electric field lines’?

• 4)

The electric field lines never intersect Justify.

• 5)

What is an equipotential surface?

#### 12th Physics Quarterly Model Question Paper - by Sarojini - Tiruppur Sep 13, 2019 - View & Download

• 1)

Two identical point charges of magnitude –q are fi xed as shown in the fi gure below. A third charge +q is placed midway between the two charges at the point P. Suppose this charge +q is displaced a small distance from the point P in the directions indicated by the arrows, in which direction(s) will +q be stable with respect to the displacement

• 2)

An electric field $\vec { E } =10x\hat { i }$ exists in a certain region of space. Then the potential difference V = Vo – VA, where Vo is the potential at the origin and VA is the potential at x = 2 m is:

• 3)

An uncharged metal sphere is placed between two equal and oppositely charged metal plates. The nature of lines of force will be

• 4)

The magnitude of electric dipole moment of water molecule is

• 5)

Which of the following statement on equipotential surface is wrong?

#### 12th Standard Physics - Electromagnetic waves Book Back Questions - by Sarojini - Tiruppur Sep 03, 2019 - View & Download

• 1)

What is displacement current?

• 2)

What are electromagnetic waves?

• 3)

What is meant by Fraunhofer lines?

• 4)

The relative magnetic permeability of the medium is 2.5 and the relative electrical permittivity of the medium is 2.25. Compute the refractive index of the medium.

• 5)

Discuss briefly the experiment conducted by Hertz to produce and detect electromagnetic spectrum.

#### 12th Standard Physics - Magnetism and Magnetic Effects of Electric Current Book Back Questions - by Sarojini - Tiruppur Sep 02, 2019 - View & Download

• 1)

Th e force experienced by a particle having mass m and charge q accelerated through a potential diff erence V when it is kept under perpendicular magnetic field $\vec { B }$ is

• 2)

Th ree wires of equal lengths are bent in the form of loops. One of the loops is circle, another is a semi-circle and the third one is a square. Th ey are placed in a uniform magnetic fi eld and same electric current is passed through them. Which of the following loop confi guration will experience greater torque ?

• 3)

A non-conducting charged ring of charge q, mass m and radius r is rotated with constant angular speed ω. Find the ratio of its magnetic moment with angular momentum is

• 4)

The vertical component of Earth’s magnetic field at a place is equal to the horizontal component. What is the value of angle of dip at this place?

• 5)

A simple pendulum with charged bob is oscillating with time period T and let θ be the angular displacement. If the uniform magnetic field i switched ON in a direction perpendicular to the plane of oscillation then

#### 12th Standard Physics - Electromagnetic Induction and Alternating Current Book Back Questions - by Sarojini - Tiruppur Sep 02, 2019 - View & Download

• 1)

Th e fl ux linked with a coil at any instant t is given by $\Phi$B =10t2−50t+ 250 . The induced emf at t = 3s is

• 2)

A circular coil with a cross-sectional area of 4 cm2 has 10 turns. It is placed at the centre of a long solenoid that has 15 turns/cm and a cross-sectional area of 10 cm2. The axis of the coil coincides with the axis of the solenoid. What is their mutual inductance?

• 3)

An inductor 20 mH, a capacitor 50 μF and a resistor 40 Ω are connected in series across a source of emf v = 10 sin 340 t. The power loss in AC circuit is

• 4)

In an oscillating LC circuit, the maximum charge on the capacitor is Q. The charge on the capacitor when the energy is stored equally between the electric and magnetic fields is

• 5)

$\frac{20}{\pi^2}H$inductor is connected to a capacitor of capacitance C. The value of C in order to impart maximum power at 50 Hz is

#### 12th Standard Physics Chapter 2 Current Electricity Book Back Questions - by Sarojini - Tiruppur Aug 31, 2019 - View & Download

• 1)

A toaster operating at 240 V has a resistance of 120 Ω. Th e power is

• 2)

What is the value of resistance of the following resistor?

• 3)

In India electricity is supplied for domestic use at 220 V. It is supplied at 110 V in USA. If the resistance of a 60W bulb for use in India is R, the resistance of a 60W bulb for use in USA will be

• 4)

The temperature coefficient of resistance of a wire is 0.00125 per °C. At 300 K, its resistance is 1 Ω. The resistance of the wire will be 2 Ω at

• 5)

A piece of copper and another of germanium are cooled from room temperature to 80 K. The resistance of

#### 12th Standard Physics Unit 1 Electrostatics Book Back Questions - by Sarojini - Tiruppur Aug 30, 2019 - View & Download

• 1)

Two identical point charges of magnitude –q are fi xed as shown in the fi gure below. A third charge +q is placed midway between the two charges at the point P. Suppose this charge +q is displaced a small distance from the point P in the directions indicated by the arrows, in which direction(s) will +q be stable with respect to the displacement

• 2)

An electric dipole is placed at an alignment angle of 300 with an electric fi eld of 2 × 105 NC-1. It experiences a torque equal to 8 N m. Th e charge on the dipole if the dipole length is 1 cm is

• 3)

The total electric fl ux for the following closed surface which is kept inside water

• 4)

Two identical conducting balls having positive charges q1 and q2 are separated by a center to center distance r. If they are made to touch each other and then separated to the same distance, the force between them will be

• 5)

Rank the electrostatic potential energies for the given system of charges in increasing order.

#### 12th Standard Physics Chapter 4 Electromagnetic Induction and Alternating Current One Mark Question with Answer Key - by Sarojini - Tiruppur Aug 29, 2019 - View & Download

• 1)

In a series resonant RLC circuit, the voltage across 100 Ω resistor is 40 V. The resonant frequency ω is 250 rad/s. If the value of C is 4 μF, then the voltage across L is

• 2)

An inductor 20 mH, a capacitor 50 μF and a resistor 40 Ω are connected in series across a source of emf v = 10 sin 340 t. The power loss in AC circuit is

• 3)

The instantaneous values of alternating current and voltage in a circuit are $i=\frac { 1 }{ \sqrt { 2 } } sin\left( 100\pi t \right)$ A and v=$=\frac { 1 }{ \sqrt { 2 } } sin\left( 100\pi t+\frac { \pi }{ 3 } \right) V.$The average power in watts consumed in the circuit is

• 4)

In an oscillating LC circuit, the maximum charge on the capacitor is Q. The charge on the capacitor when the energy is stored equally between the electric and magnetic fields is

• 5)

$\frac{20}{\pi^2}H$inductor is connected to a capacitor of capacitance C. The value of C in order to impart maximum power at 50 Hz is

#### 12th Standard Physics Chapter 3 Magnetism and Magnetic Effects of Electric Current One Mark Question and Answer - by Sarojini - Tiruppur Aug 29, 2019 - View & Download

• 1)

The BH curve for a ferromagnetic material is shown in the figure. The material is placed inside a long solenoid which contains 1000 turns/cm. The current that should be passed in the solenonid to demagnetize the ferromagnet completely is

• 2)

Two short bar magnets have magnetic moments 1.20 Am2 and 1.00 Am2 respectively. They are kept on a horizontal table parallel to each other with their north poles pointing towards the south. They have a common magnetic equator and are separated by a distance of 20.0 cm. The value of the resultant horizontal magnetic induction at the mid-point O of the line joining their centers is (Horizontal components of Earth’s magnetic induction is 3.6 x 10-5 Wb m-2 )

• 3)

The vertical component of Earth’s magnetic field at a place is equal to the horizontal component. What is the value of angle of dip at this place?

• 4)

A flat dielectric disc of radius R carries an excess charge on its surface. The surface charge density is σ. The disc rotates about an axis perpendicular to its plane passing through the center with angular velocity ω. Find the magnitude of the torque on the disc if it is placed in a uniform magnetic field whose strength is B which is directed perpendicular to the axis of rotation

• 5)

A simple pendulum with charged bob is oscillating with time period T and let θ be the angular displacement. If the uniform magnetic field i switched ON in a direction perpendicular to the plane of oscillation then

#### 12th Standard Physics - Current Electricity One Mark Question and Answer - by Sarojini - Tiruppur Aug 28, 2019 - View & Download

• 1)

A toaster operating at 240 V has a resistance of 120 Ω. Th e power is

• 2)

A carbon resistor of (47 ± 4.7 ) k Ω to be marked with rings of diff erent colours for its identifi cation. Th e colour code sequence will be

• 3)

When 'n' resistors of equal resistance (R) are connected in series and in parallel respectively,then the ratio of their effective resistance is

• 4)

Which of the following has negative temperature coefficient of resistance?

• 5)

The temperature co-efficient of resistance for alloys is

#### 12th Standard Physics Unit 1 Electrostatics One Mark Question and Answer - by Sarojini - Tiruppur Aug 27, 2019 - View & Download

• 1)

An electric dipole is placed at an alignment angle of 300 with an electric fi eld of 2 × 105 NC-1. It experiences a torque equal to 8 N m. Th e charge on the dipole if the dipole length is 1 cm is

• 2)

The total electric fl ux for the following closed surface which is kept inside water

• 3)

An electric field $\vec { E } =10x\hat { i }$ exists in a certain region of space. Then the potential difference V = Vo – VA, where Vo is the potential at the origin and VA is the potential at x = 2 m is:

• 4)

The value of constant 'K' in coulomb law is

• 5)

The electrostatic force is always greater in magnitude than gravitational force for_________ object.

#### 12th Standard Chapter 5 Electromagnetic Waves One Mark Question Paper with Answer - by Sarojini - Tiruppur Aug 27, 2019 - View & Download

• 1)

Consider an oscillator which has a charged particle and oscillates about its mean position with a frequency of 300 MHz. Th e wavelength of electromagnetic waves produced by this oscillator is

• 2)

Th e electric and the magnetic field, associated with an electromagnetic wave, propagating along X axis can be represented by

• 3)

In an electromagnetic wave in free space the rms value of the electric fi eld is 3 V m−1. The peak value of the magnetic field is

• 4)

Electromagnetic waves are produced by

• 5)

Which of the following electromagnetic waves has the highest frequency?

#### 12th Standard Physics Chapter 4 Electromagnetic Induction and Alternating Current One Mark Question with Answer Key - by Sarojini - Tiruppur Aug 27, 2019 - View & Download

• 1)

Th e fl ux linked with a coil at any instant t is given by $\Phi$B =10t2−50t+ 250 . The induced emf at t = 3s is

• 2)

When the current changes from +2A to −2A in 0.05 s, an emf of 8 V is induced in a coil. Th e co-efficient of self-induction of the coil is

• 3)

A step-down transformer reduces the supply voltage from 220 V to 11 V and increase the current from 6 A to 100 A. Then its efficiency is

• 4)

In a series RL circuit, the resistance and inductive reactance are the same. Then the phase difference between the voltage and current in the circuit is

• 5)

An inductor 20 mH, a capacitor 50 μF and a resistor 40 Ω are connected in series across a source of emf v = 10 sin 340 t. The power loss in AC circuit is

#### 12th Physics Unit 3 Magnetism and Magnetic Effects of Electric Current One Mark Question with Answer Key - by Sarojini - Tiruppur Aug 26, 2019 - View & Download

• 1)

A circular coil of radius 5 cm and 50 turns carries a current of 3 ampere. The magnetic dipole moment of the coil is

• 2)

A thin insulated wire forms a plane spiral of N = 100 tight turns carrying a current I = 8 m A (milli ampere). The radii of inside and outside turns are a = 50 mm and b = 100 mm respectively.
The magnetic induction at the center of the spiral is

• 3)

A wire of length l carries a current I along the Y direction and magnetic field is given by $\vec { B } =\frac { \beta }{ \sqrt { 3 } } =(\hat { i } +\hat { j } +\hat { k } )T.$ The magnitude of Lorentz force acting on the wire is

• 4)

The vertical component of Earth’s magnetic field at a place is equal to the horizontal component. What is the value of angle of dip at this place?

• 5)

When current is doubled deflection is also doubled in

#### 12th Physics Chapter 2 Current Electricity One Mark Question Paper - by Sarojini - Tiruppur Aug 24, 2019 - View & Download

• 1)

A toaster operating at 240 V has a resistance of 120 Ω. Th e power is

• 2)

A carbon resistor of (47 ± 4.7 ) k Ω to be marked with rings of diff erent colours for its identifi cation. Th e colour code sequence will be

• 3)

In India electricity is supplied for domestic use at 220 V. It is supplied at 110 V in USA. If the resistance of a 60W bulb for use in India is R, the resistance of a 60W bulb for use in USA will be

• 4)

In Joule’s heating law, when I and t are constant, if the H is taken along the y axis and I2 along the x axis, the graph is

• 5)

Which of the following has negative temperature coefficient of resistance?

#### 12th Physics Electrostatics One Mark Question Paper - by Sarojini - Tiruppur Aug 23, 2019 - View & Download

• 1)

Two identical point charges of magnitude –q are fi xed as shown in the fi gure below. A third charge +q is placed midway between the two charges at the point P. Suppose this charge +q is displaced a small distance from the point P in the directions indicated by the arrows, in which direction(s) will +q be stable with respect to the displacement

• 2)

Which charge configuration produces a uniform electric field?

• 3)

If voltage applied on a capacitor is increased from V to 2V, choose the correct conclusion

• 4)

Two metallic spheres of radii 1 cm and 3 cm are given charges of -1 x 10-2 C and 5 x 10-2 C respectively. If these are connected by a conducting wire, the final charge on the bigger sphere is

• 5)

Based on Franklin's convention amber rods are_________.

#### 12th Physics Quarterly Exam Model Two Marks Question Paper - by Sarojini - Tiruppur Aug 20, 2019 - View & Download

• 1)

What is meant by quantisation of charges?

• 2)

Give the relation between electric field and electric potential.

• 3)

Give a comparison of electrical and gravitional forces?

• 4)

The graph shows the variation of volatage 'v' across the plates of two capacitors A & Bversus increase of charge 'Q' stored on them. Which of the two capacitors has higher capacitance?
Give reason.

• 5)

State microscopic form of Ohm’s law.

#### Plus Two Quarterly Exam Model One Mark Question Paper - by Sarojini - Tiruppur Aug 05, 2019 - View & Download

• 1)

Which charge configuration produces a uniform electric field?

• 2)

An electric field $\vec { E } =10x\hat { i }$ exists in a certain region of space. Then the potential difference V = Vo – VA, where Vo is the potential at the origin and VA is the potential at x = 2 m is:

• 3)

Two points A and B are maintained at a potential of 7 V and -4 V respectively. The work done in moving 50 electrons from A to B is

• 4)

In electrostatics if the charges are in motion, another force named __________ comes into play in addition to coulomb force

• 5)

The value of constant 'K' in coulomb law is

#### Plus 2 Physics Chapter 5 Electromagnetic Waves Model Question Paper - by Sarojini - Tiruppur Aug 03, 2019 - View & Download

• 1)

The dimension of $\frac { 1 }{ { \mu }_{ 0 }{ \varepsilon }_{ 0 } }$ is

• 2)

During the propagation of electromagnetic waves in a medium

• 3)

The EM waves do not transport

• 4)

Which EM waves are used in medicine to destroy cancer cells?

• 5)

A magnetic field is produced by

#### 12th Standard Physics First Mid Term Model Question Paper - by Sarojini - Tiruppur Aug 01, 2019 - View & Download

• 1)

Four Gaussian surfaces are given below with charges inside each Gaussian surface. Rank the electric fl ux through each Gaussian surface in increasing order.

• 2)

The value of constant 'K' in coulomb law is

• 3)

For a large charge accumulation, the end of the conductor should have larger curvature that is_________.

• 4)

A carbon resistor of (47 ± 4.7 ) k Ω to be marked with rings of diff erent colours for its identifi cation. Th e colour code sequence will be

• 5)

Th ree wires of equal lengths are bent in the form of loops. One of the loops is circle, another is a semi-circle and the third one is a square. Th ey are placed in a uniform magnetic fi eld and same electric current is passed through them. Which of the following loop confi guration will experience greater torque ?

#### 11th Standard Physics Chapter 4 Electromagnetic Induction and Alternating Current Important Question Paper - by Sarojini - Tiruppur Jul 30, 2019 - View & Download

• 1)

Th e fl ux linked with a coil at any instant t is given by $\Phi$B =10t2−50t+ 250 . The induced emf at t = 3s is

• 2)

When the current changes from +2A to −2A in 0.05 s, an emf of 8 V is induced in a coil. Th e co-efficient of self-induction of the coil is

• 3)

A circular coil with a cross-sectional area of 4 cm2 has 10 turns. It is placed at the centre of a long solenoid that has 15 turns/cm and a cross-sectional area of 10 cm2. The axis of the coil coincides with the axis of the solenoid. What is their mutual inductance?

• 4)

In a transformer, the number of turns in the primary and the secondary are 410 and 1230 respectively. If the current in primary is 6A, then that in the secondary coil is

• 5)

A step-down transformer reduces the supply voltage from 220 V to 11 V and increase the current from 6 A to 100 A. Then its efficiency is

#### 12th Standard Physics Chapter 3 Magnetism and Magnetic Effects of Electric Current Important Question Paper - by Sarojini - Tiruppur Jul 26, 2019 - View & Download

• 1)

The magnetic fi eld at the center O of the following current loop is

• 2)

A circular coil of radius 5 cm and 50 turns carries a current of 3 ampere. The magnetic dipole moment of the coil is

• 3)

A thin insulated wire forms a plane spiral of N = 100 tight turns carrying a current I = 8 m A (milli ampere). The radii of inside and outside turns are a = 50 mm and b = 100 mm respectively.
The magnetic induction at the center of the spiral is

• 4)

Four wires each of length 2m are bent into four loops P, Q, R and S and then suspended into uniform magnetic field as shown in the figure same current is passed in each loop. On which loop the couple will be the highest?

• 5)

A current carrying conductor is associated with

#### 12th Physics Unit 2 Model Question Paper - by Sarojini - Tiruppur Jul 24, 2019 - View & Download

• 1)

A toaster operating at 240 V has a resistance of 120 Ω. Th e power is

• 2)

A carbon resistor of (47 ± 4.7 ) k Ω to be marked with rings of diff erent colours for its identifi cation. Th e colour code sequence will be

• 3)

Two wires of A and B with circular cross section made up of the same material with equal lengths. Suppose RA = 3 RB, then what is the ratio of radius of wire A to that of B?

• 4)

A wire connected to a power supply of 230 V has power dissipation P1. Suppose the wire is cut into two equal pieces and connected parallel to the same power supply. In this case power dissipation is P2. The ratio $\frac{P_2}{P_1}$ is

• 5)

When 'n' resistors of equal resistance (R) are connected in series and in parallel respectively,then the ratio of their effective resistance is

#### 12th Physics - Chapter 1 Electrostatics Important Question Paper - by Sarojini - Tiruppur Jul 18, 2019 - View & Download

• 1)

Two identical point charges of magnitude –q are fi xed as shown in the fi gure below. A third charge +q is placed midway between the two charges at the point P. Suppose this charge +q is displaced a small distance from the point P in the directions indicated by the arrows, in which direction(s) will +q be stable with respect to the displacement

• 2)

An electric dipole is placed at an alignment angle of 300 with an electric fi eld of 2 × 105 NC-1. It experiences a torque equal to 8 N m. Th e charge on the dipole if the dipole length is 1 cm is

• 3)

Four Gaussian surfaces are given below with charges inside each Gaussian surface. Rank the electric fl ux through each Gaussian surface in increasing order.

• 4)

The total electric fl ux for the following closed surface which is kept inside water

• 5)

Two identical conducting balls having positive charges q1 and q2 are separated by a center to center distance r. If they are made to touch each other and then separated to the same distance, the force between them will be

#### 12th Standard Physics Electrostatics Important Question Paper - by Ravikumar Jul 18, 2019 - View & Download

• 1)

Two identical point charges of magnitude –q are fi xed as shown in the fi gure below. A third charge +q is placed midway between the two charges at the point P. Suppose this charge +q is displaced a small distance from the point P in the directions indicated by the arrows, in which direction(s) will +q be stable with respect to the displacement

• 2)

The total electric fl ux for the following closed surface which is kept inside water

• 3)

The electrostatic force is always greater in magnitude than gravitational force for_________ object.

• 4)

The relative permittivity of water is _______.

• 5)

An isolated metal sphere of radius 'r' is given a charge' q'. The potential energy of the sphere is

#### twelfth standard english medium important multiple choice questions in physics - by Ravikumar Jun 27, 2019 - View & Download

• 1)

For sharper edge, the ______ is greater. This principle is used in Lightning arrestor

• 2)

For continuous charge distributions, _____ methods can be used.

• 3)

For a large charge accumulation, the end of the conductor should have larger curvature that is_________.

• 4)

Relative permittivity (εr) is also known as ________

• 5)

The energy stored per unit volume of space is defined as________.

#### frequently asked multiple choice questions twelfth standard english medium physics - by Ravikumar Jun 27, 2019 - View & Download

• 1)

A charge Q is placed at the corner of a cube. The electric flux through all the six faces of the cube is

• 2)

Which graph shows the variation of electric field E due to a hollow spherical conductor of radius R as a function of distance from the centre of the sphere?

• 3)

In a hollow spherical shell potential (V) changes with respect to distance (r) from centre.
$\left[ { V }_{ inside }=\frac { Q }{ 4\pi { \varepsilon }_{ 0 }R } r\le R,\quad { V }_{ outside }=\frac { Q }{ 4\pi { \varepsilon }_{ 0 }r } r\ge R,V\alpha \frac { 1 }{ r } \right]$

• 4)

What physical quantities may X and Y represent? [Y represents the first mentioned quantity].

• 5)

During charging a capacitor variation of potential V of the capacitor with time t as shown a

#### Most important one mark questions physics chapter one - by Ravikumar Jun 27, 2019 - View & Download

• 1)

If C is the capacitance of an air filled capacitor and C is the capacitance of dielectric filled capacitor, then

• 2)

The capacitance of a parallel plate capacitor increases from 5μf of 50μf when a dielectric is filled between the plates. The permitivity of the dielectric is

• 3)

An electric dipole placed at an angle in a nonuniform electric field experiences

• 4)

Two copper spheres A and B of same size are charged to same potential. A is hollow and B is solid. Which of the two holds more charge?

• 5)

A bird sitting an a high power line

#### Centum multiple choice questions Chepter one Physics - by Ravikumar Jun 27, 2019 - View & Download

• 1)

The expression for the electric field due to a surface of total charge 'Q' is given by

• 2)

The dipole is called point dipole when the distance

• 3)

The magnitude of torque on dipole is maximum if

• 4)

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• 5)

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#### New syllabus creative multiple choice questions physics - by Ravikumar Jun 27, 2019 - View & Download

• 1)

The figure shows tow parallel equipotential surface A and B kept at a small distance 'r' a part from each other. A point change of Q coulomb is taken from the surface A to B. The amount of net work done will be

• 2)

The given figure is a plot of lines of force due to two charges q1 & q2 find out the sign of charges

• 3)

An uncharged metal sphere is placed between two equal and oppositely charged metal plates. The nature of lines of force will be

• 4)

An isolated metal sphere of radius 'r' is given a charge' q'. The potential energy of the sphere is

• 5)

In a hydrogen atom the electron revolves around the proton in an orbit of 0.53 $\mathring { A }$. The potential produced by the electron on the nuleus is

#### Physics chapter one new syllabus Important one mark questions - by Ravikumar Jun 27, 2019 - View & Download

• 1)

Two identical point charges of magnitude –q are fi xed as shown in the fi gure below. A third charge +q is placed midway between the two charges at the point P. Suppose this charge +q is displaced a small distance from the point P in the directions indicated by the arrows, in which direction(s) will +q be stable with respect to the displacement

• 2)

Which charge configuration produces a uniform electric field?

• 3)

An electric dipole is placed at an alignment angle of 300 with an electric fi eld of 2 × 105 NC-1. It experiences a torque equal to 8 N m. Th e charge on the dipole if the dipole length is 1 cm is

• 4)

Four Gaussian surfaces are given below with charges inside each Gaussian surface. Rank the electric fl ux through each Gaussian surface in increasing order.

• 5)

Two identical conducting balls having positive charges q1 and q2 are separated by a center to center distance r. If they are made to touch each other and then separated to the same distance, the force between them will be

### TN Stateboard Education Study Materials

#### 12th Stateboard Physics 2019 - 2020 Academic Monthly Syllabus - by Sarojini - Tiruppur Aug 02, 2019 Aug 02, 2019

Physics 2019 - 2020 Academic Monthly Syllabus

#### TN StateboardStudy Material - Sample Question Papers with Solutions for Class 12 Session 2019 - 2020

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text:
# Proof that the sequence is divergent

Prove that the sequence $a_n = q^n , q \in \mathbb{R}$ is convergent or divergent .
I need some explanations for some cases:
a)If $q = -1$ (is a divergent sequence).
Suppose $a_n = (-1)^n$ is convergent.
With the Cauchy criterion:
$\forall \epsilon > 0, \exists N \in \mathbb{N},\forall n,m \geq N:$
$|a_n-a_m| < \epsilon$
Then: Choose $\epsilon = 1$ (after the solution =afs)
Now my question is why shall I choose 1 for epsilon?
And if I have choosen 1:
$\forall \epsilon > 0, \exists N \in \mathbb{N},\forall n,m \geq N:$
$|a_n-a_m| < 1$
Choose m=N+1 then $n,m \geq N$;
So I have: $|(-1)^n-(-1)^m|=2$ (ats)
=> Contradiction $2 \geq 1$
Why is m=N+1 choosen? and finally why do I only look at the case: $|(-1)^n-(-1)^m|$ is 2,
shouldn't I also write the case for n,m = even?
Thanks for explanations.

-

The choice $\epsilon=1$ is partly arbitrary: any value satisfying $0<\epsilon\le 2$ will work in this proof. $1$ is a simple, convenient number in that interval.

Similarly, the choice of $m=N+1$ is partly arbitrary: any $m\ge N+1$ will work. However, you’ve stated this part of the argument incorrectly. After setting $m=N+1$, the argument should continue by observing that $|a_N-a_m|=\left|(-1)^N-(-1)^{N+1}\right|=2\ge 1=\epsilon$. The point is to show that for this particular choice of $\epsilon$, namely, $\epsilon=1$, it is not true that $|a_n-a_m|<\epsilon$ for all $m,n\ge N$; specifically, it’s not true for $n=N$ and $m=N+1$. And this is the case no matter how big we make $N$, so this $\epsilon$ doesn’t ‘work’: it doesn’t have the property that every positive $\epsilon$ must have in order for the sequence to converge.

We don’t look at indices $m,n\ge N$ for which $|a_n-a_m|$ is less than $\epsilon$ because they don’t matter: we’re trying to show that $\epsilon=1$ is a ‘bad’ $\epsilon$, meaning that no matter how big we choose $N$, we can find at least one pair of $m,n\ge N$ such that $|a_n-a_m|\ge\epsilon$. And we’ve done that: $n=N$ and $m=N+1$ do the job. (Note in connection with your first question that they would do the job for any choice of positive $\epsilon\le 2$.)

-
Ok, and the interval for epsilon is ]0;2] because 2 is the supremum of {|a_N-a_m|}?And that means if I have an other sequence where I have to show the same, I always take a number x out of the "epsilon-interval" and then m=N+x, to prove convergence? –  phil Dec 10 '12 at 0:30
@phil: Better: because no matter what $N$ is, there is an $m\ge N$ such that $|a_N-a_m|=2$. If the supremum were not attained, you’d have to limit $\epsilon$ to the interval $(0,2)$. If a sequence diverges, your plan should be to find an $\epsilon>0$ such that there are differences $|a_m-a_n|\ge\epsilon$ arbitrarily far out in the sequence, i.e., with arbitrarily large $m$ and $n$. You won’t always be able to choose $m$ and $n$ as $N$ and $N+k$ for some $k$. –  Brian M. Scott Dec 10 '12 at 0:36
In the case of the sequence $(-1)^n$, the values bounce between $-1$ and $1$. The Cauchy criterion requires that if you go far enough out, the values all get very close together. Clearly they don't do that, because any two consecutive values are $2$ apart.
To make this argument more rigorous, if the sequence is Cauchy, then choosing $\epsilon = 1$, there must be an $N$ so that all $a_i$ after $N$ are at most $1$ apart. But for any value of $N$, $a_{N+1}$ is $2$ apart from $a_N$. So there is no such $N$, and therefore the sequence is not Cauchy.
So, why do we choose $\epsilon = 1$? Just because it's a number less than $2$ that's easy to work with. We could have chosen $1.9$ or $0.001$ and the argument would work just as well.
Why do we choose $m = N + 1$? Just because it's an example of a value that makes $a_m$ not too close to $a_N$. Other values that would also work are $m = N + 3$ or $m = N + 5$. We just have to find one example that makes the Cauchy criterion fail.


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# Put webMathematica folder outside Tomcat webapps folder?

Is it possible to put the webMathematica folder outside the usual Tomcat webapps folder and still have webMathematica work?

webMathematica does work OK if its whole folder is placed inside Tomcat's webapps — and accessed in the browser as localhost:8080/webMathematica.

(This is with Tomcat 9 under OS X 10.10.4 and using webMathematica 3.3.2 calling Mathematica 10.2.)

Configuration details would be greatly appreciated!

In asking this I have four aims:

1. Granularize things so that I can just back up the webMathematica folder, along with other documents of mine, separately from backing up Tomcat's own files such as those in its distributed webapps folder.

2. Be able to access webMathematica (and, eventually, some specific apps using it) on localhost by means of a simple URL "alias", such as webmathematica, rather than the usual complete URL localhost:8080/webMathematica — just as I can now access Tomcat's webapps by entering the URL tomcat.local.

3. [added] Still allow use of Tomcat for webapps other than webMathematica. Thus changing the entire Tomcat docBase value — if that actually worked — would not allow achieving this aim.

4. [added] Be able to run webMathematica in my browser by entering a simple URL such as webmathematica.local(without the fully-qualified URL localhost:8080/webMathematica).

• Did you try a symbolic link? – Rolf Mertig May 6 '13 at 21:00
• @Rolf Mertig: Yes, a symlink works, with URL http://localhost:8080/webMathematica! But what I'd actually like to do is use a simpler URL such as webmathematica.local, and for that obviously some kind of additional configuration of tomcat and/or apache is required. (In apache I'm passing calls to tomcat via ajp.) – murray May 6 '13 at 21:24
• For the URL part I think this and no.4 in this should get you started... – Albert Retey May 7 '13 at 6:12
• Alas, I don't comprehend how to put the "this" and "no. 4 in this" together for the situation I describe. Don't I need, also to add an entry such as 127.0.0.1 webmathematica.local to /etc/hosts and a corresponding VirtualHost entry to apache2/conf/extra/httpd-vhosts.conf? – murray Aug 14 '15 at 14:01

Yes, it's possible. So you don't have to type the webMathematica after your localhost address.
2. Create a file named ROOT.xml inside \Tomcat\conf\Catalina\localhost\
3. In this file, write: <Context docBase="C:\webMathematica" reloadable="false"/>
• No, that's not working (OS X 10.10.4) even if I keep the webmathematica folder in its default position /usr/local/apache-tomcat-7.0.37/webapps and use that location in the docBase value in the file  /usr/local/apache-tomcat-7.0.37/conf/Catalina/localhost/ROOT.xml` that you describe. After restarting Tomcat (and apache), browsing to localhost:8080 still brings up the default Tomcat page, not the webMathematica page. – murray Aug 13 '15 at 20:36


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# Find joint pdf table of two discrete independent random variables $X$ and $Y$

Given the pdfs of two discrete independent variables $$X$$ and $$Y$$, write the joint pdf. There is a property that $$if\ \ p_{XY}(x,y) = p_X(x)p_Y(y) \ \forall i,j \Rightarrow \text{X,Y are independent}$$. I am not that sure if this statement is true for $$"\Leftarrow"$$ and I cannot find another way to solve this problem.


text:
# Stirling’s log-factorial approximation

In my statistical physics course, I have frequently encountered the following approximation due to Stirling:

$\ln(N!) \approx N \ln (N) -N$

It’s very useful but my professor didn’t explain how good the approximation was. The derivation I found turns out to be very simple and so I can present it in a few lines here:

1. Note that:$\int \ln(x) dx = x \ln (x) -x \quad (1)$
2. Now, if we define$S = \sum_{n=1}^{N} \ln (n) \quad (2)$
we have an upper-Riemann sum with $\Delta x =1$.
3. So we basically have the following approximation:
$S = \sum_{n=1}^{N} \ln (n \Delta x)\Delta x \approx \int_{1}^{N} \ln(x) dx \quad (3)$
4. By the intermediate value theorem,$\forall n \in\mathbb{N} \thinspace \exists c_n \in (n-1,n), S' = \sum_{n=1}^{N} \ln (c_n \Delta x)\Delta x =\int_{1}^{N} \ln(x) dx \quad(4)$where $\Delta x =1$ as defined previously.
5. Let’s check how good this approximation is:$|S - S'| = | \sum_{n=1}^{N} \ln (n) - \sum_{n=1}^{N} \ln (c_n) |$$\Rightarrow |S - S'| \leq | \sum_{n=1}^{N} \ln (n) -\ln (n+1)| = \ln (N+1) \quad (5)$
6. This error grows very slowly. In fact, if $N = 10^{24}$ i.e. the number of molecules in a glass of water,$| \ln(N!) - (N \ln (N) -N)| < 60$ which is a minuscule error relative to the number of molecules.

Note: A pdf version of this blog post is available here.

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# generalized AM-GM inequality

As I was reading through my linear analysis notes today, there was a small
passage where our lecturer Jim Wright provided a proof of a generalisation
of the inequality of arithmetic and geometric means:

Let $A, B \geq 0$ and $0 \leq \theta \leq 1$. Then $A^{\theta} B^{1-\theta} \leq \theta A + (1-\theta) B$.

I didn’t bother to read the proof as I thought I could probably come up with a
good one myself. Indeed, I am sufficiently happy with the proof I found that
I think it’s worth sharing. Here it is:

Proof:

If $AB=0$ the proof is trivial. So I shall proceed by assuming the contrary:

1. Let’s define $\beta := \frac{A}{B} \in \mathbb{R}_{+}$ and divide by $B$. We then obtain:$\beta^{\theta} \leq \theta \beta + (1-\theta)$
2. Now, let’s define the following differentiable functions:$\begin{cases} f(x) =\theta x + (1-\theta) \\ g(x) =x^{\theta} \\ \end{cases}$
3. The derivatives of these functions are given by:$\begin{cases} f'(x) =\theta \\ g'(x) =\theta x^{\theta-1} \\ \end{cases}$
4. Now, we can quickly reach the following conclusions:a) $f(x) \geq g(x)$ for $x \geq 1$
b) For $x \in [0,1]$ $\begin{cases} f(0) \geq g(0)=0 \\ f(1)=g(1)=1 \\ \end{cases}$

and both functions are monotonically increasing on $(0,1)$ so there can’t be $x \in (0,1)$ such that $f(x) = g(x)$ unless $\theta$ equals zero or one.

It follows that $f(x) \geq g(x)$ on $(0,1)$.

I must say that this was an easy problem but I liked the method I found for
solving it. Namely, reducing the number of variables and then replacing
variables with functions that can then be readily analysed. But, we can go a bit further
and show how Hölder’s inequality follows easily.

Hölder’s inequality:

For any two sequences $(a_i)_{i=1}^n,(b_i)_{i=1}^n \subset \mathbb{N}$, we have:

$\sum_{j=1}^{n} a_j b_j = 1 \leq (\sum_{j=1}^{n} a_j^p)^{\frac{1}{p}}(\sum_{j=1}^{n} b_j^q)^{\frac{1}{q}}$

for $1 \leq p \leq \infty$ where $\frac{1}{p} + \frac{1}{q}=1$.

Proof:

If we define $a=(a_i)_{i=1}^n, b= (b_i)_{i=1}^n$, and normalize these vectors we have:

$\begin{cases} \alpha = \frac{a}{||a||_p} \\ \beta = \frac{b}{||b||_q} \\ \end{cases}$

Now, we may apply the generalised AM-GM inequality to deduce:

$\sum_{j=1}^{n} \alpha_j \beta_j \leq \frac{1}{p}(\sum_{j=1}^{n} \alpha_j^p) +\frac{1}{q}(\sum_{j=1}^{n} \beta_j^q)= ||\alpha||_p||\beta||_q$

# The Quadrupedal Conjecture

Last summer, I asked myself several related questions concerning the locomotion of quadrupeds:

1) What if the cheetah had more than four legs?
2) Why don’t hexapods gallop?
3) Have we ever had F1 cars with more than 4 wheels?

I found all these related questions interesting but after more reflection I summarised my problem mathematically:

Could four legs be optimal for rapid linear and quasi-linear locomotion on rough planar surfaces?

After searching for papers on this subject and finding none, I decided to name this problem ‘The Quadrupedal Conjecture’ and it may be informally described as saying that four legs allow a creature to travel fastest on a nearly-flat surface. However, I thought it might be interesting to tackle a simpler problem first which we may call the ‘Little Polypedal Conjecture’:

Can we show that given a polyped with $2 n$ legs having pre-defined mass and energy, that there exists $N$ such that for $2 N$ limbs or greater, its maximum linear velocity on a rough planar surface would be reduced?

I believe that there is an elegant solution to the above problem and I think that the quadrupedal conjecture has an elegant solution as well. But it is by no means guaranteed that a solution to such a problem would be simple. Finding the right degree of abstraction would be one of the main challenges as there are many different ways of approaching this problem with varying degrees of realism.

First, I must say that this is not primarily a problem in biology although this question has attracted the attention of biologists on the biology stackexchange. Second, I think this is a question that would heavily involve mathematical physics although this question has been controversial on the physics stackexchange. Further, I think that the solution to this problem would be affected by mass scaling. By this I mean that the optimal number of legs would probably vary with the range of masses available to the polyped.

Finally, I think that this question is highly relevant to roboticists who build legged robots as a thorough investigation of this question would probably lead to better models for polypedal locomotion.

That’s all I can reasonably say for now.

Note: I thought I’d add a touch of melodrama to this blog post as a friend of mine told me that my blog posts can be a bit dry.

# Computing the Omega constant

I) Solving a simple-looking equation:

Last Sunday I was trying to compute a complicated integral suggested to me by a friend and was led to consider the problem of solving the following innocent-looking equation:

$x\log(x)=1 \quad (1)$

Little did I know that this little problem would lead me to transcendental functions and the Lambert W function in particular. But, before saying more on those two topics I’ll go through what I tried:

1. Given that $x \in (1,e)$, I tried $x:=e^\alpha$ for $\alpha \in [0,1]$ so that I obtain $\alpha e^\alpha=1$. This resembles the familiar exponential function $f(t)=e^{\alpha t}$.

2. I tried to recover some useful relations and I found the following:

$\forall t \in [0,1]\quad f^n(t)f^{n+1}(t)=\alpha^{2n} \quad (2)$

Using the mean-value theorem:

$\int_{0}^{1} f^n(t)dt=\alpha^{n-1}(\frac{1}{\alpha}-1) \quad (3)$

which isn’t too difficult to show given that

$latex \begin{cases} \exists c_n \in (0,1) \quad f^{n+1}(c_n)=f^n(1)-f^n(0)=\alpha^n f'(c_0) \\ c_0 = \frac{1}{\alpha}\log(\frac{1}{\alpha^2}-\frac{1}{\alpha})\\ \end{cases} &s=1$

After this I still couldn’t figure out how to solve for $\alpha$ analytically. I wondered whether there might be a relation which I failed to discover but at this point it was a bit late so I thought I might want to try a numerical method.

3. Using Newton’s method:

$\large t_{n+1}=t_n-\frac{g(t_n)}{g'(t_{n+1)}}=t_n-\frac{t_n\ln(t_n)+t_n^2}{1+t_n} \quad (4)$

where $g(t)=\log(t e^t)$

I found that $\alpha \approx 0.567143$ and wondered whether it might be related to other numbers I knew. I tried a few calculations which got me nowhere and then I became more interested in whether this problem could be solved without resorting to numerical methods so I asked this on the math stackexchange. From that forum I learned that $\alpha$ is actually known as the Omega constant and it appears as $W(1)$ where $W(x)$ is the Lambert W function, a transcendental function.

II) An equation that can’t be solved analytically:

A transcendental function, $f(x)$ is basically a function that transcends algebra in the sense it can’t be expressed in terms of a finite number of algebraic operations: addition, subtraction, multiplication, division, raising to a power, and taking roots. In particular it can’t satisfy:

$\sum_{k=1}^{n} c_k(x) (f(x))^k = 0 \quad (5)$

where $c_k(x)$ are rational functions(i.e. a ratio of polynomials).

Now, it isn’t too difficult to show that:

$t(x) =e^{x\log(x)}=x^x \quad (6)$

is actually transcendental if we suppose that $t(x)$ actually satisfies (5). If we choose $x \geq 1$, $|t(x)| \geq 1$ we find that

$latex \begin{cases} |c_n(x)| |t(x)|^n \leq b(x) |t(x)|^{n-1} \\ b(x) =\sum_{k=1}^{n} |c_k(x)| \\ \end{cases} \quad (6) &s=1$

It then follows that

$\forall x \in \mathbb{R}, |t(x)| \leq \frac{b(x)}{|c_n(x)|} \quad (7)$

However, in (7) the expression on the left grows exponentially while the expression on the right has at most polynomial growth so $t(x)$ can’t be algebraic and hence can’t be solved analytically.

I’ll write a separate blog post on the Lambert W function as I think this blog post is getting a bit long and that function is worth it’s own post as it has many applications.


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# Bdeogale jacksoni — Details

## Overview

### Distribution

#### Range Description

Known only from central and southern Kenya, south-eastern Uganda, and from the Udzungwa Mountains (900 km to the south) in Tanzania (Van Rompaey and Kingdon in press). The Tanzanian population seems to be highly localized: camera-trapping conducted over three years in 76 sites throughout the Udzungwa Mountains recorded Jackson’s mongoose in only five sites, all within Matundu Forest (De Luca and Rovero 2006). It is possible that this species occurs in and around other massifs within the Eastern Arc Mountains, including the Ulugurus, Ngurus and Usambaras. Recorded to montane elevations in the Aberdares, Mt. Kenya and Mt Elgon up to 3,300 m asl.

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Source: IUCN

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## Ecology

### Habitat

#### Habitat and Ecology

Habitat and Ecology
Recorded from lowland forest as well as in montane forests and bamboo zones. Kingdon (1997) suggested that the presence of remains of rodents of the genera Dasymus and Otomys in stomachs suggests that they hunt frequently in the thick herbaceous growth around swamps.

Systems
• Terrestrial

© International Union for Conservation of Nature and Natural Resources

Source: IUCN

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## Conservation

### Conservation Status

#### IUCN Red List Assessment

Red List Category
NT
Near Threatened

Red List Criteria

Version
3.1

Year Assessed
2008

Assessor/s
Van Rompaey, H., De Luca, D., Rovero, F. & Hoffmann, M.

Reviewer/s
Duckworth, J.W. (Small Carnivore Red List Authority) and Hoffmann, M. (Global Mammal Assessment Team)

Justification
Listed as Near Threatened as the species is believed to have undergone a decline of 20-25% over the last 15 years (assuming a generation length of 5 years) due to the ongoing loss of habitat in its range. However, several populations are in protected areas, and the species probably is more widely distributed than currently known. Almost qualifies as threatened under criterion A2cd.

History
• 1996
Vulnerable
(Baillie and Groombridge 1996)
• 1994
Insufficiently Known
(Groombridge 1994)
• 1990
Insufficiently Known
(IUCN 1990)
• 1988
Insufficiently Known
(IUCN Conservation Monitoring Centre 1988)

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Source: IUCN

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#### Population

Population
There is no reliable information, but they are believed to be rare.

Population Trend
Decreasing

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### Threats

Major Threats
Given its apparent dependence on forest habitat, the main threat to this species is likely to be ongoing forest loss in its range (Van Rompaey and Kingdon in press).

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Source: IUCN

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### Management

#### Conservation Actions

Conservation Actions
In Kenya, Jackson’s Mongoose is confirmed in the Aberdare N.P. and Mt. Kenya N.P., and is also likely to occur in Mt. Elgon N.P. (Van Rompaey and Kingdon in press). In Tanzania, all confirmed localities lie within protected areas, including the Udzungwa Mountains N.P. De Luca and Rovero (2006) recommend full protection of forests adjacent to the Udzungwa Mountains N.P., and that other East African groundwater-dependant forests (including those near Matundu and Nambiga Forest Reserves) be surveyed for the presence of Jackson’s Mongoose.

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## Wikipedia

#### Jackson's mongoose

Jackson's mongoose (Bdeogale jacksoni) is a species of mongoose belonging to the genus Bdeogale. Discovered in 1889 by Frederick John Jackson, Oldfield Thomas in 1894 described it as Galeriscus jacksoni. It is most closely related to the black-footed mongoose of the same subgenus Galeriscus and both are sometimes united in a single species.

With a head and body length of more than 50 cm (20 in) and a body weight of 2–3 kg (4.4–6.6 lb), it is a large mongoose. Its long and dense fur is grizzled black and white, the cheeks, the throat and the sides of the neck are very yellowish, the legs are dark brown or black, and the bushy tail is white.

Jackson's mongoose feeds on rodents and insects, especially on army ants, and is mainly nocturnal and crepuscular and possibly solitary. Its distributional range in Kenya, Uganda and Tanzania is limited to some arboreous mountain regions. It appears to be rare, and in 2008, the IUCN classified it as Near Threatened.

## Body features

Jackson's mongoose is a large mongoose with a bushy tail. Its head and body length is 50.8–57.1 cm (20.0–22.5 in), its tail length is 28.3 to 32.4 centimeters, its hind foot length is 8.6 to 10.8 cm, its ear length is 2.3 to 3.5 cm and its body weight is 2-3 kg.[2] Young but already breeding animals may be markedly smaller than adults. From the black-footed mongoose, it is distinguishable by its much longer fur, especially on the tail, and yellowish tints on the neck and the throat.[3]

The long and dense dorsal fur is grizzled black and white. The dorsal hairs are 20 mm (0.79 in) long with black and white rings. The muzzle and the chin are brownish white and the cheeks, the throat and the sides of the neck are very yellowish. The legs are dark brown or black and the tail is white. The ventral side is light gray and the underfur is dense and woolly. The pinnae are round and broad and the muzzle is blunt. The rhinarium is large, and the hairless extension of the median groove divides the upper lip. The fore and hind feet have only four digits. A hallux and pollex are absent as is common with Bdeogale. The soles are naked, and the claws are thick and strong.[2]

The dentition of Jackson's mongoose is typical for mongooses. Three incisors, one canine, four premolars and two molars are on either side of each jaw. The total number of teeth is 40 and the dental formula is $Upper: 3.1.4.2, lower: 3.1.4.2, total: 40$.[2]

## Habitat and habits

The habitats of Jackson's mongoose are montane forests, bamboo zones, and lowland forests in mountain vicinity. Its population density is low.[3] It probably hunts frequently in the thick herbaceous growth around swamps.[1]

Jackson's mongoose is carnivorous, insectivorous and myrmecophagous. In the Aberdare Mountains, the volume of food items in 40 scats was over 50% rodents, including Otomys, Dasymys and Praomys and 40% insects, mostly army ants of the genus Anomma, but also weevils, other beetles and caterpillars. Millipedes, snails, lizards, and the eggs of snakes were also consumed. About 80% of the juveniles' diet were rodents, including Otomys, brush-furred mice, true mice and Praomys. They also fed on beetles, lizards, birds, and a few ants.[2][3] Coping with columns of army ants may depend on maturity and learning, which suggests a recent evolutionary adaptation to this diet.[3]

Jackson's mongoose is mainly nocturnal and crepuscular.[3] In the Udzungwa Mountains, 73% of 25 photos taken by camera traps during the night were shot between 7 p.m. and midnight. It is possibly solitary, but often is seen in pairs and occasionally in groups of four. Nothing is known about its reproduction.[2]

## Distribution

The distributional range of Jackson's mongoose is very limited.[3] It is known from central and southern Kenya, southeastern Uganda and the Udzungwa Mountains 900 km to the south in central Tanzania,[1] where it was first recorded in 2001/02.[4] In the Aberdare Range, the Mount Kenya massif, and the Mount Elgon massif, it occurs at elevations up to 3300 m above sea level.[1] In the Udzungwa Mountains, it seems highly localized and has only been recorded within Matunda Forest at a maximum of 2.65 km apart.[5] It possibly inhabits other massifs of the Eastern Arc Mountains, including the Uluguru Mountains, the Nguru Mountains and the Usambara Mountains.[1]

## Classification

Jackson's mongoose is usually considered a species of the genus Bdeogale. It is most closely related to the black-footed mongoose (Bdeogale nigripes),[6] from which it is distinguished by skull and skin differences,[7] as whose mountain isolate, it is often treated.[3] Both are sometimes united in the species Bdeogale nigripes[7] or in the subgenus Galeriscus,[8] or they are separated from Bdeogale as distinct genus Galeriscus.[6]

Jackson's mongoose is treated as a separate species by Allen (1939),[9] Rosevear (1974),[10] Corbet and Hill (1980),[11] Honacki et al. (1982),[12] Nowak and Paradiso (1983),[13] Corbet and Hill (1986),[14] Schliemann (1988),[15] Corbet and Hill (1991),[16] Nowak (1991),[17] Wozencraft (1993),[18] Kingdon (1997),[3] Nowak (1999),[8] Pavlinov (2003),[19] Wozencraft (2005),[7] Van Rompaey et al. (2008)[1] and Gilchrist et al. (2009).[2] Kingdon (1977) treats it as a subspecies of the black-footed mongoose[20] and Dücker (1972) also considers them conspecific.[21]

No subspecies of Jackson's mongoose have been described.[7]

## Nomenclature

Oldfield Thomas in June 1894 described Jackson's mongoose as type species Galeriscus jacksoni of the new genus Galeriscus.[10] He specified the type locality as Mianzini in the land of the Maasai at an elevation of 8000 ft (2438 m) above sea level. Reginald Ernest Moreau, George Henry Evans Hopkins and Robert William Hayman in 1945/46 restricted the type locality at Mianzini to a few miles east-southeast of Lake Naivasha, on the southern end of the Kinangop Plateau and to an elevation of 9000 ft (2743 m).[7]

Frederick John Jackson discovered Jackson's mongoose in 1889[22] and in 1894 sent a skin without skull to the British Museum in London.[23] Led astray by the poor and incomplete type specimen, Thomas originally believed it to be related to the grisons.[24] Paul Matschie in 1895 named it Massaimarder (German, "Maasai marten"),[22] but was the first to recognize it as a Bdeogale.[25] Reginald Innes Pocock in 1916, too, considered Galeriscus a synonym of Bdeogale.[23] Ned Hollister in 1918 assumed it to be a geographic subspecies of the black-footed mongoose.[26]

## Population and conservation

Jackson's mongoose occurs in isolated populations and appears to be rare.[2] There are no reliable data on its population, though. The International Union for Conservation of Nature in 2008 classified it as Near Threatened, almost qualifying as Threatened. This was justified with its assumed decline in population size of 20 to 25% within the last 15 years due to habitat destruction. Given its dependence on forest habitat, its main threat is likely to be ongoing forest loss. It was classified as Insufficiently Known in 1988, 1990 and 1994, and as Vulnerable in 1996.[1]

Several populations of Jackson's mongoose are in protected areas, including Aberdare National Park, Mount Kenya National Park, probably Mount Elgon National Park and Udzungwa Mountains National Park. In Tanzania, all confirmed localities are within protected areas. It is also probably more widely distributed than currently known. De Luca and Rovero (2006) recommend the full protection of forests adjacent to the Udzungwa Mountains National Park and a survey of other East African groundwater-dependent forests for its presence.[1]

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## Rocket Camera, CMOS vs. CCD

I'm looking for a small camera to use in my rocket. It's about a 7' rocket, 4" diameter and 3" long viewing window. I've heard that the choices are either cmos or ccd but I dont know the difference between the two. I'm only looking to spend up to $100 (really looking in the$50 range). Any specific cameras you've already worked with? How easy is it to modify?

Thanks.
 Well, for starters, Oskee-wow-wow! Second, I doubt you are going to find much in that price range. I was just talking about the differences between CMOS and CCD with a colleague earlier in the day today actually, and it seems the biggest difference is that a CCD can operate faster from what I could tell. You may try Googling it though since I am not an expert.

 Tags camera, cmos, payload, rocket, rocketry


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# Tag Info

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### Why are ambiguous grammars bad?

Consider the following grammar for arithmetic expressions: $$X \to X + X \mid X - X \mid X * X \mid X / X \mid \texttt{var} \mid \texttt{const}$$ Consider the following expression: $$a - b - c$$ ...
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### Why are ambiguous grammars bad?

In contrast to the other existing answers [1, 2], there is indeed a field of application, where ambiguous grammars are useful. In the field of natural language processing (NLP), when you want to parse ...
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### does every CFL have an ambiguous CFG?

Every nonempty context-free language has an ambiguous grammar. Consider any context-free grammar for the language with starting symbol $S$. We add new non-terminals $S',A',B'$, make $S'$ the new ...
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### Is every unambiguous grammar regular?

The following grammar is unambiguous yet generates a non-regular language: $$S \to aSb \mid \epsilon$$
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### How do I reconstruct the forest of syntax trees from the Earley vector?

I am using terminology and notations from Earley's paper. It is possible that the description you read is different. It seems frequent that general CF parsing algorithms are first presented in the ...
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### When is the concatenation of two regular languages unambiguous?

Hint: Given DFAs for $A$ and $B$, construct an NFA which accepts words in $AB$ having at least two different decompositions. The NFA keeps track of two copies of the standard NFA for $AB$ (formed by ...
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### Why are ambiguous grammars bad?

Even if there’s a well-defined way to handle ambiguity (ambiguous expressions are syntax errors, for example), these grammars still cause trouble. As soon as you introduce ambiguity into a grammar, a ...
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### unambiguous grammar that produce equal number of a and b

The problem with $S\to aSbS\mid bSaS\mid \varepsilon$ is that you're just making sure you match each $a$ with a $b$ (where we consider an $a$ and a $b$ to be matched iff they appeared during the same ...
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### Why are ambiguous grammars bad?

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### Prove that regular expression is unambiguous

This answer has 3 parts: I first answer the question as asked, sketching the beginning of a proof concerning the example given. A full proof is too long and tedious to be given here. Then I give a ...
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### Method for Creating Any Unambiguous Grammar?

No, and there is indeed a reduction to the undecidability of ambiguity. First, note that ambiguity is (for context-free grammars) semi-decidable: just keep generating words by different left-...
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