Solutions to Homework 1

11.1

#3 North to south: Starting at 65 go up to 100, passing Boise at 72. West to east: starting at 65 go up to 75, back down to 65, reach Boise at 72, dip briefly back into the 60’s, then stay in the 70’s, next go briefly into the 80’s before ending at 78.

#10 [5]

a) g decreases as h increases, because the gravitational force gets weaker the farther away you are from the planet. It’s a decreasing function in h.

b) g increases as m increases because the more massive the planet is the stronger its gravitational pull will be. It’s an increasing function in m.

#20

a) The graphs all look like half a period of a sine function, starting at 0 for x = 0 going to A when x = π/2 and back to 0 when x = π.

(i) t = 0: A = 1 (function goes up then down) (ii) t = π/4: A = 1/\sqrt{2} (function goes up then down) (iii) t = π/2: A = 0 (function is constant 0) (iv) t = 3π/4: A = −1/\sqrt{2} (function goes down then up)

(v) t = π: A = −1 (function goes down then up)

#21 [5] They represent snapshots of the whole string (x varies from 0 to π) at times t = 0, t = 1.

#22 They represent the movement of the points x = 0 (endpoint=no movement) and x = 1 (point somewhere in the middle=up and down movement) as time proceeds.

11.2

#2 A is closest to the yz-plane, B is on the xz-plane and C is farthest from the xy-plane.

#4 [5] Your final point is (1,-1,1): In front of the yz-plane, to the left of the xz-plane, and above the xy-plane.

#12 [5] Label the points A,B,C. Distances: d(A, B) = \sqrt{8}, d(B, C) = \sqrt{6}, d(A, C) = \sqrt{10}. So B and C are closest.

#18 Come to my office and I’ll show it to you.