Week 5: June 9 $^$th - June 15 $^$th

Suggested Problems

Problems you may find instructive, or that I find interesting.

4.10

#18, 23, 29 & 39

(These problems were suggested last week.)

4.11

#5, 8, 17, 20, 25, 27 & 31

(Questions #27 & 31 may help your understanding of 3.1)

Note: question #25 asks you to find $\epsilon_{rr} > 0$ so that,

$$x \in\left(x_0-\epsilon_{rr},x_0+\epsilon_{rr}\right) \Rightarrow -0.1 < \frac{dA-\Delta A}{A} < 0.1,$$

where $x_0$ is the (unknown) true diameter. This type of calculation is common in physics.

4.12

#5, 7, 9, 13 & 18

Reminder: $\frac{d}{dx} \ln(x) = \frac{1}{x}$ and $\frac{d}{dx}e^x = e^x$

11.5

#7, 9, 17, 25, 29, 37, 45, 47 & 57

(Please do not attempt #8, $30\rightarrow 33$ , 36 & $49\rightarrow 54$ )

11.6

#8, 16, 19, 20, 23, 25, 43, 53, 59 & 65

(Please do not attempt #13, 29, 46, 54, 61 & 62)

4.1

#4, 9, 12, 17, 22, 29, 30, 35, 42 & 45

(Hint: for #17 use 3.1 to find the derivative.)

(Question #42 shows up again in Chapter 12; #45 is used to prove L'Hôpital's Rule.)

4.2

#3, 9, 17, 35, 43, 45, 53, 57, 59 & 60

(Hint: for #45 use MVT; for #59 use induction)

4.3

#9, 19, 23, 27, 35, 36, 43 & 51

(You do not need to graph #51.)

Assigned Problems

Due June 16 $^$th , in lecture.

1. Let $f(x) = 1 - \frac{1}{x}$ . Prove that Newton's Method fails to find the solution if $x_1 < 0$ or $x_1 > 2$ .

   Hint: If $x_n<0$ , show that $x_{n+1} < x_n$ . For $x_1 > 2$ , run the first iterate.

2. Show that if $\alpha>0$ then,
   $$\lim_{\theta\to \infty}\theta\left(\alpha^\frac{1}{\theta}-1\right) = \ln(\alpha).$$

   This is 11.5 #44.

3. Prove that $\sqrt{x+5} < \frac{x}{10}+3$ for all $x>20$ .