Homework assignments will be posted on this page as they become available. There will be two components to the graded homework for this course - daily
problem sets and weekly written homework assignments. More details on these assignments may be found below. Extra practice problems can be accessed through Dartmouth's WeBWorK system, the Khan Academy website, and by choosing related problems in the textbook.
When turning in an assignment, include your name, write your solutions clearly and completely, staple together multiple-page assignments, and visibly mark the different problems.
Remember that complete problem solutions incorporate both the final answer as well as all the steps that lead to it. We reserve the right to reject any papers that are illegible.
Homework announcements will be posted below:
Solutions to the fourth quiz have been posted.
All homework assignments through Daily Homework #24 (due 11/9) have been posted.
Write the first 5 terms of the sequence: \( a_n=7-3\cdot n \). Is this sequence increasing? decreasing? bounded?
Write the first 5 terms of the sequence: \( \{\frac{(-1)^n}{n}\}_{n=1}^\infty \). Is this sequence increasing? decreasing? bounded?
Write the first 5 terms of the sequence: \( a_n=a_{n-1}+a_{n-2}, \quad a_0=1 \quad a_1=2 \). Is this sequence increasing? decreasing? bounded?
9/22 (x-hour)
Solve these problems:
Consider the sequence that starts at two where each successive term is equal to three less than four times
the previous term.
Write the next 5 terms of this sequence.
Write a formula for this sequence.
Create your own sequence that is not increasing, decreasing, or bounded.
Write the first 5 terms of your sequence.
Write a formula for your sequence.
Explain why your sequence is not increasing, decreasing, or bounded.
Using the following three functions: \(f(x)=\sin(x)\quad g(x)=x^3-3x+9\quad h(x)=\dfrac{x^2-x-1}{x+3}\)
compute these function compositions (simplify where possible):
\(\qquad f\circ h\)
\(\qquad g\circ f\)
\(\qquad h\circ g\)
Look at the graph given in Stewart 1.1 #56 to answer the following questions:
What is the domain of the function?
What is the range of the function?
Is this function even, odd, or neither?
On what intervals is this function increasing?
On what intervals is this function decreasing?
What are the intercepts of this function?
Write an expression for the function that gives this graph. Hint:
Write it as a piecewise function with three parts.
Given any function defined on \(\mathbb{R}\) we can construct a sequence by plugging in
the positive integers. For example, using the function \(s(x)=x^3\) gets us the sequence \(\{1,8,27,64,\ldots\}\).
We can also plug in other sets of integers, like the even numbers or squares.
Use the function \(f(x)=|x|\cdot(-2)^x\) for the problems below:
Write a formula for the sequence given by \(\{f(n)\}_{n=1}^\infty\).
Write a formula for the sequence given by \(\{f(2n)\}_{n=1}^\infty\).
Write a formula for the sequence given by \(\{f(2n+1)\}_{n=1}^\infty\).
Write a new function \(g\) defined on \(\mathbb{R}\) so that \(\{g(2n+1)\}_{n=1}^\infty\) is an increasing sequence.
Consider the two functions \(f(x)=x+2\) and \(g(x)=2x-4\), where the domain of \(f\) is \([-3,1]\)
and the domain of \(g\) is \([-1,4]\).
Draw a graph with \(f\) and \(g\) drawn on the same set of axes
What is the range of \(f\)?
What is the range of \(g\)?
On a new set of axes, draw what you think the graph of the function \(f+g\) should look like. Write a paragraph
below explaining what you think it means to add two functions whose domains are not the same.
Every month your company makes \(m^2+7m-5\) dollars of revenue and spends \(m^2+5\) dollars in costs, where \(m\)
is the number of the month (January=1, February=2, etc.). Calculate the amount of profit your company makes each year.
Prove that the sequence \(\left\{\dfrac{1}{e^n}\right\}_{n=1}^\infty\) converges to 0.
Find the limit of the sequence \(\left\{\dfrac{n+1}{n}\right\}_{n=1}^\infty\), and prove that it converges.
Consider the sequence \(\{a_n\}_{n=1}^\infty=\{3^n\}_{n=1}^\infty\). Find a sequence \(\{b_n\}_{n=1}^\infty\) such that the product sequence
\(\{a_nb_n\}_{n=1}^\infty\) converges to \(0\). Find another sequence \(c_n\) such that the quotient sequence
\(\left\{\dfrac{a_n}{c_n}\right\}_{n=1}^\infty\) converges to \(1\).
Consider the function \(f(x)=\dfrac{x^2-1}{x-1}\) defined on \((-\infty,1)\cup(1,\infty)\). Compute the first
five terms of the sequences \(\{a_n\}_{n=1}^\infty=\{f(1+\frac1n)\}_{n=1}^\infty\) and \(\{b_n\}_{n=1}^\infty=\{f(1-\frac1n)\}_{n=1}^\infty\). What are
the limits of these two sequences?
Find \(\displaystyle{\lim_{x\to 1^+} \dfrac{\sqrt{x}-1}{x-1}}\) and \(\displaystyle{\lim_{x\to 1^-} \dfrac{\sqrt{x}-1}{x-1}}\). What can you conclude about \(\displaystyle{\lim_{x\to 1} \dfrac{\sqrt{x}-1}{x-1}}\)?
Wrtie down a function \(f\) (not just a graph!) with the following properties:
A vertical asymptote at \(x=0\).
A jump discontinuity at \(x=2\).
\(\displaystyle{\lim_{x\to 4} f(x) = 5}\).
Find \(\displaystyle{\lim_{x\to -2} \dfrac{\sqrt[3]{x+5}+6}{x^3+2x-5}}\) using limit laws. Write down which limit laws you use and when you use them.
Find functions \(f\) and \(g\) so that \(\displaystyle{\lim_{x\to 1}g(x) = 2}\) but \(\displaystyle{\lim_{x\to 1} f(g(x)) \neq f(2)}\). Hint: choose \(f\) to be a piecewise function.
A Tibetan monk leaves his monastery at 7:00 AM and walks until he arrives at the top of The Mountain at 7:00 PM. After spending the night,
he begins his descent at 7:00 AM the next morning, retracing his path back down to the monastery and arriving at 7:00 PM.
Explain why there is a point on the path that the monk will cross at exactly the same time both days.
Let \(f(x) = \begin{cases} \sqrt{-x} & x<0\\ x+1 & 0\leq x\leq 2\\ \frac{x}{x-2} & 2< x < \infty\end{cases}\). Identify any vertical asymptotes and points of discontinuity of \(f\).
Is \( h(x)=\begin{cases} 3x+4&x\leq 1\\ 9x-2&x>1\end{cases}\) differentiable at 1? Continuous at 1? Why or why not?
Create a function \(f\) with the following properties:
\(f\) is not differentiable at \(0\)
\(f'(2)=0\)
Suppose the value of your comapany's stock on day \(t\) is given by \(f(t)=(x-2)^2+4\).
Sketch the graph of \(f\).
How much is the stock worth initially?
Sketch the graph of \(f'(x)\)
Is the stock price increasing or decreasing at \(t=1\) day?
Is the stock price increasing or decreasing at \(t=4\) days?
Is the stock price changing faster at \(t=1\) or \(t=4\) days?
Is the stock price ever not changing? When?
Exam Review
10/28
Stewart 2.3 #23
Stewart 2.3 #25
Stewart 2.3 #31
Stewart 2.3 #39
Stewart 2.3 #51
10/30
(Due Monday 11/2)
Stewart 2.4 #1
Stewart 2.4 #3
Stewart 2.4 #5
Stewart 2.4 #13
11/2
Stewart 6.4 #10
Stewart 6.4 #17
Stewart 6.4 #28
Stewart 6.4 #35
11/3 (x-hour)
Find the derivative of \(x^2e^x\ln(x)\).
Prove \(\frac{d}{dx}\sec(x) = \sec(x)\tan(x)\).
Hint: Use the quotient rule and the fact that \(\frac{d}{dx}\cos(x) = -\sin(x)\).
Differentiate \(5\ln(x) + \pi\tan(x) + x^2e^x\).
Differentiate \(\dfrac{x^3+3x^2-4x}{x^2-x}\).
Find the derivative of \(\dfrac{3\sec(x)\tan(x)}{e^x}\).
Let \(p(x)\) be a polynomial of degree \(n\). How many derivatives of \(p(x)\) do you need to take to get the constant 0 function? Hint: Try this with some examples first and see if you can find a pattern.
The plotting software for Newton's Method is at: this link
11/13
Section 2.9: #1
Section 2.9: #3
Section 2.9: #7 Verify the linear approximation, but you don't need to find the values of \(x\) that are accurate to within .1 of \(f(0)\).
Section 2.9: #11
Section 2.9: #17
The plotting software for linear approximation is at: this link
11/16
Compute the fourth order Taylor Polynomials for the functions below:
\(f(x)=e^x \qquad\qquad\qquad a=0\)
\(f(x)=\cos(x) \qquad\qquad\qquad a=\pi\)
\(f(x)=x^6+4x^4+3x^2-2x \qquad a=0\qquad{}\)
\(f(x)=\sqrt{x} \qquad\qquad\qquad a=1\)
The plotting software for Taylor Polynomials is at: this link
Write two questions that you think might be on the final exam.
11/17 (x-hour)
Use Newton's Method to find a positive zero of \(f(x)=2\cos(3x)-x^2\), accurate to at least 4 decimal places.
Let \(g(x)=4^x+x^4\). Compute the linearization of \(g\) at \(a=4\) and use it to estimate \(g\) at \(3.9\) and \(4.1\). Whichof these estimates is more accurate? Why?
Compute \(\Delta y\) and \(dy\) for \(g\) at \(a=\frac12\) for \(\Delta x=dx=\pm.1\).
Compute the 5th order Taylor polynomial for \(\ln(x^2)\) at \(a=1\).
Compute the derivative of \(\arcsin(\pi^x)\).
Compute the derivative of \( \dfrac{x^3\cot(x)}{e^{2x}}\).
Compute the \(y'\) for \(y+xy^2=\cos(x^2y)\).
Final Exam Review
Daily Problem Sets
These homework problems will be assigned after almost every lecture (on Mondays, Wednesdays, and Fridays - not after x-hours) and they are due at the beginning of the next lecture period. For example, daily homework that
is assigned on Monday is due at the beginning of class on Wednesday.
The purpose of these problems is to encourage you to practice applying the concepts that have been presented in lecture. Accordingly, each problem set
will contain approximately 10 problems and they should encompass a variety of types of questions. However, it is unlikely that these problems will be enough to master the material. We strongly encourage you to find similar problems in the book (such as the odd-numbered problems at the end of each section) to help solidify your understanding.
The daily assignments also provide an opportunity to receive feedback from your instructor regarding the clarity of your solutions and apparent understanding of the material. There will be 28 total daily assignments, each worth 2 points, and we will drop the three lowest scores at the end of the term. Each assignment should take about an hour to complete. Please inform your instructor if you are finding that this is not the case. Late homework will not be accepted.
Written Homework
Longer written assignments will be assigned on Wednesdays and they will be due the following Tuesday during the x-hour period. Homework solutions should be written legibly in pen or pencil or typed.
These weekly assignments will be based on the material that is covered during the week that they are assigned, covering the concepts at a deeper level than the daily homework. This may be done by solving more difficult or conceptual questions, but we may also ask open-ended questions. You are encouraged to work together on these problems
although each student must write up solutions in their own words.
There will be a total of 9 written homework assignments, each worth 15 points, and we will drop the lowest score at the end of the term.