Introductory Writing Assignment (link)

Solve these problems:

1. Stewart 1.1 #20


2. Stewart 1.1 #21


3. Stewart 1.1 #47


4. Stewart 1.1 #55


5. Stewart 1.3 #29


6. Stewart 1.3 #33


7. Stewart 1.3 #51


8. Write the first 5 terms of the sequence: \( a_n=7-3\cdot n \). Is this sequence increasing? decreasing? bounded?


9. Write the first 5 terms of the sequence: \( \{\frac{(-1)^n}{n}\}_{n=1}^\infty \). Is this sequence increasing? decreasing? bounded?


10. 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?

Solve these problems:

1. Consider the sequence that starts at two where each successive term is equal to three less than four times the previous term.
   1. Write the next 5 terms of this sequence.
   2. Write a formula for this sequence.


2. Create your own sequence that is not increasing, decreasing, or bounded.
   1. Write the first 5 terms of your sequence.
   2. Write a formula for your sequence.
   3. Explain why your sequence is not increasing, decreasing, or bounded.
   
   
3. 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):
   1. \(\qquad f\circ h\)
   2. \(\qquad g\circ f\)
   3. \(\qquad h\circ g\)


4. Look at the graph given in Stewart 1.1 #56 to answer the following questions:
   1. What is the domain of the function?
   2. What is the range of the function?
   3. Is this function even, odd, or neither?
   4. On what intervals is this function increasing?
   5. On what intervals is this function decreasing?
   6. What are the intercepts of this function?
   7. Write an expression for the function that gives this graph. Hint: Write it as a piecewise function with three parts.


5. 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:
   1. Write a formula for the sequence given by \(\{f(n)\}_{n=1}^\infty\).
   2. Write a formula for the sequence given by \(\{f(2n)\}_{n=1}^\infty\).
   3. Write a formula for the sequence given by \(\{f(2n+1)\}_{n=1}^\infty\).
   4. Write a new function \(g\) defined on \(\mathbb{R}\) so that \(\{g(2n+1)\}_{n=1}^\infty\) is an increasing sequence.


6. 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 \([-1,4]\).
   1. Draw a graph with \(f\) and \(g\) drawn on the same set of axes
   2. What is the range of \(f\)?
   3. What is the range of \(g\)?
   4. 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.


7. 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.

Solve these problems:

1. Stewart 1.2 #1


2. Stewart 1.2 #2


3. Stewart 1.2 #19


4. Stewart 1.2 #25

1. Stewart 1.2 #7


2. Stewart 1.2 #9


3. Stewart 1.2 #13


4. Stewart 1.4 #1


5. Stewart 1.4 #5


6. Stewart 1.4 #7


7. Use Lagrange interpolation to find a polynomial \(f\) so that \(f(1)=0, f(2) = 3, f(3)=8\).

1. Stewart 1.3 #1


2. Stewart 1.3 #3


3. Stewart 1.3 #5


4. Stewart 1.3 #39


5. Stewart 1.3 #41


6. Stewart 1.3 #45


7. Stewart 1.3 #49


8. Stewart 1.3 #59


9. Stewart 1.3 #61

Weekly Assignment #2 (link)

1. Stewart 6.1 #3


2. Stewart 6.1 #7


3. Stewart 6.1 #17


4. Stewart 6.1 #21


5. Stewart 6.2 #7


6. Stewart 6.2 #15


7. Stewart 6.2* #3


8. Stewart 6.2* #5


9. Stewart 6.3* #3


10. Stewart 6.3* #5

1. Stewart Appendix D #1


2. Stewart Appendix D #9


3. Stewart Appendix D #25


4. Stewart Appendix D #29


5. Stewart Appendix D #37


6. Stewart Appendix D #65


7. Stewart Appendix D #77


8. Stewart Appendix D #81

1. Stewart 6.6 #3


2. Stewart 6.6 #7


3. Stewart 6.6 #11


4. Stewart 6.6 #13


5. Stewart 6.6 #15

Weekly Assignment #3 (link)

Nothing Due!

Nothing Due!

Nothing Due!

1. For what values of \(x\) does the sequence \(\{0,x^2,0,x^4,0,x^6,\ldots\}\) converge?


2. Write a formula for each of the following sequences:
   1. \(\{(\sqrt{2}-\sqrt{3}),(\sqrt{3}-\sqrt{4}),(\sqrt{4}-\sqrt{5}), \ldots\}\)
   
   
   2. \(\{\frac12,\frac34,\frac56,\frac78, \ldots\} \)
   
   
   3. \(\{0,\frac14,\frac29,\frac{3}{16}, \ldots\} \)
   
   


3. Prove that the sequence \(\left\{\dfrac{1}{e^n}\right\}_{n=1}^\infty\) converges to 0.



4. Find the limit of the sequence \(\left\{\dfrac{n+1}{n}\right\}_{n=1}^\infty\), and prove that it converges.


5. 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 \(1\).


6. 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?

1. Stewart 1.5 #1


2. Stewart 1.5 #5


3. Stewart 1.5 #7


4. Stewart 1.5 #9


5. Stewart 1.5 #15


6. Stewart 1.5 #17

1. Stewart 1.6 #1


2. Stewart 1.6 #7


3. Stewart 1.6 #11


4. Stewart 1.6 #13


5. Stewart 1.6 #21


6. Stewart 1.6 #23


7. Stewart 1.6 #31


8. Stewart 1.6 #41

1. Stewart 1.8 #1


2. Stewart 1.8 #3


3. Stewart 1.8 #7


4. Stewart 1.8 #11


5. Stewart 1.8 #13


6. Stewart 1.8 #17


7. Stewart 1.8 #21


8. Stewart 1.8 #25


9. Stewart 1.8 #31


10. Stewart 1.8 #51

1. 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 \(\displaystyle{\lim_{x\to 1} \dfrac{\sqrt{x}-1}{x-1}}\)?


2. Wrtie down a function \(f\) (not just a graph!) with the following properties:
   • A vertical asymptote at \(x=0\).
   • A jump discontinuity at \(x=2\).
   • \(\displaystyle{\lim_{x\to 4} f(x) = 5}\).


3. 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.


4. 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.


5. 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.


6. 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 \(f\).

1. Stewart 1.6 #2


2. Stewart 1.8 #23


3. Stewart 1.8 #57

1. Stewart 2.1 #1


2. Stewart 2.1 #5


3. Stewart 2.1 #9(a, b)


4. Stewart 2.1 #11(a)


5. Stewart 2.1 #27

1. Stewart 2.2 #3


2. Stewart 2.2 #21


3. Stewart 2.2 #35

1. Stewart 2.3 #1


2. Stewart 2.3 #5


3. Stewart 2.3 #9


4. Stewart 2.3 #11


5. Stewart 2.3 #21


6. Stewart 2.3 #52

1. Find the derivative of \(f(x)=(x+1)^3-x^2+\pi\) using the definition of the derivative.


2. Find the tangent line to the graph of \(g(x)=x^{\frac{5}{6}}+x^2+4\) at \(x=1\).


3. Find the derivatives of each of the following (using any method):
   1. \(f(x)=4x^3+x^2+5x\)
   
   
   2. \(f(x)=\frac{2}{x^2}+\sqrt[3]{x}\)
   
   
   3. \( f(x)=\begin{cases} 3x^2+x&x\leq 3\\ \frac13x^3+\frac32x^2+x+\frac95&x>3\end{cases}\)
   
   
4. 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?


5. Create a function \(f\) with the following properties:
   • \(f\) is not differentiable at \(0\)
   • \(f'(2)=0\)


6. Suppose the value of your comapany's stock on day \(t\) is given by \(f(t)=(x-2)^2+4\).
   1. Sketch the graph of \(f\).
   
   
   2. How much is the stock worth initially?
   
   
   3. Sketch the graph of \(f'(x)\)
   
   
   4. Is the stock price increasing or decreasing at \(t=1\) day?
   
   
   5. Is the stock price increasing or decreasing at \(t=4\) days?
   
   
   6. Is the stock price changing faster at \(t=1\) or \(t=4\) days?
   
   
   7. Is the stock price ever not changing? When?
   
   

1. Stewart 2.3 #23


2. Stewart 2.3 #25


3. Stewart 2.3 #31


4. Stewart 2.3 #39


5. Stewart 2.3 #51

(Due Monday 11/2)

1. Stewart 2.4 #1


2. Stewart 2.4 #3


3. Stewart 2.4 #5


4. Stewart 2.4 #13

1. Stewart 6.4 #10


2. Stewart 6.4 #17


3. Stewart 6.4 #28


4. Stewart 6.4 #35

1. Find the derivative of \(x^2e^x\ln(x)\).


2. Prove \(\frac{d}{dx}\sec(x) = \sec(x)\tan(x)\).
   Hint: Use the quotient rule and the fact that \(\frac{d}{dx}\cos(x) = -\sin(x)\).


3. Differentiate \(5\ln(x) + \pi\tan(x) + x^2e^x\).


4. Differentiate \(\dfrac{x^3+3x^2-4x}{x^2-x}\).


5. Find the derivative of \(\dfrac{3\sec(x)\tan(x)}{e^x}\).


6. 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.

1. Stewart 2.5: #5


2. Stewart 2.5: #7


3. Stewart 2.5: #17


4. Stewart 2.5: #29


5. Stewart 2.5: #41


6. Stewart 2.5: #51

1. Stewart 2.6: #1


2. Stewart 2.6: #3


3. Stewart 2.6: #5


4. Stewart 2.6: #13


5. Stewart 6.6: #23


6. Stewart 6.6: #31

The plotting software for implicit equations is at: this link

1. Section 6.8: #1


2. Section 6.8: #3


3. Section 6.8: #7


4. Section 6.8: #9


5. Section 6.8: #13


6. Section 6.8: #23

1. Find the derivative of \(\cot(3x^2+5)\).


2. Find \(f'(x)\) for \(f(x) = \ln(5^{\cos(x)}+1)\).


3. Use implicit differentiation to find \(y'\) for the equation \(\cos(x+y) = x^2y\).


4. Find the equation of the tangent line to \(x^2 + 2xy+y^2 = 9\) at the point \((1,-4)\).


5. Find \(\displaystyle{\lim_{x\to 0} \sin(x)\ln(x)}\).


6. Find \(\displaystyle{\lim_{t\to 0} \dfrac{8^t-5^t}{t}}\).

1. Section 3.8: #5


2. Section 3.8: #7


3. Section 3.8: #11


4. Plot the equation in Section 3.8: #21


5. Section 3.8: #29

The plotting software for Newton's Method is at: this link

1. Section 2.9: #1


2. Section 2.9: #3


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 \(f(0)\).


4. Section 2.9: #11


5. Section 2.9: #17

The plotting software for linear approximation is at: this link

Compute the fourth order Taylor Polynomials for the functions below:

1. \(f(x)=e^x \qquad\qquad\qquad a=0\)


2. \(f(x)=\cos(x) \qquad\qquad\qquad a=\pi\)


3. \(f(x)=x^6+4x^4+3x^2-2x \qquad a=0\qquad{}\)


4. \(f(x)=\sqrt{x} \qquad\qquad\qquad a=1\)

The plotting software for Taylor Polynomials is at: this link

Write two questions that you think might be on the final exam.

1. Use Newton's Method to find a positive zero of \(f(x)=2\cos(3x)-x^2\), accurate to at least 4 decimal places.


2. 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 \(\Delta x=dx=\pm.1\).


3. Compute the 5th order Taylor polynomial for \(\ln(x^2)\) at \(a=1\).


4. Compute the derivative of \(\arcsin(\pi^x)\).


5. Compute the derivative of \( \dfrac{x^3\cot(x)}{e^{2x}}\).


6. Compute the \(y'\) for \(y+xy^2=\cos(x^2y)\).