# Course:MATH110/Archive/2010-2011/003/Homework

## Homework 19 - due Friday April 1

The homework consists of two parts, all of which are due by Friday, April 1, 8:00am, no exceptions.

First part - WebWork

Complete the set Homework19.

Second part - Written Homework

The written homework is what prepares you best for harder problem solving questions. Those are the most advanced problems you'll be solving, taking a genuine effort into working on those problems is a guarantee to develop deeper understanding of the material. The general writing guidelines are found here and contain some indications on how points relating to the presentation are given or taken away.

Solve the following two optimization problems. Remember you're not getting marks for just the answer, the quality of your explanation is the key here. Setting up the problems is by far the most difficult part of these problems, if you just let someone else tell you how to "get started" you don't get any guarantees that you will be able to solve such problems on your own on the Final Exam, so watch out for what looks like the easy way.

The following problem is a little bit longer (not harder) than the two above and hence will constitute a bonus question counting towards helping your homework grade.

## Homework 18 - due Friday March 25

The homework consists of two parts, all of which are due by Friday, March 25, 8:00am, no exceptions.

First part - WebWork

Complete the set Homework18.

Second part - Written Homework

The written homework is what prepares you best for harder problem solving questions. Those are the most advanced problems you'll be solving, taking a genuine effort into working on those problems is a guarantee to develop deeper understanding of the material. The general writing guidelines are found here and contain some indications on how points relating to the presentation are given or taken away.

Solve the following three optimization problems. Remember you're not getting marks for just the answer, the quality of your explanation is the key here. Setting up the problems is by far the most difficult part of these problems, if you just let someone else tell you how to "get started" you don't get any guarantees that you will be able to solve such problems on your own on the Final Exam, so watch out for what looks like the easy way.

## Homework 17 - due Friday March 18

The homework consists of two parts, all of which are due by Friday, March 18, 8:00am, no exceptions.

First part - WebWork

Complete the set Homework17.

Second part - Written Homework

The written homework is what prepares you best for harder problem solving questions. Those are the most advanced problems you'll be solving, taking a genuine effort into working on those problems is a guarantee to develop deeper understanding of the material. The general writing guidelines are found here and contain some indications on how points relating to the presentation are given or taken away.

Problem 1

A Gaussian function is a function of the type

${\displaystyle f(x)=a\cdot e^{-{\frac {(x-b)^{2}}{2c^{2}}}}}$

where ${\displaystyle a}$, ${\displaystyle b}$ and ${\displaystyle c>0}$ are parameters. Gaussian functions are used in many disciplines, from statistics to image processing.

One of the most common use is to have them describe probability distributions (for example, the probability distribution of the height of Canadians). Then the parameters ${\displaystyle a}$, ${\displaystyle b}$ and ${\displaystyle c}$ are constrained by the expected value ${\displaystyle \mu }$ (the Greek letter mu) and the variance ${\displaystyle \sigma ^{2}}$ (the Greek letter sigma) by the following equations:

${\displaystyle a={\frac {1}{\sigma {\sqrt {2\pi }}}},\quad b=\mu ,\quad c=\sigma }$

Consider the Gaussian probability distribution with expected value 3 and variance 4. Find the coordinates of the global maximum of the function and the coordinates of the inflexion points and then sketch the function.

Problem 2

Prove that a Gaussian function of type

${\displaystyle f(x)=a\cdot e^{-{\frac {(x-b)^{2}}{2c^{2}}}}}$

will always have inflexion points at ${\displaystyle x=b+c}$ and ${\displaystyle x=b-c}$.

Problem 3

This is a review of the Winter Final. We say that two curves intersect at a right angle if they intersect at a point at which their respective tangent lines are perpendicular.

• Draw a picture/diagram which describes visually what it means for two curves to intersect at a right angle.
• Verify that the curves ${\displaystyle y={\frac {1}{6}}{\sqrt {2x^{2}+7}}}$ and ${\displaystyle y=-2x^{2}+7x-{\frac {13}{6}}}$ intersect at ${\displaystyle x=3}$.
• At this point, do they intersect at a right angle?

Consider now the curve ${\displaystyle y=x^{2}}$. We would like it to intersect at a right angle a curve of the type ${\displaystyle y=ax^{2}+2x+c}$, for some specific values of ${\displaystyle a}$ and ${\displaystyle c}$.

• For which values of ${\displaystyle a}$ and ${\displaystyle c}$ does ${\displaystyle y=x^{2}}$ intersect the curve ${\displaystyle y=ax^{2}+2x+c}$ at a right angle at ${\displaystyle x=2}$?

## Homework 16 - due Friday March 11

The homework consists of two parts, all of which are due by Friday, March 11, 8:00am, no exceptions (unless discussed by email prior to the deadline).

First part - WebWork

Due to the feedback from last week, there are less WebWork problems assigned this week. There are 15 problems, of which the first 6 are about computing derivatives and playing with the second derivative. The remaining 9 are about finding critical points and extremums.

A useful tip: When computing derivatives of polynomials that are factored, it is sometimes best to use the product rule on the factors instead of simply expanding the polynomials, this allows for an easier factorization of the derivative (which is very desirable if you plan on computing a sign table).

As usual now, 10% of the webwork grade will consist of the Success Indicator. Basically, this takes your performance and ponders by the number of attempts. The more you attempt a problem, the lower your success indicator.

Complete the set Homework16.

Second part - Written Homework

The written homework is what prepares you best for harder problem solving questions. Those are the most advanced problems you'll be solving, taking a genuine effort into working on those problems is a guarantee to develop deeper understanding of the material. The general writing guidelines are found here and contain some indications on how points relating to the presentation are given or taken away.

Problem 1

The function

${\displaystyle P(t)=300e^{{\frac {1}{3}}t^{3}-6t^{2}+35t}}$

describes the population of fish in a pond for the next 8 years. So ${\displaystyle t=0}$ corresponds to this year and ${\displaystyle t=8}$ is 2019.

According to this model, will the population always increase or will it have periods of population decrease? What important features of this model can you describe using calculus? What does those features of the model tell you about the fish population?

Problem 2

As we know, vertical parabolas have an equation of the type ${\displaystyle y=ax^{2}+bx+c}$ where ${\displaystyle a}$, ${\displaystyle b}$ and ${\displaystyle c}$ are any choice of real numbers (and of course we ask ${\displaystyle a}$ not to be zero, otherwise it isn't much of a parabola).

We know since high school that the parabola is positive (or smiling) if ${\displaystyle a}$ is positive and that it is negative (or sad) if ${\displaystyle a}$ is negative. Give a rigorous proof of this claim using your knowledge of calculus.

Problem 3

Determine the absolute and local extremas of the function

${\displaystyle f(x)=x-x^{2/3}}$ on the interval [-1,2].

## Homework 15 - due Friday March 4

The homework consists of three parts, all of which are due by Friday, March 4, 8:00am, no exceptions.

First part - WebWork

As mentioned last week, 10% of the webwork grade will consist of the Success Indicator. Basically, this takes your performance and ponders by the number of attempts. The more you attempt a problem, the lower your success indicator.

Complete the set Homework15.

Second part - Written Homework

The written homework is what prepares you best for harder problem solving questions. Those are the most advanced problems you'll be solving, taking a genuine effort into working on those problems is a guarantee to develop deeper understanding of the material. The general writing guidelines are found here and contain some indications on how points relating to the presentation are given or taken away.

Problem 1

We'll finish the proof of the Mean Value Theorem in this problem. Recall that in class, we proved a subcase of the Mean Value Theorem, mainly we proved it to be true when the average of the function is zero, this is called Rolle's Theorem.

Starting with any function ${\displaystyle f}$ which is continuous and differentiable on an interval [${\displaystyle a}$, ${\displaystyle b}$], we would like to prove that there exists a point ${\displaystyle c}$ in that interval such that its tangent line is parallel to the average of the function on the interval, this is equivalent to ask that:

${\displaystyle f'(c)={\dfrac {f(b)-f(a)}{b-a}}}$

We want to do this using our knowledge of Rolle's Theorem which almost says the same, but only for functions with an average of 0.

How to do this? Take your function ${\displaystyle f}$ and tilt it! To do so, we'll create a new function ${\displaystyle g}$ as follow:

${\displaystyle g(x)=f(x)-\left({\dfrac {f(b)-f(a)}{b-a}}(x-a)+f(a)\right)}$

Remarks

1. Notice that
${\displaystyle y={\dfrac {f(b)-f(a)}{b-a}}(x-a)+f(a)}$
is the equation of the line going through the points (${\displaystyle a}$,${\displaystyle f(a)}$) and (${\displaystyle b}$,${\displaystyle f(b)}$). You should be able to do that on your own.
1. If you think this is confusing, just pick your favourite function ${\displaystyle f(x)}$ and do the above explicitly to see what happens and what does ${\displaystyle g}$ looks like, you can ask Wolfram|Alpha to plot it for you. Then try to understand why this will work all the time and how the above is the good way to write this.#

Questions
Now, the goal of this homework is two-folds. First show that the new function ${\displaystyle g}$ has the correct properties to allow us to apply Rolle's Theorem on it. Then, show how applying Rolle's Theorem to ${\displaystyle g}$ will give us a proof of the Mean Value Theorem.

Problem 2

When proving the Mean Value Theorem, we made use of the Extreme Value Theorem (which I called the Min/Max Theorem in class, but it seems not to be the general habit, so I'll call it the Extreme Value Theorem from now on).

In this problem, we'll practice your ability to produce counter-examples. For each of the following cases, we're entirely satisfied by a decent picture that illustrates the required behaviour, no need to explicitly produce an algebraic expression for the functions.

1. Sketch the graph of a continuous function defined over the interval [3, 12] whose maximum and minimum values are none of the endpoints.
2. Sketch the graph of a continuous function defined over the interval [3, 12] which attains its maximum value twice and whose minimum value is at one of the endpoint.
3. We'll show that the condition on the Extreme Value Theorem asking for the function to be continuous is absolutely necessary. For this, sketch the graph of a non-continuous function on the interval [3, 12] which does not have a maximum value or minimum value. Explain in your own words why this counter-example accomplishes what was intended.
4. We'll show that the condition on the Extreme Value Theorem asking for the function to be continuous on a closed interval is absolutely necessary. For this, sketch the graph of a continuous function on the open interval (3, 12) which does not have a maximum value or minimum value. Explain in your own words why this counter-example accomplishes what was intended.

Note: What is mainly interesting in all these problems is clearly not the answers but how you can explain how you got them and why they are correct and make sense. That's the main reason behind you submitting your work to be marked so you get the feedback on how you attack problems and solve them.

Third part - Midterm corrections

The team work is substituted this week with remedial work on the midterm. As for the rest of the homework, you're invited to work in groups (that doesn't mean get the answers from a friend) but you have to submit your own work.

Second Midterm, Question 2

HOP-62 and NCI-H226 are two cell lines maintained by the National Cancer Institute in the United States. Under ideal laboratory conditions, a population of HOP-62 cells doubles every 39 days, and a population of NCI-H226 cells doubles every 61 days.

• How long does it take for a population of HOP-62 cells to triple?
• If you start with 10 HOP-62 cells and 200 NCI-H226 cells, after how many days are the populations equal? (Note: This is not the original question form the midterm, here the initial populations of cells have been switched so that there is an answer in the future).

Remarks
If in the first question you've answered anything like 39 + 0.5 x 39 (or even worse 40 + 20) ~ 60 days, you might think it's a good rough answer but it actually highlights deep misunderstanding about how such exponential functions behave. If you want to check if you have some understanding, you should be able to find out without any computations if the actual answer is going to be smaller or larger than 60 (or more precisely 39 + 0.5 x 39 = 58.5.

Second Midterm, Question 6
• Use an appropriate linear approximation to estimate ${\displaystyle {\sqrt {48.9}}}$.
• Is your estimate in the first part greater than, less than, or equal to ${\displaystyle {\sqrt {48.9}}}$.

Remarks
If you've answered roughly 7 that's clearly a deep misunderstanding. Linear approximations are really really not about that. If in the second part you answered that the linear approximation is larger than the actual value because the approximation is done at 49 which is slightly larger than 48.9, then you are terribly wrong. Rethink of the question, what is really asked, draw a picture and once you understand this better, look back at your initial answer to see what went wrong in your thinking.

## Homework 14 - due Friday February 25

The homework consists of three parts, all of which are due by Friday, February 25, 8:00am, no exceptions.

First part - WebWork

As a way to keep pushing you to think more about problem solving, from now on 10% of the webwork grade will consist of the Success Indicator which is computed as follow:

${\displaystyle 100\cdot \left({\frac {\text{total number of correct problems}}{\text{total number of problems}}}\right)^{2}\cdot {\frac {1}{\text{average number of attempts per problem}}}}$

(Which is clearly a number between 0 and 100.) Basically, this takes your performance and ponders by the number of attempts. The more you attempt a problem, the lower your success indicator.

Complete the set Homework14.

Second part - Written Homework

The written homework is what prepares you best for harder problem solving questions. Those are the most advanced problems you'll be solving, taking a genuine effort into working on those problems is a guarantee to develop deeper understanding of the material. The general writing guidelines are found here and contain some indications on how points relating to the presentation are given or taken away.

Problem 1

The Space Shuttle launches vertically. In the initial part of its ﬂight, its altitude in kilometres after ${\displaystyle t}$ seconds is given by

${\displaystyle h(t)={\dfrac {t^{2}}{300}}}$.

The launch is supervised from the Launch Control Centre (LCC), 5 kilometres away from the launch pad. Determine the rate at which the distance between the LCC and the Space Shuttle is increasing 30 seconds after liftoff.

Note: What is mainly interesting in all these problems is clearly not the answers but how you can explain how you got them and why they are correct and make sense. That's the main reason behind you submitting your work to be marked so you get the feedback on how you attack problems and solve them.

Third part - Midterm corrections

The team work is substituted this week with remedial work on the midterm. As for the rest of the homework, you're invited to work in groups (that doesn't mean get the answers from a friend) but you have to submit your own work.

Second Midterm, Question 1

For each derivative, compute the actual answer and explain what you didn't do correctly in your midterm (if you did your computation correctly, that's great).

Compute the derivative of each of the following functions.

• ${\displaystyle f(x)=\displaystyle {3^{x}}}$
• ${\displaystyle g(x)=\ln \left({\frac {3x+3}{x^{2}+1}}\right)}$
• ${\displaystyle h(x)=\ln(x\cdot x^{2}\cdot x^{3}\cdot x^{4}\cdot x^{5}\cdot x^{6})+e^{x\cdot x^{2}\cdot x^{3}\cdot x^{4}\cdot x^{5}\cdot x^{6}}}$
Second Midterm, Question 5

As a woman walks away from a streetlight, her shadow lengthens. Prove that it does so at a rate which depends on her speed but not on her position.

Explain in details how to solve this problem and what you have learned from this problem. What could you have done differently during the midterm to solve the problem? Can you use some of those new ideas to get a good start at the problem in the second part of the homework?

HINT: The question 5 of this week's webwork should help you get started if you're stuck.

## Homework 13 - due Friday February 4

The homework consists of three parts, all of which are due by Friday, February 4, 8:00am, no exceptions.

First part - WebWork

Complete the set Homework13.

Second part - Written Homework

This second part focuses on mathematical exposition. The general writing guidelines are found here and contain some indications on how points relating to the presentation are given or taken away.

Problem 1

A nuclear weapon detonated at an altitude emits a blast wave that expands radially in three dimensions, like a sphere. The speed of the blast wave is not constant: it is approximately ${\displaystyle 30/r^{2}}$ kilometres per second, where ${\displaystyle r}$ is the distance (in kilometres) from the centre of detonation. Explain why the rate of expansion of the volume contained within the blast wave is constant at any time after the detonation.

Problem 2

A company has paid some external consultants to model the demand over time of their leading product: Product X. The consultants gave them the function

${\displaystyle Q(t)=10\left(1+{\frac {1}{10}}\sin \left({\frac {t}{2}}\right)\right)e^{\frac {t}{25}}}$

Where ${\displaystyle Q}$ is the demand in ${\displaystyle t}$ years measured in thousands of units.

Having done some data analysis over the past, the company knows that the cost of producing Product X (in thousands of dollars) is linked to its demand (again in thousands of units) in the following way: the marginal cost of demand is always 2% of the cost. We also know that it costs 20 thousand dollars to produce 10 thousands units of Product X.

Now the big question: Out of all that information, what the company wants to know is their predicted marginal cost in 10 years from now.

Because there will be some nasty numbers involved here, you may of course use a calculator for your computations. You're even welcome to use WolframAlpha to check your graphing and your derivatives (I've created some nice widgets just for that on a test page, it's here tell me what you think of them). But I want to point out that you should be able to do all the derivatives by hand on your own). What matters most here is the quality of your explanation, how you solve the problem, how you deal with the assumptions and so on. Just getting the good answer won't grant a lot of points, that's a promise.

Note: The economic purist will have noticed that we are making the assumption here that the company exactly produces its demand. I'm more willing to get rid of that assumption in the future, but it will make things more complicated.

Note: What is mainly interesting in all these problems is clearly not the answers but how you can explain how you got them and why they are correct and make sense. That's the main reason behind you submitting your work to be marked so you get the feedback on how you attack problems and solve them.

Third part - Team Work

Team Problem

Pick one of the topic offered below and then explain in your own words what it means that these concepts work on a logarithmic scale. Create a wiki page with all your explanations. The length of that page is up to you, but it should feel like the work of four people thinking about a topic and trying to make sense of it. If you drafted something and would like some comments (within 24 hours), send me an email and a link where to look at, I'll post comments).

Topics:

Note: If you have any question about any part of this homework assignment, start a thread in the discussion page or send me an email.

## Homework 12 - due Friday January 28

The homework consists of three parts, all of which are due by Friday, January 28, 8:00am, no exceptions.

First part - WebWork

Complete the set Homework12. You can help each other on the new version of the Math Forum page. Ask questions and give hints as much as possible, not full answers please.

Webwork FAQ help
Note: you're only allowed 5 tries for most questions in the system. Make it your own goal to not only answer correctly on your first try, but build strategies to guarantee that your answer is correct. You're not asking the machine to tell you if you got it right, you should aim at just "letting the machine know" what the good answer is.

Second part - Written Homework

This second part focuses on mathematical exposition. The general writing guidelines are found here and contain some indications on how points relating to the presentation are given or taken away.

Please make sure you do the following:

• Write your solutions clearly and neatly in ink (no pencil).
• Write your name, student number and group number at the top of the page.
• Leave a margin on one side for comments.

As a general guideline, think of this as sort of an essay that you work on and the write-up, note worksheets that are passed on to us. You don't have to type your homework, hand writing (if legible) is fine. Some of you asked/suggested to write the homework on a page of yours on the wiki (in the Sandbox or as a subpage of your userpage) that's fine, as long as you then print it and hand it in class. You're allowed to work collaboratively on the solutions, but the write-up should be your own.

Problem 1

Prove that the function

${\displaystyle P(t)={\frac {1}{1+e^{-t}}}}$

Satisfies the equation

${\displaystyle P'(t)=P(t)\left(1-P(t)\right)}$

Note: such an equation involving both a function and its derivative is an example of what is called a differential equation.

Then try to interprete this equation. What does it say about the instantaneous rate of change and how does that make sense when looking at the graph of the function?

Problem 2

1. If a patient ingests a 200mg tablet of Ibuprofen at 1pm, at what time will there be less than 1mg of Ibuprofen left in his body?
2. How much time would it take to eliminate 99.9% of an ingested tablet of Ibuprofen?
Problem 3

On your profile page within the wiki, write an essay describing a particular use of calculus in your ﬁeld of study, or in a ﬁeld of interest to you. Feel free of course to add pictures, links and/or videos on the page (that's the advantage of writing online versus on paper). Your essay should be at least 500 words long (think a nice and interesting full page of text).

Note: What is mainly interesting in all these problems is clearly not the answers but how you can explain how you got them and why they are correct and make sense. That's the main reason behind you submitting your work to be marked so you get the feedback on how you attack problems and solve them.

Third part - Team Work

Team Problem

Starting with the function

${\displaystyle P(t)={\frac {1}{1+e^{-t}}}}$

Your goal is to modify the function so that we can use it to model a real-life problem. We want to be able to control the following things:

• Change the height of the horizontal asymptote on the right, we'll denote it by ${\displaystyle K}$.
• Change the ${\displaystyle y}$-intercept to any number between ${\displaystyle 0}$ and ${\displaystyle K}$

BONUS (just the point below, not what comes after)

• Change the slope of the curved part. Find a way so that the slope can go from very close to zero to almost vertical. (If you graph it, it should be quite clear).

Once you've played with the function enough, try to find an application of the graph to model something. It can be anything which starts at a value and then goes to another one (think for a population, it goes from 0 to it's carrying capacity). Explain what you are modelling and how you decide to attribute a numerical value to each of the 2 or 3 parameters that you researched just above. Then use the model to make a prediction. For example, if your model is suppose to describe a population for which you have its initial population and carrying capacity (potentially its rate of increase if you solved the bonus part), then use that data to make a prediction for the population in 20 years, or use the model to predict when will the population reach 95% of its carrying capacity).

When doing this last part, explain well where you're taking your data from (real data or imagined data), what it is that you're modelling and how you are doing the math to answer a predictive question.

This should all be done on a dedicated page of the wiki whose address should look like

• wiki.ubc.ca/Course:MATH110/Archive/2010-2011/003/Teams/YOURTEAMNAME/Homework_12
• wiki.ubc.ca/Course:MATH110/Archive/2010-2011/003/Teams/YOURTEAMNAME/Homework/Homework_12

Note: If you have any question about any part of this homework assignment, start a thread in the discussion page or send me an email.

## Homework 11 - due Wednesday January 19

The homework consists of three parts, all of which are due by Wednesday, January 19, 8:00am, no exceptions.

First part - WebWork

Complete the set Homework11. You can help each other on the new version of the Math Forum page. Ask questions and give hints as much as possible, not full answers please.

Webwork FAQ help
Note: you're only allowed 5 tries for most questions in the system. Make it your own goal to not only answer correctly on your first try, but build strategies to guarantee that your answer is correct. You're not asking the machine to tell you if you got it right, you should aim at just "letting the machine know" what the good answer is.

Second part - Written Homework

This second part focuses on mathematical exposition. The general writing guidelines are found here and contain some indications on how points relating to the presentation are given or taken away.

Please make sure you do the following:

• Write your solutions clearly and neatly in ink (no pencil).
• Write your name, student number and group number at the top of the page.
• Leave a margin on one side for comments.

As a general guideline, think of this as sort of an essay that you work on and the write-up, note worksheets that are passed on to us. You don't have to type your homework, hand writing (if legible) is fine. Some of you asked/suggested to write the homework on a page of yours on the wiki (in the Sandbox or as a subpage of your userpage) that's fine, as long as you then print it and hand it in class. You're allowed to work collaboratively on the solutions, but the write-up should be your own.

Problem 1

Consider the following operations:

• Sketch the graph of the function ${\displaystyle e^{x}}$.
• Pick any value on the ${\displaystyle x}$-axis and denote it by ${\displaystyle k}$.
• Find the value on the function ${\displaystyle e^{x}}$ at your value ${\displaystyle x=k}$ and call that point ${\displaystyle A}$. In other words, the point ${\displaystyle A}$ has coordinate ${\displaystyle (k,e^{k})}$.
• Denote by ${\displaystyle B}$ the point ${\displaystyle (k-1,0)}$ and add it to your picture.
• Now draw the line passing through ${\displaystyle A}$ and ${\displaystyle B}$.
• We claim that this line is tangent to the graph of the function ${\displaystyle e^{x}}$.

What do you think about the above? If you think it is correct, provide a solid justification for it (among other things, it should work for any ${\displaystyle x}$-value that I pick). If you believe it doesn't necessarily work all the time, describe when it does and when it doesn't. Provide some precise counter-examples.

Problem 2

Compare the family of functions which look like ${\displaystyle b^{x}}$ for any positive values of ${\displaystyle b}$ to the family of functions which look like ${\displaystyle e^{rx}}$ for any (negative, zero or positive) values of ${\displaystyle r}$. What can you say?

Note: What is mainly interesting in all these problems is clearly not the answers but how you can explain how you got them and why they are correct and make sense. That's the main reason behind you submitting your work to be marked so you get the feedback on how you attack problems and solve them.

Third part - Team Work

Write a linear model to predict the cost of producing flags of your team's Canton under the assumptions that the marginal cost is $7 per unit and that at the current production level of 20 items, the cost is$100.

• What does your model predict for a production of 150 items?
• According to your model, what happens to the average cost per item as production levels increase?
• Finally, find some other models (not necessarily linear) for which you get other behaviours such as:
• The average cost remains constant as production increases.
• The average cost diminishes as production increases.
• The average cost increases as production increases.
• You obtain an economy of scale. This means that starting at some specific production level, the marginal cost is always less than the average cost.
• Any other interesting properties that you can think of and create a model for. Bonus points can be obtained for very interesting ideas.

Recall that this is a team project question. Your answer is to be posted as a subpage of your team page as a comprehensive essay. Clarity of exposition will be an integral part of the grade. During the weekend, I will post some comments on the discussion page of your team page if I do find some content worth commenting. It is up to your team to decide if you want to make use of this offer.

## Homework 10 - due Wednesday January 12

For the first week, the homework assignment consists of only two parts. Both parts are due by Wednesday, January 12, 8:00am, no exceptions.

First part - WebWork

Complete the set Homework10. You can help each other on the new version of the Math Forum page. Ask questions and give hints as much as possible, not full answers please.

Webwork FAQ help
Note: you're only allowed 5 tries for most questions in the system from now on, make the best of those tries.

Second part - Written Homework

This second part focuses on mathematical exposition. The general writing guidelines are found here and contain some indications on how points relating to the presentation are given or taken away.

Please make sure you do the following:

• Write your solutions clearly and neatly in ink (no pencil).
• Write your name, student number and group number at the top of the page.
• Leave a margin on one side for comments.

As a general guideline, think of this as sort of an essay that you work on and the write-up, note worksheets that are passed on to us. You don't have to type your homework, hand writing (if legible) is fine. Some of you asked/suggested to write the homework on a page of yours on the wiki (in the Sandbox or as a subpage of your userpage) that's fine, as long as you then print it and hand it in class. You're allowed to work collaboratively on the solutions, but the write-up should be your own.

Problem 1

Consider the parabola of equation ${\displaystyle y=x^{2}-4x+5}$ and the point (5,2). Do the following:

• Find the vertex of the parabola and draw a decent sketch of it for values of ${\displaystyle x}$ between -2 and 6.
• Draw the point (5,2) as well.

We would like to know if it is possible that a tangent line to the parabola passes through the point (5,2).

• Use your sketch to give an educated guess for the answer. Justify.
• Solve the problem by creating an equation which tells you which points on the parabola have their tangent lines going through the point (5,2).
Problem 2

In this problem, we'll try to see how to make an informed decision. You got yourself into a time machine and end up back in 2004, on August 19 to be more precise. As you walk in the streets, a newspaper's headline indicates that today is Google's initial public offering (IPO). The shares are selling for $85. Knowing that these shares sold for$280 a piece on August 19 2005, there's a neat amount of money to be made there. But for this, you need cash to be able to buy those shares and as it happens, you have none right now.

After a quick search, the only way that you can get your hands on cash today is to deal with a loan shark. This guy asks for 100% interest per year! And to top it off, he likes to compound it often.

For recall, if you borrow $1000 for one year at 100% interest per year, you owe the loan shark$2000 at the end of the year (you give him back his $1000 and you add another$1000 in interests). But if he decides to compound your interest twice a year, it means that half way through the year he'll compound you 50% (half of his interest rate) which means $500 in interest and at the end of the year, he'll compound the remaining 50% but of the amount you owed him at this point which is now the initial$1000 and the extra $500 of interest computed half way through. This means that you owe him now an additional 50% of$1500 which is $750 and the total amount you owe him is now$2250 (the initial $1000, the first compounded interest of$500 and the second one of $750). You can work out for yourself what happens if the interest is compounded every month or any other frequency. The loan shark will lend you the money and offers two different options to pay him back. The first one is simple: pay him back$3000 in one year. The second option is slightly more tricky. He will charge you 100% interest per year, but he gets to decide how often he will compound it.

You clearly have three choices:

A) Get the $1000 in loan and pay back$3000 in one year.
B) Get the \$1000 in loan and accept to pay back one year later at 100% interest per year compounded to the loan shark's choice.
C) Not get the loan, you don't want to risk losing money to the loan shark.

This problem involves several intermediate steps, the more steps you do, the better of course. Study each choice carefully and justify your answer. This problem is not trivial, get started early enough in the week to guarantee you'll have opportunities to work at it several times and ask questions around if necessary.

Note: What is mainly interesting in all these problems is clearly not the answers but how you can explain how you got them and why they are correct and make sense. That's the main reason behind you submitting your work to be marked so you get the feedback on how you attack problems and solve them.

## Homework 9 - due Monday November 29

The homework assignment consists of three parts. The first two parts are due by Monday, November 29, 8:00am, no exceptions.

First part - WebWork

This week's WebWork will train you with trigonometric functions and their derivatives. The chain rule will be necessary for some of the questions. Make it a personal challenge to answer computational problems as quickly as possible but with the constraint of giving the correct answer on the first try. (Remember you can always hit preview to see if your input is typed correctly so you're not losing a try just because there is a typo in your answer).

Complete the set Week12. You can help each other on the Math Forum page. Ask questions and give hints as much as possible, not full answers please.

Webwork FAQ help
Note: you're only allowed 5 tries for most questions in the system from now on, make the best of those tries.

Second part - Written Homework

This second part focuses on mathematical exposition. The general writing guidelines are found here and contain some indications on how points relating to the presentation are given or taken away.

Please make sure you do the following:

• Write your solutions clearly and neatly in ink (no pencil).
• Write your name, student number and group number at the top of the page.
• Leave a margin on one side for comments.

As a general guideline, think of this as sort of an essay that you work on and the write-up, note worksheets that are passed on to us. You don't have to type your homework, hand writing (if legible) is fine. Some of you asked/suggested to write the homework on a page of yours on the wiki (in the Sandbox or as a subpage of your userpage) that's fine, as long as you then print it and hand it in class. You're allowed to work collaboratively on the solutions, but the write-up should be your own.

Problem 1

For a person whose height is ${\displaystyle x}$ in centimeters (the following model is only valid for people whose height is between 75 and 185 centimeters), the average number ${\displaystyle p}$ of heartbeats per minutes is modelled by

${\displaystyle p(x)={\dfrac {938}{\sqrt {x}}}}$
1. Compare the instantaneous rate of change of the average number of heartbeats per minute of a person whose height is 162.5 cm and a person whose height is 163.5 cm.
2. According to this model, do children have a higher instantaneous rate of change of their average number of heartbeats per minute than adults?
Problem 2

Sketch a single function ${\displaystyle f}$ which has the following properties:

• The function is defined on the interval ${\displaystyle (-\infty ,5)\cup (7,+\infty )}$
• The function has a vertical asymptote at ${\displaystyle x=-5}$
• The function is continuous everywhere on its domain except at ${\displaystyle x=-5}$ where it has a jump discontinuity and at ${\displaystyle x=12}$ where it has a removable discontinuity.
• The function is differentiable everywhere on its domain except at ${\displaystyle x=1}$ and ${\displaystyle x=3}$.
• ${\displaystyle \lim _{x\rightarrow -\infty }f(x)=-\infty }$
• ${\displaystyle \lim _{x\rightarrow +\infty }f(x)=-2}$
• ${\displaystyle \lim _{x\rightarrow 12^{-}}f(x)=5}$
• ${\displaystyle \lim _{x\rightarrow -5^{+}}f(x)=-\infty }$
• ${\displaystyle \displaystyle {f(-9)=6}}$
• ${\displaystyle \displaystyle {f'(-9)=0}}$
• ${\displaystyle \displaystyle {f'(14)=0}}$
• ${\displaystyle \displaystyle {f'(0)=1}}$
Problem 3

When the tortoise raced the hare, the former maintained a constant pace of one kilometer per hour throughout the race, while the latter, being overconfident, wasted much time and averaged only a half kilometer per hour for the first half of the course.

How fast must the hare run over the second half of the course in order to win?

Note: What is mainly interesting in all these problems is clearly not the answers but how you can explain how you got them and why they are correct and make sense. That's the main reason behind you submitting your work to be marked so you get the feedback on how you attack problems and solve them.

Third part - Group work on Basic Skills Project At this point, you should have most of your content on your wiki pages. Tutorials, videos, examples, bits of theory; add it all. We'll spend the last week doing most of the final editing and moving things on the actual project page.

## Homework 8 - due Friday November 19

The homework assignment consists of three parts. All three parts are due by Friday, November 19, 8:00am, no exceptions.

First part - WebWork

This week's WebWork aims at helping you train to take derivatives efficiently (and correctly). Make it a personal challenge to answer the problems as quickly as possible but with the constraint of giving the correct answer on the first try. (Remember you can always hit preview to see if your input is typed correctly so you're not losing a try just because there is a typo in your answer).

Complete the set Week11. You can help each other on the Math Forum page. Ask questions and give hints as much as possible, not full answers please.

Webwork FAQ help
Note: you're only allowed 5 tries for most questions in the system from now on, make the best of those tries.

Second part - Written Homework

This second part focuses on mathematical exposition. The general writing guidelines are found here and contain some indications on how points relating to the presentation are given or taken away.

Please make sure you do the following:

• Write your solutions clearly and neatly in ink (no pencil).
• Write your name, student number and group number at the top of the page.
• Leave a margin on one side for comments.

As a general guideline, think of this as sort of an essay that you work on and the write-up, note worksheets that are passed on to us. You don't have to type your homework, hand writing (if legible) is fine. Some of you asked/suggested to write the homework on a page of yours on the wiki (in the Sandbox or as a subpage of your userpage) that's fine, as long as you then print it and hand it in class. You're allowed to work collaboratively on the solutions, but the write-up should be your own.

Problem 1

Write a proof for the fact I claimed while proving the Power Rule which states that:

The derivative of   ${\displaystyle \displaystyle {h(x)=xf(x)}}$   is   ${\displaystyle \displaystyle {h'(x)=f(x)+xf'(x)}}$ .

Problem 2

Salted water containing 5 grams of salt per liter is being poured at a rate of 10 liters per hour into a tank that initially contains 10 liters of pure unsalted water. Give an expression for the function that gives the concentration of salt in the tank after ${\displaystyle t}$ hours and then explain what will the concentration of salt in the tank be after a very long period of time.

Problem 3

This problems follows on the one from last week but doesn't require you to have solved the previous one to be able to solve this one.

The silver currency of the Kingdom of Bonoria consists of glomeks, nindars and morms. Four golmeks are equal in value to seven nindars; and one glomek and one nindar together are worth thirty-three morms.

On my last visit to Bonoria, I entered a bank, handed the teller some glomeks and nindars and asked him to change them into morms. He told me that if I had twice as many glomeks, he could give me 120 morms; and if I had twice as many nindars he could give me 114 morms.

How many morms did I get in the end?

Note: What is mainly interesting in all these problems is clearly not the answers but how you can explain how you got them and why they are correct and make sense. That's the main reason behind you submitting your work to be marked so you get the feedback on how you attack problems and solve them.

The solution of this homework is here.

Third part - Group work on Basic Skills

The Basic Skills Project spans Homework 5, 6, 7, 8, 9 and 10 and will be ended on December 3rd.

## Homework 7 - due Friday November 12

The homework assignment consists of three parts. All three parts are due by Friday, November 5, 8:00am, no exceptions.

First part

This week, the webwork set is slightly longer and in two parts. Complete sets Week10A and Week10B by Friday morning (I recommend starting to work on them earlier though). You can help each other on the Math Forum page. Ask questions and give hints as much as possible, not full answers please.

Webwork FAQ help
Note: you're only allowed 5 tries for most questions in the system from now on, make the best of those tries.

Second part

This second part focuses on mathematical exposition. The marker wrote some remarks following on the previous homework that you can read here. The general writing guidelines are still here.

Important note: Starting from this week, you are being asked to improve even more the quality of your handed homework. Please make sure you do the following:

• Write your solutions clearly and neatly in ink (no pencil).
• Write your name, student number and group number at the top of the page.
• Leave a margin on one side for comments.

As a general guideline, think of this as sort of an essay that you work on and the write-up, note worksheets that are passed on to us. You don't have to type your homework, hand writing (if legible) is fine. Some of you asked/suggested to write the homework on a page of yours on the wiki (in the Sandbox or as a subpage of your userpage) that's fine, as long as you then print it and hand it in class. You're allowed to work collaboratively on the solutions, but the write-up should be your own.

Problem 1

Find the points of the curve ${\displaystyle y=x^{3}+x^{2}}$ for which the tangent line passes through the origin.

Problem 2

The paper currency of the Kingdom of Bonoria bears the pictures of the country's monarchs. The one-bonor note carries the picture of Queen Griselda the Good. Other notes not exceeding 100 bonors bear the pictures of King Randolph the Rotten, Queen Carrie the Charming, Queen Bonita the beautiful, King Gerlad the Gross, King Waldo the Wicked, and King Hilary the Hairy. We know that:

• Together the notes on which Bonita's and Carrie's pictures appear are worth 102 bonors.
• One Gerald, Waldo, and Randolph together give 73 bonors.
• Hilary and Bonita give 22 bonors.
• Hilary and Carrie give 120 bonors.
• Hillary, Gerald, and Randolph give 43 bonors.
• And Carrie, Waldo, and Randolph give 168 bonors.

On what size note does Waldo's picture appear?

The solution of this homework is here.

Third part - Group work on Basic Skills

The Basic Skills Project spans Homework 5, 6, 7, 8, 9 and 10 and will be ended on December 3rd.

Since the Webwork packages are still giving me a hard time, we'll give them a few days. In the meantime, for Friday's homework, your mission is to make me an offer as a group. My Basic Skills Learning Guide Project is still crying for contributions. Have a look at the project once again, and using the list of basic skills you created last week, make me a contribution offer. More precisely, I want you to create a wiki page (under your group page somewhere) titled Basic skills project and there make me an offer of how you plan, as a group, to contribute to the project. This could be an overview of the topic or worked-out examples, there are many suggestions on the Basic Skills Learning Guide Project page itself. This is will work the same way public offers work and I will only allow give points to offer that seem realistic and valuable. The best contributions will participate in redeeming points for the Skills Test (in the case you failed it, if not, we can work out an arrangement). I want the offers online for Friday morning, same deadline as the rest of this homework. Feel free to email me for questions (but I won't be much around on Thursday, so Wednesday is the day to do so).

## Homework 6 - due Friday November 5

The homework assignment consists of three parts. All three parts are due by Friday, November 5, 8:00am, no exceptions.

First part

WeBWorK problem set. Go on the website, click on this section's link and enter your CWL login. Once in there, click on Week9 to see the homework set and type your answers directly in the system. The option to get the pdf version is just there in case you want to look at the problems away from a computer. Everything should be fairly explicit over there. The system will close the assignment on Friday morning just before class, so make sure you finish your session before this.

Note: you're only allowed 5 tries for most questions in the system from now on, make the best of those tries.

If you have trouble typing the math in the system, check out the FAQ and add to it if necessary.

You can help each other on the Math Forum page. Ask questions and give hints as much as possible, not full answers please.

Second part

This second part focuses on mathematical exposition; you will be graded primarily on the clarity and elegance of your solutions. Solve the following two problems:

Problem 1

The distance covered on the runway by a plane which is about to take off is given by the function ${\displaystyle d(t)=t^{2}}$ where ${\displaystyle t}$ is measured in seconds from the moment the plane starts to move; and ${\displaystyle d(t)}$ is measured in metres from the starting point. If the takeoff speed of the plane is of 200km/h, at what distance from the starting point does the plane take off?

Problem 2

The floor of a square room is to be tiled with black and white tiles according to the pattern showed in the figure here below.

Both white sections are themselves square and the larger white square has exactly eight tiles more on each side than the smaller one. If 1000 white tiles are needed in all, how many black tiles are required?

The solution of this homework is here.

Third part - Group work on Basic Skills

The Basic Skills Project spans Homework 5, 6, 7, 8, 9 and 10 and will be ended on December 3rd.

## Homework 5 - due Friday October 29

The homework assignment consists of three parts. All three parts are due by Friday, October 29, 8:00 am.

First part

WeBWorK problem set. Go on the website, click on this section's link and enter your CWL login. Once in there, click on week8 to see the homework set and type your answers directly in the system. The option to get the pdf version is just there in case you want to look at the problems away from a computer. Everything should be fairly explicit over there. The system will close the assignment on Friday morning just before class, so make sure you finish your session before this.

Note: you're only allowed 5 tries in the system from now on, make the best of those tries.

If you have trouble typing the math in the system, check out the FAQ and add to it if necessary.

You can help each other on the Math Forum page. Ask questions and give hints as much as possible, not full answers please.

Second part

This second part focuses on mathematical exposition; you will be graded primarily on the clarity and elegance of your solutions. Solve the following two problems:

Problem 1

A Rock-Paper-Scissors competition is played with four candidates: Adrian, Bruno, Charlotte and Diana. They each play against each other in a match where the winner is the first to get five points (wins give you 1 point, loss or draws give 0 points). We only got access to the following information:

• Charlotte won all three of her games and her opponents scored a total of only 2 points against her.
• Bruno scored a total of 10 points and his opponents also scored 10.
• Adrian and Diana both scored 8 points, but Adrian allowed his opponents 15 points whereas Diana's opponents only scored 14 points.
• Finally, Diana scored more points against Charlotte than Adrian did.

What was the outcome and the score of each head to head match?

Problem 2

Sketch a single function ${\displaystyle f}$ which has the following properties:

• The function is defined on the interval ${\displaystyle (-2,+\infty )}$
• The function has a vertical asymptote at ${\displaystyle x=-2}$
• The function has a horizontal asymptote on the right at ${\displaystyle y=5}$
• The function is continuous everywhere on its domain except at ${\displaystyle x=3}$ where it has a jump discontinuity and at ${\displaystyle x=5}$ where it has a removable discontinuity.
• ${\displaystyle \lim _{x\rightarrow 5^{-}}f(x)=3}$

Third part - Group work on Basic Skills

The Basic Skills Project spans Homework 5, 6, 7, 8, 9 and 10 and will be ended on December 3rd.

On your group page, have a look at the learning objectives for the basic skills and create a list that classifies these into three categories:

• Those that pose no problem to anyone in the group.
• Those that some of you have issues with, but not everyone in the group.
• Those that no one in the group knows how to handle.

## Homework 4 - due Wednesday October 20

The homework assignment consists of three parts. All three parts are due by Wednesday, October 13, 8:00 am.

First part

WeBWorK problem set. Go on the website, click on this section's link and enter your CWL login. Once in there, click on A4 to see the homework set and type your answers directly in the system. The option to get the pdf version is just there in case you want to look at the problems away from a computer. Everything should be fairly explicit over there. The system will close the assignment on Wednesday morning just before class, so make sure you finish your session before this.

If you have trouble typing the math in the system, check out the FAQ and add to it if necessary.

You can help each other on the Math Forum page. Ask questions and give hints as much as possible, not full answers please.

Second part

From the Calculus: Early Transcendentals text, complete question 54 from section 2.5 and question 86 from section 2.6.

Third part

For the third part, we'll continue to train on problem-solving skills. Have a look at the description of the problems here. Once again, those problems are assigned to you as a group, you can split the work or work all together, but I want all the solutions displayed on your group pages. If a member of your group is not active, send him some emails. If you don't have that person't email, please, contact me, I'll see what I can do.

Make sure to put some order in your group pages, you're allowed to create subpages as you want (the help pages will tell you all about page creations in less than 5 minutes). It is up to you to keep your group pages in good order and maintain a decent presentation.

## Homework 3 - due Wednesday October 13

As usual, the homework assignment consists of three parts. All three parts are due in class by Wednesday, October 13, 8:00 am.

First part

WeBWorK problem set. Go on the website, click on this section's link and enter your CWL login. Once in there, click on A3 to see the homework set and type your answers directly in the system. The option to get the pdf version is just there in case you want to look at the problems away from a computer. Everything should be fairly explicit over there. The system will close the assignment on Wednesday morning just before class, so make sure you finish your session before this.

If you have trouble typing the math in the system, check out the FAQ and add to it if necessary.

If you have trouble with doing the math, you can help each other on the Math Forum page. Ask questions and give hints as much as possible, not full answers please.

Second part

This part of the assignment is drawn directly from the course texts. It focuses on mathematical exposition; you will be graded primarily on the clarity and elegance of your solutions.

From the Calculus: Early Transcendentals text, complete questions 2 from section 2.2 and 80 from section 2.3.

Please have a look at the following guidelines when writing your homework.

Don't forget to staple your homework if needed (folded papers don't hold well) and add your student number. Simply put your homework in your group's folder and bring me the folder at the end of class. Late homework won't be considered.

Third part

The third part consists of the group project. This week, we'll train your problem-solving skills. I've created a new page here which describes Pólya's method and offer 25 problems to practice on. On your wiki page (or a subpage), solve all the problems as described . Whether you split the problems among yourselves or collaborate in your groups is up to you, but I want all the problems to be solved and their solutions displayed. Notice how I said their solutions and not their answers, that's a big difference.

For this part of the homework, I let you work it out among your group. Remember, your group page and subpages are yours, feel free to create pages, change the content, as long as I can find my way there and that you keep track of your group members list, all is game, so if you feel like creating your own Math Forum pages, it's really up to you. If all of you are stuck, feel free to add me to the conversation by asking me a question on my talk page.

## Homework 2 - due Monday October 4

The second homework assignment consists of three parts. If you have any questions about the homework, ask it here.

First part

WeBWorK problem set. We'll be using this online homework tool throughout the year. Go on the website, click on this section's link and enter your CWL login. Once in there, click on A2 to see the homework set and type your answers directly in the system. The option to get the pdf version is just there in case you want to look at the problems away from a computer. Everything should be fairly explicit over there. The system will close the assignment sometime early in the morning on Monday, so make sure you finish your session before this.

If you have trouble typing the math in the system, check out the FAQ and add to it if necessary.

If you have trouble with doing the math, you can help each other on the Math Forum page. Ask questions and give hints as much as possible, not full answers please.

Second part

This part of the assignment is drawn directly from the course texts. It focuses on mathematical exposition; you will be graded primarily on the clarity and elegance of your solutions.

From the Calculus: Early Transcendentals text, complete questions 55 from section 1.1 and 60 from section 1.2.

Please have a look at the following guidelines when writing your homework.

For practical reasons, I will collect these two problems at the end of the class on Monday; this doesn't mean class time is there for you to finish your homework, it is your responsibility to finish your work in time.

Third part

The third part consists of the group project. On paper, describe the result of your research during the week. This should contain:

• The questions that you studied,
• The main results that you found and proved,
• The main conjectures that you believe in and some arguments for them,
• The proofs of your results,
• The list of all sources used in the project.

The project is due by in class at 8am on Monday October 4. This project will be marked on the quality of all the above points, both on the mathematical content and the presentation.

## Homework 1 - due Monday September 20

The first homework assignment consists of three parts.

First part

WeBWorK problem set. We'll be using this online homework tool throughout the year. Go on the website, click on this section's link and enter your CWL login. Everything should be fairly explicit over there. The system will close the assignment sometime early in the morning on Monday, so make sure you finish your session before this.

Second part

This part of the assignment is drawn directly from the course texts. It focuses on mathematical exposition; you will be graded primarily on the clarity and elegance of your solutions.

From the Just-In-Time text, complete the following questions: section 3.3, question 30; and section 4.2, question 18.

For practical reasons, I will collect these two problems at the end of the class on Monday; this doesn't mean class time is there for you to finish your homework, it is your responsibility to finish your work in time.

Third part

Log into this wiki, then click on your username that will appear at the top right of the window. This opens your profile page on the wiki. If it's your first time, the page will not exist. In this case, you'll see a tab at the top (the third from the left I think) saying create. Hit this and in the new page simply write a few words about yourself, what you are studying and anything else you feel like. Then hit the save page button at the bottom.

You should then be able to contemplate your first new page on the wiki! Feels a bit empty isn't it? Well, let's have you add more to your user page then. The third part of your assignment is to write a short essay of at least 150 words (I'm counting the words on the actual page, not the code that you type when in edit mode) on one of the following topics:

I expect you to do a little bit of research, not only feed me paraphrasing of the wikipedia articles. I hope it is clear to everyone that I will not tolerate academic misconduct. You are free to inspire yourself from what you find in any resource of your choice, but your essay has to be yours and any citation or reference must be appropriately mentioned.

As you'll easily see, there is a very convenient history tab on each wiki page to track back previous versions of the page. It is useful if you delete content by mistake, if someone vandalizes your page (very unlikely and I would add that it is customary on wikis not to edit someone else's user page). The history tab is also very convenient for me and my marker to make sure we'll only be reading the last version of your page before Monday, September 20, 8 am. Any edits past this time will not be considered.

Final step, go to the Class List page and add your name there, otherwise, I'll never find your page and won't be able to mark it!

Here's an online word count tool.

If you have a question about this homework, send me an email, or add a comment in the discussion tab of this page. Make sure you also visit the help page of this wiki if you need help formatting, adding pictures, links or anything of the like.