Problem Points Score
M161, Test 1, Spring 2008
0
1
1
14
2ab
20
2cb
20
3
15
4
15
5
15
∑
100
Please circle where you took M160 (or equivalent):
I CSU
AP
Community College
other College
You get 1 point for this indication.
Name:
Section:
Instructor:
You may not use calculators on this exam
cos2 θ =
1+cos 2θ
2
sin2 θ =
1−cos 2θ
2
Theorem (The Derivative Rule for Inverses) If f has an interval I as domain and f 0 (x) exists and
is never zero on I, then f −1 is differentiable at every point in its domain. The value of ( f −1 )0
at a point b in the domain of f −1 is the reciprocal of the value of f 0 at the point a = f −1 (b):
1
.
( f −1 )0 (b) = 0 −1
f ( f (b))
1)
a)Determine tan(acos(
√
2
2 )).
b) Determine all real numbers x for which e2x − ex − 6 = 0. (Hint: Set y = ex and rewrite the
equation in y.)
2) Z Evaluate the following integrals. Show your work.
8xe9x
a)
Z
b)
2 +4
dx
1
dx
(1 + x)2
Z
c)
Z
d)
1
(1 + 7 log x)1/3 dx
x
1
x2 + 2x + 17
dx
3)
Solve the separable differential equation
du
= e6u+3t
dt
for the initial condition u(0) = 15.
4) Let f be a continuous, differentiable one-to-one function on an interval I ⊂ R with range R.
Assume that f 0 (x) 6= 0 for all x ∈ I.
Show that the inverse f −1 of f is one-to-one on R. Your answer should consist of complete sentences.
Hint: What can you tell about f 0 and ( f −1 )0 ?
5) For each of the following statements indicate whether it is true or false. (You do not need to
give a proof.)
Each correct anwer is worth 3 points, each incorrect answer is counted as −2 points. (Unanswered
questions are 0 points.) You cannot get less than 0 points in this problem.
True
a)
If x > 1 and x log(x) = 5 log(5) then x = 5.
b)
The function f (x) = Z
sin(x) is one-to-one on the interval [− 23 π, 0].
c)
The function f (x) =
x
log(t)dt is one-to-one on the interval [2, 10].
5
d)
The function y(x) = x log(x) − x is a solution to the differential equation
xy0 (x) − x − y(x) = 0
e)
For every real number x we have that asin(sin(x)) = x.
False