Math 30G – Calculus – Sample Exam Fall 2012

Math 30G – Calculus – Sample Exam
Fall 2012
The Harvey Mudd College Honor Code applies to this exam. That means
it is a violation of the Honor Code to make this exam accessible in any
format to any person not currently enrolled in this class this semester.
1
20
2
20
3
10
4
10
5
10
Total
70
1
Problem 1 (2 pts each). True or False (with no explanations needed, no partial credit).
(a)
If f and g are differentiable, then
d
[f (x)g(x)] = f 0 (x)g 0 (x).
dx
(b)
If f and g are differentiable, then
d
[f (g(x))] = f 0 (g(x))g 0 (x).
dx
(c)
There exists a function f : R → R such that f is continuous nowhere but |f | is
continuous everywhere.
(d)
If a function f is continuous on [0, ∞), then it has an absolute minimum on this
interval.
(e)
The slope of the tangent line at a point (a, f (a)) of the graph of a differentiable
function f is equal to f 01(a) .
(f)
The sum of the series
∞
X
22k
k=0
Suppose we know that
∞
X
5k
is
1
.
5
ak converges to 0.8. We are given no other information about
k=1
the infinite series. For the remaining parts, determine if the following statements are
• true (i.e., must be true),
• or false (i.e., must be false).
(g)
(h)
(i)
(j)
lim ak = 0.8.
k→∞
lim ak = 0.
k→∞
lim
k→∞
ak+1
> 1.
ak
lim Sn = 0.8, where Sn = a1 + a2 + · · · + an .
n→∞
2
Problem 2 (5 pts each, 20 pts total). Short answer.
(a) The function
(
if x < 1
x + b if x ≥ 1
f (x) =
is continuous for every x if b =
x2 −4
x−2
.
(b) Assume we have two differentiable functions f and g. We are given that f 0 (x) = g 0 (x),
f (1) = 0, and g(1) = 5. Are f and g related? Can you find an explicit relationship
between f and g?
√
3
1+h−1
by relating it to the value of a derivative. (This means you should
h→0
h
not use l’Hˆopital’s rule here.)
(c) Find lim
1
= 0 means the following (fill in blanks appropriately):
n→∞ n2
(d) To say that lim
For every
we have
, there is an index N such that for
,
< .
To satisfy the above definition, if =
1
,
100
then the smallest integer N can be is
3
.
Problem 3 (10 pts).
(a) Suppose f (x) is a differentiable function. Show that at any x,
f (x + h) − f (x − h)
= f 0 (x).
h→0
2h
lim
Hint: You may find it helpful to add and subtract a value to the numerator in the limit.
(b) Use the limit given in part (a) to find f 0 (x) for the function f (x) = x2 .
4
Problem 4 (10 pts). Let f be a function having derivatives of all orders for all real numbers.
The third-degree Taylor polynomial for f at the point a = −2 is given by
3
1
T3 (x) = 2 − (x + 2)2 − (x + 2)3 .
8
12
(a) Find f (−2), f 0 (−2), and f 00 (−2).
(b) Determine whether f has a local minimum, a local maximum, or neither at the point −2.
Justify your answer.
(c) Use T3 (x) to find an approximation for f (−1).
(d) Find a best linear approximation to the function f at a = −2.
5
Problem 5 (10 pts). Use the mean value theorem to prove that
| sin y − sin z| ≤ |y − z| for all y, z ∈ R.
6