Math 120: Assignment 4 (Due Tue., Oct. 9 at start of class)
Suggested practice problems (from Adams, 6th ed.):
2.5:
2.6:
2.7:
2.8:
5,9,11,15,17,19,33,35,41,47,51,55,57
3,5,9,17,20
11,15,27,29
1,5,11,13,19,30,32
Problems to hand in:
1. Find the following:
(a)
(b)
d
dt cos(sec(t))
d
2
2
dx [csc (x) − cot (x)]
(and simplify it – what’s going on here?)
(c) the first, second, and third derivatives of f (x) = sec(x)
(d) the n-th derivative, f (n) (x), of f (x) = sin(kx) (k a constant)
(e) the n-th derivative of f (x) = (1 − 3x)−1/2
(f) limx→0
1−cos(x)−x2 /2
x2
(hint: use limt→0
sin(t)
t
= 1 and a trig identity)
2. Sketch the graph of y = cot(x). Show that it has no horizontal tangents. Find the
points where it has slope −2
3. Find the intervals of increase and decrease of f (x) = x3 − 4x + 1.
4. Let f (x) = x2/3 . Show there are no points c such that
this not contradict the mean-value theorem?
f (1)−f (−1)
1−(−1)
= f 0 (c). Why does
5. Suppose two shapes – a ball of radius r, and a cylinder with radius 1 and height h
– always have the same volume. The volume starts at 1, and then begins to change.
Which changes faster, r or h?
6. Suppose f 000 (x) exists on an interval I and f vanishes (equals 0) at 4 distinct points
in I. Show that f 000 must vanish at some point in I.
Oct. 1
1