Week 1: May 12 $^$nd - May 18 $^$th

Suggested Problems

1.2

#28, 31, 32, 33, 71, 73, 74 & 76

1.4

#5, 7, 16, 33, 34, 43, 46 & 65

1.5

#17, 18, 28, 29, 42, 43, 52, 56, 62 & 76

1.7

#14, 27, 30, 34, 35, 39, 43 & 59

(Use #57 & 58 for #59)

1.6

#8, 10, 29, 30, 36, 38, 59, 61, 75, 82, 83, 84 & 87

1.3

#5, 13, 39, 41, 45, 46, 48, 55 &58

1.8

#5, 10, 11, 17 & 19

(#19 is hard)

Assigned Problems

Due May 19 $^$th , in lecture.

1. Prove that,
   $$\frac{1}{1-\sin\theta} + \frac{1}{1+\sin\theta} = 2\sec^2\theta.$$

2. Find the (shortest) distance between $(-1,3)$ and the line $y=4x-5$ .

3. The Fibonacci numbers are defined recursively:
   $$\begin{aligned} {\mathcal F}_1 = 1, &&{\mathcal F}_2 = 1, &&{... ..._{n-1}+{\mathcal F}_{n-2} \text{ for all } n\geq 3. \end{aligned}$$
   (The first few Fibonacci numbers are: $1, 1, 2, 3, 5, 8, 13, 21, \dots$ )

   Prove that for all $j\geq 1$ , ${\mathcal F}_{4j}$ is divisible by $3$ .