Homework 2, due June 29 at beginning of class.

  1. Comput the following limits:
    1. $\lim\limits_{x\rightarrow\infty}\frac{e^{2x}}{e^{2x+5}+1}$
    2. $\lim\limits_{x\rightarrow \infty} \frac{x^{3}+2}{x^{2}+1}$
    3. $\lim\limits_{x\rightarrow\infty} \frac{2x^{10}+5x+3}{x^{2}+x^{10}+15}$
    4. $\lim\limits_{x\rightarrow\infty} (\sqrt{2x+1}-\sqrt{2x-1})$ ( Hint: you will need to multiply and divide by something, and remember $(a-b)(a+b)=a^2-b^2$)
    5. $\lim\limits_{x\rightarrow\infty}\frac{x}{\sqrt{x^{2}+1}}$

  2. Compute the following limits (these all use the squeeze/sandwich theorem):
    1. $\lim\limits_{x\rightarrow\infty} e^{-x}(\cos x+\sin x)$
    2. $\lim\limits_{x\rightarrow 0}\frac{x}{1+\sin x}$
    3. $\lim\limits_{x\rightarrow \infty} \frac{1}{x+\cos x}$
    4. $\lim\limits_{x\rightarrow \infty}2^{-x+\cos x}$
    5. $\lim\limits_{x\rightarrow\infty} \frac{\cos(\sin(\sqrt{x}))}{x}$

  3. Intermediate Value Theorem:
    1. Without solving, show that $x^{3}+5x-2$ has a root in $[-1,1]$.
    2. Show that for some $x\in [0,1]$, $\cos x=x$ (Hint: Consider $f(x)=\cos x-x$).

  4. Compute the derivatives of the following function using the definition of a derivative (you can use any limits we've covered so far, e.g. $\lim\limits_{h\rightarrow 0}\frac{\sin h}{h}=1$)
    1. $10$
    2. $x^{3}$
    3. $x^{2}+2x+2$
    4. $\frac{1}{\sqrt{x}}$
    5. $\frac{1}{\sin x}$ ( Hint: At some point, you will require the formula $\sin(a)\cos(b)+\sin(b)\cos(a)=\sin(a+b)$)

  5. Find the derivative at the given point:
    1. $f(x)=3x+2$, $x=5$
    2. $f(x)=5x^{2}+2x$, $x=1$
    3. $f(x)=-2x^{2}+8$, $x=2$
    4. $f(x)=\frac{1}{x}$, $x=1$
  6. Let \[f(x)=\left\{ \begin{array}{cc} x^{2}\sin\frac{1}{x} & x\neq 0 \\ 0 & x=0\end{array}\right\}.\] Compute the derivative of $f(x)$ at $x=0$ (Hint: when computing the derivative using the definition of a derivative, use the squeeze theorem to compute the limit).

  7. Bonus Riddle: Suppose it takes Jack all of Monday (starting at midnight) to climb a mountain (so by midnight of Tuesday, he is at the top). Suppose also that it takes all of Tuesday (starting at midnight) to return to the base of the mountain. Show that there is a time of day when Jack is at the same spot on Monday and Tuesday (for example, if the time were 3pm, this would mean that he would be at that spot at 3pm Monday when he's on his way up, and at that same spot at 3pm on Tuesday when he's on his way back down).