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## INTRODUCTION

Malaria is a disease that threatens the lives of 3.2 billion people throughout the world and has a cascading impact on pregnant women and children under the age of five1. Malaria is one of the leading causes of illness, mortality, poverty, and low productivity, among the majority of people living in low- to middle-income countries (LMICs)2. Malaria increases the risk of serious pregnancy complications for the mother, fetus, and newborn3. Insecticide-treated bed nets (ITNs) are a type of personal protection that has been proved to minimize malaria infection, severe disease, and death4. The effective use of ITNs has reduced malaria sickness by 50% in children aged <5 years, and reduced mortality by 17%5. The WHO Roll Back Malaria (RBM) project, which began in 1998, intended to prevent and control malaria during pregnancy via the use of ITNs and other measures4. In a meta-analysis of malaria-prevention datasets from several African countries, it was discovered that ITNs are critical in avoiding malaria in pregnancy3. Miscarriages and stillbirths are reduced by roughly 33% when ITNs are used throughout pregnancy6.

In Ethiopia, 39% of pregnant women did not have a positive attitude towards ITNs7. In the Democratic Republic of Congo, about 82% reported having an ITN in their household, of these only 78.4% used ITNs the night before the study survey8. In Sudan, the ownership of ITNs was nearly 57%, however, only 11.5% used ITNs frequently9. About 92% demonstrated sufficient knowledge on the cause of malaria, and 60% exhibited appreciable knowledge on the reasons why ITNs are used. Despite the above, only 22.1% knew the correct usage of ITNs9.

Ghana is one of the world’s fifteen countries with the greatest malaria infection rate10. Malaria infection is among the ten topmost diseases in Ghana11. Malaria affects all people, but it is common with expectant mothers and babies with very dire consequences12. Ghana saw the biggest increase in absolute case counts (about 0.5 million new cases) between 2017 and 2018, indicating a 5% increase over 2017 levels (from 213 to 224 per 1000 at risk)2. In Sub-Saharan Africa, Ghana has the highest incidence of Intermittent Preventive Treatment in Pregnancy (IPTp) for pregnant women (78%). The percentage of pregnant women taking the third dosage of IPTp climbed from 39% to 60% in 2016, to 61% in 201913. IPTp involves giving all pregnant women a curative dose of an effective antimalarial medicine (now sulfadoxine-pyrimethamine) without first determining whether or not they have malaria. It is recommended that pregnant women take at least three doses of IPTp starting from second trimester till birth14.

The intentional campaign and intervention on ITNs have resulted in a rise in household ownership of ITNs from 49% in 2011 to 68% in 20146,15. In Ghana, the usage of ITNs among pregnant women grew from roughly 33% in 2011 to 43% in 2014. Lack of availability of ITNs, as well as a lack of information and perception about ITNs and malaria, have previously been identified as significant hurdles to ITN adoption in Africa16. Indication from Ghana has shown that over 40% of ITNs available in the households go unused17. A study conducted in the middle belt of Ghana showed that pregnant women knew how crucial ITNs are to preventing malaria. Some pregnant women due to financial constraints, and missed chance during the free distribution of ITNs, do not own a bed net and do not sleep under one. Aside from the above genuine excuses, some of the pregnant women were also not using ITNs because of discomfort resulting from heat, the smell of the net, and difficulty in hanging the net18.

Despite the progress achieved in carrying out active preventive and treatment measures, malaria is still very prevalent in Ghana. Malaria accounts for 38.1% of all outpatient department cases and more than half of all hospital admissions in Ghana among children aged <5 years. Malaria infection is responsible for about half of all fatalities among children aged <5 years in Ghana19. Available literature in Ghana has demonstrated that the most households who own bed nets, do not sleep under them20. In addition, research performed in seven districts of Ghana’s Upper East region found that while 79% of pregnant women had bed nets, only 62% utilized them the night before the survey15. In addition to the above, there is a vast difference with regard to the utilization of bed nets between urban and rural areas of Ghana20.

Ghana is still working to achieve universal ITN coverage (defined as use by 80% or more of the population in an endemic region) to provide the best possible protection4. As a result, continuous monitoring and assessment of access to and use of treated bed nets are required to guide malaria control policy and practice, particularly in high-risk parts of the nation. There are vast studies on ITNs utilization in Ghana but most are often in the urban settings17,20-22 with limited studies in deprived areas23. Malaria prevention especially among pregnant women and children aged <5 years is a topmost priority for the Ghana government. As such, studies across the countries are required to inform policy decisions aimed at eradicating or reducing malaria infection among the general population. The Kassena-Nankana East Municipality is mostly rural with no study conducted in this area on ITN usage. To bridge this gap, this study aims to assess the utilization of Insecticides Treated Mosquito Nets (ITNs) among pregnant women in Kassena-Nankana East Municipality in the Upper East Region of Ghana.

## METHODS

### Study setting

The study setting was conducted in the Kassena-Nankana East Municipality, which was upgraded by legislative instrument (LI) 2106 from the District level in 2012. The municipality has Navrongo as its political and administrative capital. The study setting was chosen because the municipality is predominately rural with no study conducted on ITN.

### Study design

The study employed a descriptive cross-sectional survey with a quantitative approach to explore the utilization of ITNs among pregnant women in Kassena-Nankana East from May to July 2021.

Inclusion and exclusion criteria

The study included pregnant women who resided in the Kassena-Nankana East Municipality, who were of sound mind, and agreed to participate in the study voluntarily. All other persons who were outside these criteria were excluded including pregnant women who were on admission.

Sample size determination, sampling techniques, and procedure

The sample size was computed using the Snedecor and Cochran24 formula for a point estimate sample. The utilization of ITNs among pregnant women was 62%23. From the Snedecor and Cochran formula, the sample size was calculated as 362. All communities with a hospital, health center or community-based health planning and service (CHPS) center, were all written on pieces of paper, and ten health facilities were chosen at random without replacement. The study used simple random sampling in recruiting the sample unit in each facility. Thus, for each of the communities (health facilities) visited, 30 pregnant women were recruited at random, except the District Hospital where 92 pregnant women were recruited. The District Hospital serves as a referral center for all the other facilities and most pregnant women choose to go there for their ANC irrespective of the distance.

### Data collection tools and procedures

The questionnaire was adapted from existing literature15,25,26 and modified to suit the study objectives (Supplementary file Questionnaire). The questionnaire was structured for the specific objectives of the study consisting of Section A: Sociodemographic data of the respondents; Section B: knowledge on ITNs; Section C: utilization of ITNs; and Section D: barriers to the use of ITNs as a preventive and control tool against malaria among pregnant women. The questions were both open and close-ended.

The study explored the use of self-administered questionnaire techniques. A total of six field assistants were trained on the data collection tool to gather the data within the shortest possible time. Data collectors were also trained on the simple random sampling procedure for the selection of respondents.

### Reliability and validity of the study

To ensure that the data collected were reliable and valid, the field enumerators were trained on the use of data collection tools. Thirty pregnant women in Paga (a community similar to the study area) were interviewed during pretesting (piloting). Pre-testing helped in restructuring the questionnaire to elicit the right response for the specific objective. It also offered the field enumerators the chance to familiarize themselves with the data collection process and to rectify all difficulties before the main data collection process. For the validity of this study, the questionnaires were asked in plain language to solicit the right response. The questionnaire was also shown to experienced researchers in the area for face validity. The Supplementary file Dataset contains a dataset from the pre-testing (piloting). Internal consistency (reliability) of the measurement scales of the instrument was computed using Cronbach’s alpha. The overall alpha of the instrument was α=0.81 while the alpha values for the scales of knowledge, utilization and barriers of ITNs were α=0.86, α=0.89 and α=0.68, respectively. The overall alpha was considered as good.

### Data analysis and presentation

Data were analyzed using SPSS version 25.0. Data are presented using descriptive and inferential statistics. Chi-squared analysis was used to determine the statistical association between background characteristics and ITN utilization. All statistics were performed at a 95% confidence level, and p<0.05 was considered significant.

## RESULTS

### Sociodemographic characteristics

Most (46.4%) of the respondents were aged 30–39 years. Over 90% of the respondents had some level of education, with only 9.10% not having gone to school. A majority, 64.9%, 87.6%, and 52.2% of the respondents were Christians, married, and resided in rural areas, respectively (Table 1).

##### Table 1

Sociodemographic characteristics of the participants

Characteristicsn%
Age (years)
<207219.90
20-299225.40
30-3916846.40
≥40308.30
Education level
No formal education339.10
Junior high school and below13737.80
Senior high school9125.10
Tertiary10127.90
Religion
Christianity23564.90
Islam12534.50
Marital status
Single339.10
Cohabiting20.60
Married31787.60
Divorced61.70
Separated41.10
Occupation
Farmer8222.70
Teacher6818.70
Health worker277.50
Housewife4111.30
Seamstress164.40
Students143.90
Hairdresser92.50
Other92.50
Number of children
1–225971.50
3–48523.50
5–6174.70
>610.30
Area of residence
Rural18952.20
Urban17347.80
Average monthly income (GHS)
<50018150.00
500-10009827.10
1001-15006217.10
1501-2000154.10
>200061.70
Ethnicity
Frafra5816.00
Nankam7921.80
Kassena17448.10
Builsa164.40
Mamprusi113.10

### Association between sociodemographic characteristics and utilization of ITNs

With regard to place of stay, 96.8% of respondents in rural and 92.5% in urban areas used ITNs, while 3.2% and 7.5% of respondents in rural and urban areas, respectively, did not use the treated nets. The study revealed a significant association between area of residence and utilization of ITNs (χ2=3.4; p=0.04). Also, increasing average monthly income was associated with utilization of ITNs (χ2=46.4; p<0.001) (Table 5).

##### Table 5

Association between sociodemographic characteristics and utilization of insecticides treated mosquito bed nets (ITNs)

VariablesTotalDo you use ITNs?
Statistical test
Yes n (%)No n (%)
Age (years)
<207271 (98.6)1 (1.4)χ2=5.4
20–299289 (96.7)3 (3.3)p=0.14
30–39168156 (92.9)12 (7.1)
≥403027 (90.0)3 (10.0)
Education level
No formal education3332 (97.0)1 (3.0)χ2=3.1
Junior high school and below137128 (93.4)9 (6.6)p=0.38
Senior high school9189 (97.8)2 (2.2)
Tertiary10194 (93.1)7 (6.9)
Religion
Christianity235220 (93.6)15 (6.4)χ2=1.8
Islam125121 (96.8)4 (3.2)p=0.41
Marital status
Single230 (90.9)3 (9.1)χ2=3.1
Cohabiting22 (100.0)0 (0.0)p=0.54
Married317302 (95.3)15 (4.7)
Divorced65 (83.3)1 (16.7)
Separated44 (100.0)0 (0.0)
Area of residence
Rural189183 (96.8)6 (3.2)χ2=3.4
urban173160 (92.5)13 (7.5)p=0.04
Average monthly income (GHS)
<500181170 (93.9)11 (6.1)χ2=46.4
500–10009892 (93.9)6 (6.1)p<0.001
1001–15006260 (96.8)2 (3.2)
1501–20001515 (100.0)0 (0.0)
>200066 (100.0)0 (0.0)

[i] GHS: 1000 Ghanaian Cedis about US\$160.

## DISCUSSION

This study aimed to assess the use of ITNs among pregnant women in the Kassena-Nankana East Municipality. The study revealed that almost all respondents had heard about ITNs. This is consistent with other studies conducted elsewhere. For instance, in the Ho municipality, Ghana, a study showed that 98.7% of pregnant women had heard about bed nets26. Amara27 revealed that, in the Greater Accra Region of Ghana, over 60% of the mothers with children aged <5 years had heard about ITNs. Also, in the Nanumba South District in the Northern Region of Ghana, over 80% of pregnant mothers had heard about bed nets10. In Nigeria, 93.2% of pregnant women were aware of bed nets28. These similarities are justified by the fact that, in Ghana and in most African countries, there have been focused attempts by the Ministry of Health, through the support of WHO and UNICEF, to provide education on malaria prevention. This is because malaria is among the key causes of maternal anemia. Maternal anemia is one of the leading causes of infant and maternal morbidity and mortality in Sub-Saharan Africa (SSA). Insecticides treated mosquito nets are the simplest most effective way of preventing malaria22. This, therefore, explains why most respondents in Africa are aware of ITNs.

The majority of the pregnant women in this study reported having obtained their ITNs from a health facility. According to Nungbaso et al.25, the majority of the respondents in the Tamale metropolis benefitted from the free distribution of bed nets from the health facilities. Even though the respondents are different, the policies regarding the free distribution of ITNs are often targeted at the general population with interest among pregnant women and children aged <5 years29.

About 90% of the respondents knew that ITNs were used to prevent malaria. Available literature in Ghana supports the current findings. For instance, 91.1% and 97.8% of the respondents in Ho26 and Greater Accra27, respectively, knew about the general usage of ITNs. However, in Nigeria, Musa30 showed that only 36% of the respondents knew the correct usage of ITNs. This difference could be attributed to the geographical difference, beliefs, and sociocultural characteristics of the respondents.

The ownership of ITNs was determined as 97.8% in this study. The current findings are higher than the 71% reported in Western Kenya by Atieli et al.31 and 64.9% reported in Nigeria by Ezire et al.28. Though these studies indicate that the majority of their respondents owned ITNs, they were observed to be lower than the findings of the current studies. This suggests that the Ministry of Health of Ghana has adopted a robust method to distribute the ITNs.

The majority of the respondents indicated that they believed ITNs were effective. This is consistent with a study by Abokyi et al.32 where over 95% of respondents believed ITNs were very effective in preventing malaria. This is crucial in determining the utilization of ITNs because, if the population does not believe in the efficacy of ITNs, the patronage would be lower. This is evident by the 94.8% of pregnant women in this study using ITNs. In other studies, the majority of respondents said that they were using ITNs10,31. However, Amara27 reported very low usage of ITNs. This could be attributed to the differences in government policies on free ITN distribution and behavioral and personal preferences of the respondents.

Though the use of ITNs was observed to be high in the study, only 47.8% were using ITNs every day. The actual utilization of ITNs is varied across the globe. For example, Atieli et al.31 showed that the majority used ITNs throughout the year, Bukari10 reported 34.4% actual utilization, and 19% were using ITNs in another study27. What this tells us is that ownership of ITNs does not always result in utilization due to sociocultural and logistic reasons. Some of the women having ITNS, do not use them all year round. For some of the respondents in this study, ITNs were used frequently only in the rainy season. This is because the cases of malaria often increase during rainy periods25. Some also reported dissatisfaction with the use of ITNs and as such resort to other preventive measures. This is supported by available literature, where some people prefer other methods of protection such as mosquito coils, repellents, electric fans, etc33. From the foregoing, the use of other preventive measures could be a reason for the difference between ownership and actual utilization of ITNs.

The study revealed a wide variety of challenges such as ITNs being inaccessible, side effects, ITNs being warm, among others. This is similar to a study where respondents currently using ITNs complained of scarcity of new nets, difficulty in getting chemicals for re-treatment of nets, non-availability of quality ITNs for sale, resulting in disuse of ITNs26. Also, the cost of the nets was featured among the barriers to effective utilization of ITNs. In Ghana, there are free distributions of ITNs at antenatal care (ANC) units, child welfare clinics (CWC), postnatal care (PNC) service centers as well as household distributions. The authors do not understand why issues of cost and accessibility are being raised by the respondents. These revelations by the respondents are sensitive and as such require further studies on the reasons for not using ITNs, using a mixed-method approach to bring in-depth understanding to these issues especially among those who cite cost as a barrier to ITN utilization.

The study also showed a significant association between area of residence and use of ITNs. This is similar to in the Upper West Region, Ghana where pregnant women in urban areas tend to use ITNs more than their counterparts in the rural setting34. In relation to the above, increasing average monthly income is associated with utilization of ITNs. This is consistent to a study in Northern23 and a systematic review in Sub-Sahara Africa35 where higher wealth index is associated with higher ITN ownership and usage, and vice versa. However, Garcia-Basteiro et al.36 did not establish any association between wealth index and ITN ownership and use. This means that, depending on the unique characteristics of each area and the tactics employed to deploy ITNs, the impoverished and relatively well-off may gain differentially. Often, the person with higher wealth index stays in the urban areas which are accessible and could be reached during distribution of ITNS. Also, people with a high socioeconomic class typically have access to other measures for preventing malaria. Whilst it is important to evenly distribute ITNs, extra efforts are required to reach the hard-to-reach rural folk and the poor.

### Limitations

As with self-reported surveys, our study is subject to information bias coupled with our inability to confirm the responses given. The study does, however, provide population-based data on ITN use among pregnant women in the study area. This can be relied on by agencies, including the Ministry of Health, Ghana Health Services, and other stakeholders for informed targeted public health interventions.

## CONCLUSIONS

The study revealed a high awareness and knowledge of ITNs. Though the majority of the respondents owned Insecticides Treated Mosquito Nets (ITNs), it did not translate to effective utilization of ITNs among pregnant women. Thus, distribution of ITNs to pregnant women is not enough, there is the need to establish practical measures to ensure that persons who have the bed nets sleep under them. The free distribution of ITNs for the majority means that most people obtained ITNs. However, some persons whose residence were hard to reach due to bad roads are often left out of the free distribution of ITNs. They are therefore compelled to buy ITNs, which they think are very expensive. To resolve this, we recommend to the Ghana Health Service to train volunteers at local level to aid in the distributions of bed nets.


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# Universal Gravitation and Gravitational Field Question

## Homework Statement

The earth has a mass of 5.98x10^24 kg and the moon has a mass of 7.35x10^22 kg. The distance from the centre of the moon to the centre of the earth is 3.84x10^8 m. A rocket with a total mass of 1200 kg is 3.0x10^8 m from the centre of the earth and directly in between earth and the moon. Find the net gravitional force on the rocket from the earth and moon.

Fg = Gm1m2 / r^2

## The Attempt at a Solution

I think the above is the equation I'm suppose to be using. I have hard time grasping this concept :S.

And this is what I did

Fg on the rocket to the earth:

Fg = (6.67x10^-11)(5.98x10^24)(1200) / (3.0x10^8)^2

Fg = 5.3182 N

Fg on the rocket to the moon:

Fg = (6.67x10^-11)(7.35x10^22)(1200) / (3.0x10^8 + 3.84x10^8)^2

Fg = (6.67x10^-11)(7.35x10^22)(1200) / (6.84 x10^9)^2

Fg = 1.2574 x 10^-4 N

Fgnet = 5.3182 N + 1.2574 x 10^-4 N

Fgnet = 5.3183 N

I'm not sure if this is correct or not :S. From my understanding it seems to be. Please help.

Doc Al
Mentor
And this is what I did

Fg on the rocket to the earth:

Fg = (6.67x10^-11)(5.98x10^24)(1200) / (3.0x10^8)^2

Fg = 5.3182 N
Looks good.

Fg on the rocket to the moon:

Fg = (6.67x10^-11)(7.35x10^22)(1200) / (3.0x10^8 + 3.84x10^8)^2
You want the distance from rocket to moon. How might you figure that out? Draw a diagram!

Fg = (6.67x10^-11)(7.35x10^22)(1200) / (6.84 x10^9)^2

Fg = 1.2574 x 10^-4 N

Fgnet = 5.3182 N + 1.2574 x 10^-4 N
Why did you add the forces? Do they point in the same direction?

Hi EE123. The gravitational force due to the Earth seems to be correct. But for the moon, the equation $F=\frac{Gm_1m_2}{r^2}$ where r is the separation of mass 1 and mass 2. The rocket is between the Earth and the Moon and you know the distance from the Earth to the rocket and from the Earth to the moon, so you should be able to get r from that, it isn't 3.0x10^8 + 3.84x10^8. Also another thing is the force on the rocket due to the moon is in the opposite direction to the force on the rocket due to the Earth, so the net force won't be adding both values.

Hi EE123. The gravitational force due to the Earth seems to be correct. But for the moon, the equation $F=\frac{Gm_1m_2}{r^2}$ where r is the separation of mass 1 and mass 2. The rocket is between the Earth and the Moon and you know the distance from the Earth to the rocket and from the Earth to the moon, so you should be able to get r from that, it isn't 3.0x10^8 + 3.84x10^8. Also another thing is the force on the rocket due to the moon is in the opposite direction to the force on the rocket due to the Earth, so the net force won't be adding both values.

Hi guys and thank you!

And thank you very much for clarification sleepy_time, I appreciate it.

So would it be this then?

distance from rocket to the moon:

Δd = 3.84E8 - 3.0E 8
Δd = 8.4E7 m

Thus,

Fg = (6.67E-11)(7.35E22)(1200) / (8.4E7)^2

Fg = 0.83375 N

FgNet = 5.3182 + (-0.83375 N)

FgNet = 4.48445 N

FgNet = 4.50 N

Can you please explain the following: "the force on the rocket due to the moon is in the opposite direction to the force on the rocket due to the Earth"

cepheid
Staff Emeritus
Gold Member
Can you please explain the following: "the force on the rocket due to the moon is in the opposite direction to the force on the rocket due to the Earth"

Well, since the rocket lies on a straight line in between Earth and the moon, the moon pulls on the rocket in one direction (towards the moon), and the Earth pulls on the rocket in the exact opposite direction (towards the Earth). This is shown in the diagram below. In the diagram, "E" is the earth, "M" is the moon, "R" is the rocket, and arrows show the directions of the two gravitational forces on the rocket. They oppose each other.

Code:
E <--------- R ------> M

Well, since the rocket lies on a straight line in between Earth and the moon, the moon pulls on the rocket in one direction (towards the moon), and the Earth pulls on the rocket in the exact opposite direction (towards the Earth). This is shown in the diagram below. In the diagram, "E" is the earth, "M" is the moon, "R" is the rocket, and arrows show the directions of the two gravitational forces on the rocket. They oppose each other.

Code:
E <--------- R ------> M

Okay, thank you! So is my calculation correct??

Okay, thank you! So is my calculation correct??
Yes I believe it is. Just careful with rounding your answer. Rounding 4.48446.... will give 4.5N (1d.p.) or 4.48N (2d.p.)

format: plain


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# How is $\frac{dQ}{T}$ measure of randomness of system?

I am studying entropy and its hard for me to catch up what exactly is entropy.

Many articles and books write that entropy is the measure of randomness or disorder of the system. They say when a gas system is let expand the randomness increases etc. But they end up saying $\frac{\mathrm dQ}{T}$ is the measure of increase in randomness and is called the entropy.

Even if I believe that entropy is the measure of randomness of the system I don't understand:

1. How does $\frac{\mathrm dQ}{T}$ hold the information about the randomness of the system?
2. How is entropy independent property of any system. I suppose that any two parameter in the equation $PV=nRT$ should completely describe the system. Why would we need entropy?

Thank you.

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My advice to you, based on personal experience, is to accept these confusions until you learn about the definition of entropy in statistical mechanics (which is what Nathaniel refers to below). Entropy is defined for any system where the specific state is chosen from a probability distribution, and it directly measures our ignorance about which state the system is in ($S=0$ means we know the state exactly, $S$ maximal means all states have equal probability). I find it very confusing to try to understand it through $dQ/T$, since that is a phenomenological rather than microscopic description. – Guy Gur-Ari Aug 3 '12 at 20:43
@GuyGur-Ari: Is there a formal definition for the word "phenomenological", one that contrests it from the latter? – NikolajK Aug 4 '12 at 18:49
I don't know about a 'formal' definition, but a phenomenological description is a description that is at the level of the observations. We observe that a system has certain properties that we can measure, like temperature, pressure, and so on. And we observe certain relations between these properties. When we combine these into a (hopefully simple) model -- that is a phenomenological description. For example, the ideal gas law and Ohm's law are both phenomenological models. – Guy Gur-Ari Aug 4 '12 at 22:29
A microscopic description (like statistical mechanics) considers the underlying degrees of freedom, and should explain how the observed 'macroscopic' phenomena emerge from the detailed model. If you derive the ideal gas law by postulating tiny atoms bouncing around -- that is a microscopic description. Of course a microscopic model at one level may become phenomenological once you go to more detailed observations, so this distinction depends on one's point of view. People who study optics would not call particle physicists 'phenomenologists' -- but strings theorists would. – Guy Gur-Ari Aug 4 '12 at 22:33

This answer is somewhat hand-wavy, but I do believe it should help to grasp the concepts on an intuitive level.

First of all, entropy is not a measure of randomness. For an isolated system in equilibrium under the fundamental assumption of statistical mechanics, the entropy is just $$S=k\ln\Omega$$ where $\Omega$ is the number of microstates - microscopic system configurations - compatible with the given macrostate - macroscopic equilibrium state characteristed by thermodynamical variables.

It follows from the second law $$\delta Q = T\mathrm{d}S=T\mathrm{d}(k\ln\Omega)=kT\frac1\Omega\mathrm{d}\Omega$$ or equivalently $$\mathrm{d}\Omega = \Omega\frac{\delta Q}{kT}$$ The energy $kT$ is related to the average energy per degree of freedom, so this formula tells us that the transfer of heat into a system at equilibrium opens up a new number of microstates proportional to the number of existing ones and the number of degrees of freedom the transferred energy may excite.

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This answer effectively complements what has been already said, and succintly shows the relation between added heat and the increase in the number of possible microstates corresponding to the same macrostate; and does it without assuming no more than an elementary background of statistical mechanics. +1 – Mono Aug 4 '12 at 17:26

In my opinion, it isn't strictly correct to say that entropy is "randomness" or "disorder". The entropy is defined in statistical mechanics as $-k_B \sum_i p_i \log p_i$, where $k_B$ is Boltzmann's constant (which is only there to put it into physically convenient units) and $p_i$ is the probability that the system is in state $i$. These probabilities do not mean that the system is "randomly" jumping from one state to another (although quite often it is), they just mean that you, as an experimenter observing the system, don't know exactly which state is in, but you think some are more likely than others. Since Shannon (1948) and Jaynes (1957), this formula for the entropy has been interpreted in terms of the information that an experimenter has about a system: the less information, the more entropy. (Those links are just for completeness - I wouldn't recommend reading them as your first introduction to the subject.) The amount of information an experimenter has about a system can decrease for many reasons, but the only way it can increase is if the experimenter makes a new measurement. This is the reason for the second law of thermodynamics.

It should be noted that there are many different perspectives on the meaning of entropy and the second law, and not everyone agrees with the one I outlined above. However, I will try to answer your two questions from this point of view.

1. From a modern perspective, it's better to view $dS = \frac{dQ}{T}$ as a definition of $Q$ rather than of $S$. After all, $S$ already has a definition in terms of the probabilities. If we view $dQ$ as being defined as $TdS$ we can see that it's equal to $dU + PdV - \sum_i \mu_i dN_i$ (by rearranging the fundamental equation of thermodynamics), which is equal to the total change in energy minus the energy that's transferred in the form of work. (Here I've defined work as "mechanical work" $PdV$ plus "chemical work" $-\mu_i dN_i$. You can also add terms for electrical work, etc.)

2. There are several reasons we need to consider the entropy of an ideal gas. One is that $T$, which appears in the ideal gas law, is defined as $T=\frac{\partial U}{\partial S}$, so the $S$ comes in that way. Another is that the equation $PV = nRT$ does't tell you how the temperature changes when you add energy to the system. For that you need to know the heat capacity, which is closely related to the entropy. Finally, the concept of entropy is extremely useful in understanding why you can't build a perpetual motion machine.

If this point of view sounds like it might make sense to you, it might be worthwhile reading this paper by Jaynes, which takes a historical perspective, explaining how entropy was first discovered (and defined in terms of $dQ/T$), and how it then unexpectedly turned out to be all about information.

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We can't really use this equation to define Q because it can not be extended to systems which are not in equilibrium. For such systems, the heat flux and entropy are well defined but not necessarily T. Temperature is a macroscopic parameter which happens to be equal with the average kinetic energy for systems in equilibrium. For systems out of equilibrium, everything is possible: from a multiple temperature to no temperature at all. I don't see any simple explanation to the question out of statistical physics. – Shaktyai Aug 4 '12 at 20:09
@Shaktyai I disagree that the heat flux can be well defined for systems with no definable $T$. The energy flux is always definable, but if there's no $T$ then there's no meaningful way to partition it into work versus heat. Or at least, I don't know of an example where this can be done. If you can show me one I'll change my answer. – Nathaniel Aug 4 '12 at 22:57
In non equilibrium statistical mechanics, the heat flux is just the moment of order 3 of the velocity: =int(1/2*m*(V-v)^2*vf(r,v,t)) where V is the average velocity V=int(vf(r,v,t)) If the system is not in LTE, then f(r,v,t) is not a Maxwellian and T is not defined. In collisional radiative models (star's atmosphere or fusion plasmas) it is very common to encounter distribution function with two or no temperature. – Shaktyai Aug 5 '12 at 17:42

A wealth of meaningful info is contained in the above answers. However, a short and simple intuitive picture still seems missing.

The bottom line is that temperature measures the energy per degree of freedom, and hence $\frac{dQ}{T}$ measures nothing more than the number of degrees of freedom over which the energy has spread. The number of degrees of freedom describes the microscopic complexity (as others have remarked, the term 'randomness' many consider less appropriate) of the system - the amount of information needed to specify the system down to all its microscopic details. This quantity is known as the (statistical) entropy.

You might like this blog that discusses the subject.

-

It's worth noting that your definition of an elemental change in the entropy of a system, namely:

$dS=\displaystyle\frac{\delta Q}{T}$

It's just valid for an internally reversible change. This is not a technicism which can be omitted; I think part of your question might be related to the notion of heat (a measurable amount of energy transferred) and statistical uncertainty (which is, up to alternative and equivalent interpretations, the intrinsic meaning of entropy).

In an internally reversible process which involves heat addition or substraction from a system, that T under de heat (inexact) differential must be a uniform temperature across the system's spatial extension up to it's boundaries, so that at every moment the temperature of system's boundaries is equal to it's bulk temperature (and unique). That means that there are no temperature gradients inside the system of interest, and because of that very fact, there aren't any possible heat exchanges inside the system's boundaries. That is because, for a system to exchange heat with something else, there must be a difference in temperature between them, and if the difference is zero (they are equal) then no heat will be transferred. If you think about it this is a sound argument: a cold glass of water gets increasingly hotter when you leave it in a room, but when it reaches the same temperature of the air around it, then there's no more change and it stays there indefinitely.

Going back to the original equation, you can now interpret the RHS as telling you that, at situations where the system's temperature is uniform at every moment, the ratio of the infinitesimally small amount of heat added or substracted to the system by it's environment, and the unique temperature at every point of the system (which is nothing more but a measure of the mean kinetic energy of individual molecules which make it up), is equal to it's change in entropy. And what is entropy? Well, macroscopically talking, you can take what I've written above as a definition of entropy, and you can thermodynamically deduce that it is indeed a state function (it only depends on the point properties of the system, like it's pressure and temperature) and it doesn't depend upon the chain of events by which that state was reached.

On the other hand, statistical mechanics (which is a more recent way of addressing what we see macroscopically as thermodynamical properties, like entropy, starting from a mechanical description at the molecular level) gives us more details on the nature of entropy. I think it's better to think about it not as a measure of randomness but as the (macroscopic) uncertainty of the (microscopic) state of the system.

I'll give you a simple example: imagine you had a pool table with it's top totally covered by an opaque fabric, with just one open end for introducing the cue stick. Assume now that you know (by some means) that the eight balls are distributed in the table forming a straight line with an equal spacing between them, but you don't know where exactly this line stands in the table's rectangular area; and that, for the purpose of the experiment, the white one is just next to the hole (and of course you know it). Now, you take the cue stick, introduce it in the fabric's hole left open, and strike the cue ball. After a few seconds of (hearing) collisions, you can be sure that movement stopped under the fabric. What happened to your knowledge about the system?

Well, you don't know where does each ball gone (we've sealed the pockets, of course!) but you didn't knew it before the strike, did you? But then, you at least knew they were forming a line, and that information is now gone. From your outside point of view, your prior information about the positions of the balls and the energy and momentum you introduced in the system trough the strike isn't enough to rule out a huge number of possible actual distributions of the balls. At the begining of the experiment, you could at least write down the number of possible positions of the line of balls (perhaps by drawing a grid over the table's area, with each cell's side length equal to a ball's diameter, and counting the number of longitudinal cell lines) but now the number of possible positions has multiplied. Before and after you only have partial knowledge of the system's configuration (all you can do is count the possible ones, based on what you know about the system from the outside, which restrict the possibilities) but that knowledge has decreased after the experiment. It has nothing to do with the physics of the collisions between the balls: it has to do with the fact that you can't see the balls from your point of view, and all you can do is retrieve partial information through indirect measurements.

The analogy with the example above in a statistical system is that by measurements of macroscopic observables (like temperature, pressure, density, etc) we only measure mean microscopic properties. For example, temperature is a measure of the mean molecular kinetic energy, and pressure is a measure of the mean rate of momentum transferred by striking molecules per area unit. Measuring them gives us partial knowledge of it's microscopic configuration (like the original information you held about the positions of the pool balls). And any change in the macroscopic observables is correlated to a change in the possible (i.e. not ruled out) microscopic configurations, and then that causes a change in our knowledge about it. It turns out that those changes can be measured, and that's indeed entropy variation, in the sense that an entropy increase correlates to an uncertainty increase, or a knowledge decrease. Showing that this relation holds, starting from a mechanical framework, is the whole point behind statistical mechanics.

Finally, I hope you can see now that what $\displaystyle\frac{\delta Q}{T}$ is just analogue to the energy introduced by the strike in the experiment in relation to the previous knowledge of the position of the balls (lower temperatures imply less molecular translational, roational and vibational molecular movements, and vice versa, so it is actually a "partial measure" of their positions). So:

1. It doesn't hold the information about the randomness of the system, it is just a measure of the increase in uncertainty from a macroscopic perspective, and only holds for reversible processes (in general, entropy can increase without adding energy to a system).

2. As other answers have stated, entropy is needed to define some of the terms in any state equation (like the Ideal Gas law), and by the way, state equations are just approximations to the actual behavior of real substances (something pretty clear in the "ideal" part of the the law you cite), so it's natural for them to be based on more fundamental concepts (like entropy).

EDIT: As Nathaniel rightly pointed out below, my original statement that the validity of the macroscopic definition of entropy in terms of heat and temperature depended on the (tacitly) total reversibility of the process, was flawed. The only requirement for it to be valid is that the heat exchange process must be internally reversible, becasue we're only measuring this way the change in entropy inside the system (and so external irreversibilities associated with the process are irrelevant).

-

You should think of the equation

$$dS = {dQ\over T}$$

As the definition of temperature, not of entropy. Entropy is more fundamental--- it's the size of the phase space, the log of the number of possible states. The temperature is a derivative of this with respect to energy.

To understand why this makes sense, put two systems side by side. If the energy flows from hot to cold, the loss of entropy in the hot system is more then compensated by the gain in entropy of the cold system. So energy will flow from hot to cold, statistically, on average.

It is not the properties of temperature which make $dQ\over T$ an entropy change, rather it is the properties of the entropy which makes the coefficient of $dS\over dQ$ an inverse-temperature.

-

An microscopic approach to entropy has lead to great insight and is in detail explained in the given answers.

To understand the concept of entropy there is an equally valid but macroscopic approach that might complement the given answers. The idea has been developed on the basis of 'adiabatic accessibility' and the authors Elliott H. Lieb and Jakob Yngvason have done an excellent job explain this concept, although a little heavy on the mathematical side (arxiv link). Their work has been summarized in the book The Entropy principle by André Thess.

So for whoever is interested in a different approach to rigorously define entropy should give this concept a closer look.

-

"How does dQT hold the information about the randomness of the system"

The answer lies in the microscopic definition of the heat. The velicity of any particle can be writen: V=Vb+v . Vb is the bulk velocity and vi the "random" velocity: =0. The kinetic energy associated to vi is the heat. So measuring the heat is nothing else than measuring the degree of randomness of the molecules in the system. If all the molecules fly in the same direction then vi=0 and V=Vb: the kinetic energy is the macroscopic kinetic energy Ec=1/2*m*Vb^2, if all the directions are equiprobable Vb=0 and the kinetic energy is purely heat.

" I suppose that any two parameter in the equation PV=nRT should completely describe the system. Why would we need entropy?" Take two gases (P1,V1,T1) and (P2,V2,T2) put them in contact. You can't predict how the temperature evolves without the entropy.

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Home

#### The Six Pillars of Calculus

The Pillars: A Road Map
A picture is worth 1000 words

#### Trigonometry Review

The basic trig functions
Basic trig identities
The unit circle
Addition of angles, double and half angle formulas
The law of sines and the law of cosines
Graphs of Trig Functions

#### Exponential Functions

Exponentials with positive integer exponents
Fractional and negative powers
The function $f(x)=a^x$ and its graph
Exponential growth and decay

#### Logarithms and Inverse functions

Inverse Functions
How to find a formula for an inverse function
Logarithms as Inverse Exponentials
Inverse Trig Functions

#### Intro to Limits

Overview
Definition
One-sided Limits
When limits don't exist
Infinite Limits
Summary

#### Limit Laws and Computations

Limit Laws
Intuitive idea of why these laws work
Two limit theorems
How to algebraically manipulate a 0/0?
Indeterminate forms involving fractions
Limits with Absolute Values
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Limits of Piece-wise Functions
The Squeeze Theorem

#### Continuity and the Intermediate Value Theorem

Definition of continuity
Continuity and piece-wise functions
Continuity properties
Types of discontinuities
The Intermediate Value Theorem
Summary of using continuity to evaluate limits

#### Limits at Infinity

Limits at infinity and horizontal asymptotes
Limits at infinity of rational functions
Which functions grow the fastest?
Vertical asymptotes (Redux)
Summary and selected graphs

#### Rates of Change

Average velocity
Instantaneous velocity
Computing an instantaneous rate of change of any function
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The Derivative of a Function at a Point

#### The Derivative Function

The derivative function
Sketching the graph of $f'$
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Notation and higher-order derivatives

#### Basic Differentiation Rules

The Power Rule and other basic rules
The derivative of $e^x$

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#### Derivatives of Trig Functions

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#### The Chain Rule

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A hybrid chain rule

#### Implicit Differentiation

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Formulas and Examples
Logarithmic Differentiation

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#### Linear Approximation and Differentials

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#### Differentiation Review

How to take derivatives
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Product and Quotient Rules
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Logarithmic Differentiation
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#### Absolute and Local Extrema

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Critical Numbers
Steps to Find Absolute Extrema

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Rolle's Theorems
The Mean Value Theorem
Finding $c$

#### $f$ vs. $f'$

Increasing/Decreasing Test and Critical Numbers
Process for finding intervals of increase/decrease
The First Derivative Test
Concavity
Concavity, Points of Inflection, and the Second Derivative Test
The Second Derivative Test
Visual Wrap-up

#### Indeterminate Forms and L'Hospital's Rule

What does $\frac{0}{0}$ equal?
Examples
Indeterminate Differences
Indeterminate Powers
Three Versions of L'Hospital's Rule
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Strategies
Another Example

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The Idea of Newton's Method
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When NM doesn't work

#### Anti-derivatives

Antiderivatives
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Antiderivatives are not Integrals

#### The Area under a curve

The Area Problem and Examples
Riemann Sum Notation
Summary

#### Definite Integrals

Definition of the Integral
Properties of Definite Integrals
What is integration good for?
More Applications of Integrals

#### The Fundamental Theorem of Calculus

Three Different Concepts
The Fundamental Theorem of Calculus (Part 2)
The Fundamental Theorem of Calculus (Part 1)
More FTC 1

#### The Indefinite Integral and the Net Change

Indefinite Integrals and Anti-derivatives
A Table of Common Anti-derivatives
The Net Change Theorem
The NCT and Public Policy

#### Substitution

Substitution for Indefinite Integrals
Examples to Try
Revised Table of Integrals
Substitution for Definite Integrals
Examples

#### Area Between Curves

Computation Using Integration
To Compute a Bulk Quantity
The Area Between Two Curves
Horizontal Slicing
Summary

#### Volumes

Slicing and Dicing Solids
Solids of Revolution 1: Disks
Solids of Revolution 2: Washers
More Practice

### The Area Between Two Curves

To get the area between two curves, $f$ and $g$, we slice the region between them into vertical strips, each of width $\Delta x$.  Denote by $H(x)$ the height of the area at a point $x$.  The area of each strip is roughly $H(x)\cdot \Delta x$.
Adding up the area strips, the total area is approximately $\displaystyle\sum_{i=1}^n H(x_i)\, \Delta x$.  Taking a limit, the total area is exactly $\displaystyle\int_a^b H(x)\, dx$.

Let's look at an example.  Suppose $f(x)\ge g(x)$ for all $x$ in the interval we are considering, as in the graph below, so that $H(x)=f(x)-g(x)$ on the interval $[0,10]$.

Then $$\displaystyle\int_a^b H(x)\, dx=\int_0^{10} (f(x)-g(x))\,dx=\int_0^{10}(2\cos x+8 -(3\sin x+4))\,dx=\int_0^{10}(2\cos x-3\sin x+4)\,dx.$$
DO:  Before looking ahead, evalute the previous integral.  Check your antiderivative!

$$\int_0^{10}(2\cos x-3\sin x+4)\,dx=(2\sin x+3\cos x+4x)\left|\begin{array}{c} ^{10} \\ _0 \end{array}\right .=2\sin(10)+3\cos(10)+4(10)-(2\cdot 0+3\cdot 1+4(0))$$$$=2\sin(10)+3\cos(10)+37\approx 2(-.544)+3(-.839)+37=33.395$$ (here we must round off the trig values, so our answer is no longer exact).

We look at more examples in the video below.  Notice while watching that, as was true in the example above, you must know what the graphs of the two functions look like in order to know which function is the larger.  Also notice that sometimes we are given the beginning and ending values of $x$ explicitly, and sometimes we have to figure out where two curves meet.


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# STEP Maths I, II, III 1989 solutions

Announcements Posted on
1. STEP I (Mathematics)
1:
2: Solution by kabbers
3: Solution by Dystopia
4: Solution by Dystopia
5: Solution by Dystopia
6: Solution by Swayam
7: Solution by squeezebox
8: Solution by squeezebox
9: Solution by nota bene
10:
11: Solution by Glutamic Acid
12:
13:
14:
15:
16:

STEP II (F.Maths A)
1: Solution by squeezebox
2: Solution by kabbers
3: Solution by squeezebox
4:
5: Solution by Dystopia
6: Solution by *bobo*
7:
8:
9:
10:
11:Solution by *bobo*
12:
13:
14:Solution by squeezebox
15:
16:

STEP III (F.Maths B)
1: Solution by squeezebox
2: Solution by *bobo*
3:
4:
5:
6:
7:
8: Solution by Dystopia
9: Solution by squeezebox
10: Solution by squeezebox
11:
12:
13:
14:
15:
16:

Solutions written by TSR members:
1987 - 1988 - 1989 - 1990 - 1991 - 1992 - 1993 - 1994 - 1995 - 1996 - 1997 - 1998 - 1999 - 2000 - 2001 - 2002 - 2003 - 2004 - 2005 - 2006 - 2007
2. I am no expert at STEP, so don't hesitate to correct me, it is more than likely made that I have made a mistake.

STEP I - Question 8

Using good old de Moivre's theorem;

(using

and similary;

Now, consider:

Where c is .

Now let ,

so:

Which gives;

().
These values of give distinct values of

or

()

Hence the roots of the equation are:

and .
3. STEP III - Question 10

Lets assume that the result is true for n = k;

(*)

This is the same as (*) except k+1 replaces k. Hence if the result is true for n=k, its true for n= k+1.

When n=1,

LHS of (*) = 1x2x3x4x5 = 120
RHS of (*) = (1/6)x1x2x3x4x5x6 = 120.

So (*) is true for n=1.
Hence, by induction;

.

Since;

.

Using (*);

In this case,

From the previous parts, we know that:

and clearly, also:

as ,

and

So,

Using a similar arguement, we can show that:

We have shown that the limits exist and are equal to .

Hence;

4. STEP I, Q2

The For , find

Using integration by parts ():

By approximating the area corresponding to by n rectangles of equal width and with their top right hand vertices on the curve , show that, as ,

Firstly, note that

(by the above indefinite integral)

The sum of the areas of our rectangles is going to be

Claim:

So for k + 1, we must prove that is the equation we arrive at.

Basis case (let k=1):

Inductive step:

So,

As , our sum tends to the integral . So,

is this ok? have i missed anything out or not explained each step in enough detail?
5. Question 4, STEP I

With six points, each point is attached to five other points, so at least three lines radiating from A must be of the same colour (gold, say). The points which are joined to A must then be joined to each other by silver lines, or an entirely gold triangle would be made. However, then a triangle with entirely silver edges can be made from these three points.

Diagram showing five points is attached.
Attached Thumbnails

6. Question 5, STEP I

a)
Let x = 1.

Let x = -1

As n is even. So

As their sum is

b) Suppose that

Then

Note that and that

So, upon adding these two expressions, we get

Also,

So true by induction.

7. STEP II Q1

Substituting x= y-a into the equation:

Notice that the reduced form shown in the question has no term, so we need to find the value of a which will get rid of the term.

Expanding out and collectiing terms we get:

So the value of a to get rid of the is -1.

So we now have:

(*)

Substituting y=z/b into (*);

dividing both sides by 2 and multiplying by ;

.

And so to make this equal to;

we make b = .

Lets take b = and let z =

so;

and (these give distinct values of , and hence, three distinct solutions.)

we know that; x = y - a = (z/b) - a.

Hence the solutions of the equation are:

and
8. Question 11, STEP I.

Vertical component =
Using s = ut + 1/2at^2.

Quadratic in t. Ignoring the negative root.

R = horizontal component x time.
Horizontal component =

Multiply both numerator and denominator by 2.

Using
We can simplify it.

Splitting up Mr. Big Fraction.

Ignore the first fraction for the moment, we'll work on the second one.

Unparseable or potentially dangerous latex formula. Error 4: no dvi output from LaTeX. It is likely that your formula contains syntax errors or worse.
\dfrac{2v^2 \cos \theta \sin \theta}{2g} \times \sqrt{1 + \dfrac{2hg}{v^2 \sin^2 \theta}

We can use our old friend Mr. Double Angle Formula to simplify it:

The whole fraction looks like:
Unparseable or potentially dangerous latex formula. Error 4: no dvi output from LaTeX. It is likely that your formula contains syntax errors or worse.
R = (\dfrac{v^2\sin 2\theta}{2g}) + (\dfrac{v^2 \sin 2\theta}{2g}) \sqrt{1 + \dfrac{2hg}{v^2 \sin^2 \theta}

And lo and behold, there's a common factor to both fractions. Let's take it out:

And that's what we're looking for.
*I'll finish off the last small bit in a few minutes*
9. STEP I Question 6

The normal to the curve will therefore have gradient -1/f'(x). The equation of the normal would be

(where x' is the general x coordinate of the normal).

At the point Q, x' = 0 as it cuts the y axis.

The distance PQ can be worked out using Pythagoras' theorem.

It's given that

So

(use a substitution of u = x^2 if you can't see why)

10. STEP III, Q8

Let

Then

Using an integrating factor, , we get

Applying boundary conditions:

---

I think that I shall restrict myself to a maximum of three solutions per day (at least at the start), so that everyone has a chance to answer some if they want.
11. Hang on a sec, mine was mislabelled, should be Step I not step II.. sorry about that!

edit: thanks.. now typing up II/2
12. STEP II/2

Therefore, by (*),

we can also use (*) to express 2cot2x as

Therefore,

we can compare the coefficients of the respective summations and say

for the second part, note that

we have to find an identity for 2csc2x

(using partial fractions)

Also,

, so:

Comparing coefficients of the polynomials (in x) of the expansion,

Rearranging and substituting in the earlier definition of ,

any corrections welcome !
13. STEP II, (11) attached

attachments 2&3 are in the wrong order
Attached Files
14. step1989q11.doc (540.0 KB, 95 views)
15. step1989q11p3.doc (205.5 KB, 93 views)
16. step1989q11p2.doc (158.5 KB, 86 views)
17. Step III (2)
Attached Files
18. step1989q2p1.doc (113.5 KB, 120 views)
19. step1989q2p2.doc (79.0 KB, 103 views)
20. step1989q2p3.doc (71.5 KB, 100 views)
21. STEP II (6) attached
Attached Files
22. step1989q6p1.doc (79.5 KB, 117 views)
23. step1989q62.doc (95.0 KB, 123 views)
24. STEP II Question 3

Equating real and imaginary parts:

(*)

(**)

if x=a, then:

(1)

(2)

Now, subbing (1) and (2) into:

We get:

As required.

Substituting y=b into (*) and (**):

(3)

(4).

This time, using , we end up with:

.

Since and , . Hence, () corresponds to a single hyeprbola branch, which is above the line v=0.

One point of intersection is: and .

Differentiating implicitly we get:

.

Which is equal to: at and .

Differentiating implicitly we get:

.

Which is, when and .

X

Hence the curves intersect and right angles at this point.

Regarding the sketch, I think the elipse becomes a vertical straight line from v = -1 to v = 1, and the hyperbola becomes the line v = 1.
25. STEP 1, Q3

(By the ratio theorem.)

These vectors are perpendicular, so the scalar product is zero.

(Note that the scalar product is commutative and distributive.)

Let

Minimum and maximum values occur at the endpoint of a range or at a turning point. There is a turning point when x = 1/2.

So

Note that

Where

The maximum value is occurs when

Suppose that

Then

So

By the cosine rule,
Let F be the midpoint of AB. By Pythagoras,

26. *bump* (just to remind people that this thread is still here ) - I have solutions that I can type up if no one else wants to, but I really don't want to take over the thread..
27. STEP I - Question 7

*Graph Attached*

If we work out the area of a quarter of the rectangle, which is in the top right hand bit of the axes, and maximise this, it make calculation soooo much eaiser. (Since we can forget about modulus signs and y being negative) :

Let the area of the rectangle be .

Base of smaller rectangle = 1-x
height =

so

which we want to maximise.

Which is zero when

Finding the second derivative and subsituting x = in shows is a maximum at this point.

So .

________

We can do the second bit in the same way, since its a closed curve that is symmetrical in both axes.

Let the area of the second rectangle be .
Working in the top right quadrant again;
Base = 1-x
height = .

So .

Which is zero when x = 0 or x = .

Clearly the area is a minimum when x= 0, and so is a maximum when x =.

.
Attached Files
28. STEP I Q7 graph.doc (196.0 KB, 106 views)
29. I've got a solution to I question 9 , but don't have a scanner available so will add graphs when I get the opportunity.

Question 9 STEP I

and Setting y'=0
f(2)=-2, f(-2)=2, also for graphing purposes it may be worth to note that f(0)=0 and f(4)=2 and f(-4)=-2. y=0 has solutions x=0 or f''(-2)=- i.e. local max f''(2)=+ i.e. local min
Graph, see attachment

(a) i.e. therefore the equation of y in this X-Y plane is
setting it equal to 0 gives f(1)=-2 ans f(-1)=2
For graphing it can also be good to see that f(2)=2, f(-2)=-2 and
Graph, see attachment

(b) i.e. therefore the equation of y in this X-Y plane is
Setting this equal to 0 gives . f(2)=-1 and f(-2)=1
For graphing, also see that , f(4)=1 and f(-4)=-1
Graph, see attachment.

(c) i.e. therefore the equation in the X-Y plane is
Setting this equal to 0 gives f(0)=2, f(2)=-2.
For graphing, also see that and f(-1)=-2, f(3)=2

(d) i.e. therefore the equation in the X-Y plane is
setting this equal to 0 gives f(2)=0 f(-2)=2
For graphing I'll check f(4) and f(-4) too, which give 2 and 0 respectively.

For the last part, we're looking for a graph in the X-Y plane with local min (0,0) and local max (1,1). To obtain this on the form X=ax+b and Y=cy+d, first observe that for a graph of a cubic, changing the sign means a reflection in the x-axis i.e. local max and min swap places, this is what we need to do. That means either a or c is negative (but not both!). a squeezes the graph along the X-axis and c squeezes it along the Y-axis, c moves the graph to the left/right and d moves the graph up/down. Knowing these we can see that and will produce the desired graph.

Okay, so I hope I've interpreted everything correct in the question. Seems pretty nice as a question.

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# 9.4 Sum-to-product and product-to-sum formulas  (Page 3/6)

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## Verifying the identity using double-angle formulas and reciprocal identities

Verify the identity $\text{\hspace{0.17em}}{\mathrm{csc}}^{2}\theta -2=\frac{\mathrm{cos}\left(2\theta \right)}{{\mathrm{sin}}^{2}\theta }.$

For verifying this equation, we are bringing together several of the identities. We will use the double-angle formula and the reciprocal identities. We will work with the right side of the equation and rewrite it until it matches the left side.

$\begin{array}{ccc}\hfill \frac{\mathrm{cos}\left(2\theta \right)}{{\mathrm{sin}}^{2}\theta }& =& \frac{1-2\text{\hspace{0.17em}}{\mathrm{sin}}^{2}\theta }{{\mathrm{sin}}^{2}\theta }\hfill \\ & =& \frac{1}{{\mathrm{sin}}^{2}\theta }-\frac{2\text{\hspace{0.17em}}{\mathrm{sin}}^{2}\theta }{{\mathrm{sin}}^{2}\theta }\hfill \\ & =& {\mathrm{csc}}^{2}\theta -2\hfill \end{array}$

Verify the identity $\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}\mathrm{cot}\text{\hspace{0.17em}}\theta -{\mathrm{cos}}^{2}\theta ={\mathrm{sin}}^{2}\theta .$

$\begin{array}{ccc}\hfill \mathrm{tan}\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}\mathrm{cot}\text{\hspace{0.17em}}\theta -{\mathrm{cos}}^{2}\theta & =& \left(\frac{\mathrm{sin}\text{\hspace{0.17em}}\theta }{\mathrm{cos}\text{\hspace{0.17em}}\theta }\right)\left(\frac{\mathrm{cos}\text{\hspace{0.17em}}\theta }{\mathrm{sin}\text{\hspace{0.17em}}\theta }\right)-{\mathrm{cos}}^{2}\theta \hfill \\ & =& 1-{\mathrm{cos}}^{2}\theta \hfill \\ & =& {\mathrm{sin}}^{2}\theta \hfill \end{array}$

Access these online resources for additional instruction and practice with the product-to-sum and sum-to-product identities.

## Key equations

 Product-to-sum Formulas $\begin{array}{ccc}\hfill \mathrm{cos}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\beta & =& \frac{1}{2}\left[\mathrm{cos}\left(\alpha -\beta \right)+\mathrm{cos}\left(\alpha +\beta \right)\right]\hfill \\ \hfill \mathrm{sin}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\beta & =& \frac{1}{2}\left[\mathrm{sin}\left(\alpha +\beta \right)+\mathrm{sin}\left(\alpha -\beta \right)\right]\hfill \\ \hfill \mathrm{sin}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\beta & =& \frac{1}{2}\left[\mathrm{cos}\left(\alpha -\beta \right)-\mathrm{cos}\left(\alpha +\beta \right)\right]\hfill \\ \hfill \mathrm{cos}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\beta & =& \frac{1}{2}\left[\mathrm{sin}\left(\alpha +\beta \right)-\mathrm{sin}\left(\alpha -\beta \right)\right]\hfill \end{array}$ Sum-to-product Formulas $\begin{array}{ccc}\hfill \mathrm{sin}\text{\hspace{0.17em}}\alpha +\mathrm{sin}\text{\hspace{0.17em}}\beta & =& 2\text{\hspace{0.17em}}\mathrm{sin}\left(\frac{\alpha +\beta }{2}\right)\mathrm{cos}\left(\frac{\alpha -\beta }{2}\right)\hfill \\ \hfill \mathrm{sin}\text{\hspace{0.17em}}\alpha -\mathrm{sin}\text{\hspace{0.17em}}\beta & =& 2\text{\hspace{0.17em}}\mathrm{sin}\left(\frac{\alpha -\beta }{2}\right)\mathrm{cos}\left(\frac{\alpha +\beta }{2}\right)\hfill \\ \hfill \mathrm{cos}\text{\hspace{0.17em}}\alpha -\mathrm{cos}\text{\hspace{0.17em}}\beta & =& -2\text{\hspace{0.17em}}\mathrm{sin}\left(\frac{\alpha +\beta }{2}\right)\mathrm{sin}\left(\frac{\alpha -\beta }{2}\right)\hfill \\ \hfill \mathrm{cos}\text{\hspace{0.17em}}\alpha +\mathrm{cos}\text{\hspace{0.17em}}\beta & =& 2\text{\hspace{0.17em}}\mathrm{cos}\left(\frac{\alpha +\beta }{2}\right)\mathrm{cos}\left(\frac{\alpha -\beta }{2}\right)\hfill \end{array}$

## Key concepts

• From the sum and difference identities, we can derive the product-to-sum formulas and the sum-to-product formulas for sine and cosine.
• We can use the product-to-sum formulas to rewrite products of sines, products of cosines, and products of sine and cosine as sums or differences of sines and cosines. See [link] , [link] , and [link] .
• We can also derive the sum-to-product identities from the product-to-sum identities using substitution.
• We can use the sum-to-product formulas to rewrite sum or difference of sines, cosines, or products sine and cosine as products of sines and cosines. See [link] .
• Trigonometric expressions are often simpler to evaluate using the formulas. See [link] .
• The identities can be verified using other formulas or by converting the expressions to sines and cosines. To verify an identity, we choose the more complicated side of the equals sign and rewrite it until it is transformed into the other side. See [link] and [link] .

## Verbal

Starting with the product to sum formula $\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\beta =\frac{1}{2}\left[\mathrm{sin}\left(\alpha +\beta \right)+\mathrm{sin}\left(\alpha -\beta \right)\right],$ explain how to determine the formula for $\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\beta .$

Substitute $\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}$ into cosine and $\text{\hspace{0.17em}}\beta \text{\hspace{0.17em}}$ into sine and evaluate.

Provide two different methods of calculating $\text{\hspace{0.17em}}\mathrm{cos}\left(195°\right)\mathrm{cos}\left(105°\right),$ one of which uses the product to sum. Which method is easier?

Describe a situation where we would convert an equation from a sum to a product and give an example.

Answers will vary. There are some equations that involve a sum of two trig expressions where when converted to a product are easier to solve. For example: $\text{\hspace{0.17em}}\frac{\mathrm{sin}\left(3x\right)+\mathrm{sin}\text{\hspace{0.17em}}x}{\mathrm{cos}\text{\hspace{0.17em}}x}=1.\text{\hspace{0.17em}}$ When converting the numerator to a product the equation becomes: $\text{\hspace{0.17em}}\frac{2\text{\hspace{0.17em}}\mathrm{sin}\left(2x\right)\mathrm{cos}\text{\hspace{0.17em}}x}{\mathrm{cos}\text{\hspace{0.17em}}x}=1$

Describe a situation where we would convert an equation from a product to a sum, and give an example.

## Algebraic

For the following exercises, rewrite the product as a sum or difference.

$16\text{\hspace{0.17em}}\mathrm{sin}\left(16x\right)\mathrm{sin}\left(11x\right)$

$8\left(\mathrm{cos}\left(5x\right)-\mathrm{cos}\left(27x\right)\right)$

$20\text{\hspace{0.17em}}\mathrm{cos}\left(36t\right)\mathrm{cos}\left(6t\right)$

$2\text{\hspace{0.17em}}\mathrm{sin}\left(5x\right)\mathrm{cos}\left(3x\right)$

$\mathrm{sin}\left(2x\right)+\mathrm{sin}\left(8x\right)$

$10\text{\hspace{0.17em}}\mathrm{cos}\left(5x\right)\mathrm{sin}\left(10x\right)$

$\mathrm{sin}\left(-x\right)\mathrm{sin}\left(5x\right)$

$\frac{1}{2}\left(\mathrm{cos}\left(6x\right)-\mathrm{cos}\left(4x\right)\right)$

$\mathrm{sin}\left(3x\right)\mathrm{cos}\left(5x\right)$

For the following exercises, rewrite the sum or difference as a product.

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## Post-edited  Statistical inverse problems and geometric "wavelet" construction Kerkyacharian, Gérard (Auteur de la Conférence) | CIRM (Editeur )

In the fist part of the talk, we will look to some statistical inverse problems for which the natural framework is no more an Euclidian one.
In the second part we will try to give the initial construction of (not orthogonal) wavelets -of the 80 - by Frazier, Jawerth,Weiss, before the Yves Meyer ORTHOGONAL wavelets theory.
In the third part we will propose a construction of a geometric wavelet theory. In the Euclidian case, Fourier transform plays a fundamental role. In the geometric situation this role is given to some "Laplacian operator" with some properties.
In the last part we will show that the previous theory could help to revisit the topic of regularity of Gaussian processes, and to give a criterium only based on the regularity of the covariance operator.
In the fist part of the talk, we will look to some statistical inverse problems for which the natural framework is no more an Euclidian one.
In the second part we will try to give the initial construction of (not orthogonal) wavelets -of the 80 - by Frazier, Jawerth,Weiss, before the Yves Meyer ORTHOGONAL wavelets theory.
In the third part we will propose a construction of a geometric wavelet theory. In the Euclidian case, Fourier transform ...

## Multi angle  On the relations between the $L^1$ norm of partial derivatives with very accessible proof Wojciechowski, Michal (Auteur de la Conférence) | CIRM (Editeur )

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Nuage de mots clefs ici

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Nov 11, 2019

Electrical Characterization of Wafer-Bonded Germanium-on-Insulator Substrates Using a Four-Point-Probe Pseudo-Metal–Oxide–Semiconductor Field-Effect Transistor

The electrical characteristics of wafer-bonded non-doped germanium-on-insulator (GOI) substrates were investigated using a four-point-probe pseudo-metal–oxide–semiconductor field-effect transistor. Annealing the wafer-bonded GOI substrates in vacuum strongly influenced their electrical characteristics. GOI samples annealed at temperatures below 500 °C exhibited n-channel depletion transistor operation, whereas GOI samples annealed at temperatures between 550 and 600 °C exhibited p-channel depletion transistor operation. The carrier mobility strongly depended on the sweep direction of the gate voltage; this characteristic disappeared after annealing at temperatures above 550 °C. The dependence of the electrical characteristics on the annealing temperature is explained in terms of the influence of the defect states on energy band bending near the interface.

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# Circle A has a center at (3 ,1 ) and an area of 12 pi. Circle B has a center at (5 ,2 ) and an area of 24 pi. Do the circles overlap?

Oct 6, 2016

The circles overlap

#### Explanation:

We can use the areas of the circles to compute their respective radii:

A_A = pi(r_A)²
A_B = pi(r_B)²

12pi = pi(r_A)²
24pi = pi(r_B)²

${r}_{A} = \sqrt{12}$
${r}_{B} = \sqrt{24}$

The circles will not overlap if the distance, d, between their centers is greater than $\sqrt{12} + \sqrt{24} \approx 8.36$

d = sqrt((5 - 3)² + (2 - 1)²)

d = sqrt((4)² + (1)²)

$d = \sqrt{5}$

$d \approx 2.23$

The larger circle encloses the smaller if:

$d < \sqrt{24} - \sqrt{12}$

$d < 1.44$

$1.44 < 2.23 < 8.36$, therefore, the circles overlap.