MathRoom Lesson Features
In street talk, Rolle's theorem says "if we have a chunk of a continuous function that begins and ends at the same y-value -- somewhere on that chunk, there's a horizontal tangent". Since the slope of the tangent is f '(x), and a horizontal line has slope = 0, there's a point on the chunk where the derivative is equal to 0.
Color Coded Explanations
The acronym is BEMDAS or BEDMAS
for brackets, exponents, multipication, division, addition, subtraction.
(with Multiplication & Division, Adding & Subtracting -- order doesn't matter.)
Simplify 5 [ 2x + 3(x2 - 7x + 6) ] + 3 [-2x - 4 (x2 - 8x + 1) - 7 ]
1st: multiply the inner parentheses ( ) by their coefficients + 3 and - 4.
2nd: collect like terms within the brackets [ ].
3rd: multiply the brackets [ ] by 5 and by 3, and collect like terms.
1st: 5 [2x + 3x2 - 21x + 18] + 3 [-2x - 4x2 + 32x - 4 - 7] =
2nd: 5 [3x2 - 19x + 18] + 3[- 4x2 + 30x- 11] =
3rd: 15x2 - 95x + 90 - 12x2 + 90x - 33 =
3x2 - 5x + 57
"Side by side" math and explanations
if you don't understand how to get from one step to the next, the explanation is right there beside the math -- you don't have to read through a whole paragraph to find what you need.
|-2 [3 (x + 1) 2 + 4( x 2 - 3)] + 6x - 5 =||square (x + 1), multiply 4(x2 - 3)|
|-2[3(x 2 +2x +1) +4x 2 -12] + 6x -5 =||multiply 3(x 2 + 2x + 1)|
|-2[7x 2 + 6x - 9] + 6x -5 =||collect like terms in brackets [ ]|
|-14x 2 - 12x + 18 + 6x -5 =||multiply -2[ 7x 2 + 6x - 9]|
|-14x 2 - 6x + 13||collect like terms|
Clear, Easy to Understand Explantations
Why in the world is a minus times a minus equal to a plus???
Say you owe me $10.00 but you have no money in your pocket now to pay your debt.
You have -$10.00 (negative ten dollars); you owe me $10.00 and you have nothing -- right?
I'm feeling generous for some strange reason and I say "Forget about the $10 you owe me."
Now, you have $0 , which is $10 more than you had before I took away a debt of $10.
The 1st minus sign instructs us to take away (an operator) and
the 2nd minus sign is the negative quality of the $10.00, because it is a debt.
So - (- 10) = + 10 and 5x - (- 3x) = 5x + 3x = 8x.
and 3a - 2( -5a ) = 3a + 10a = 13a
So, just as a minus sign can mean 2 things:
the operation to subtract, or
the negative quality of loss or debt or retreat (down or left)
a plus sign can also mean 2 things:
the operation to add, or
the positive quality of a gain or advance (up or right).
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