AMT 2016
Calculus Test
August 20-21, 2016
Time limit: 50 minutes.
Instructions: This test contains 10 short answer questions. All answers must be expressed in
simplest form unless specified otherwise. Only answers written inside the boxes on the answer
sheet will be considered for grading.
No calculators.
1. Let f (x) = (x − 1)3 . Find f 0 (0).
2. Suppose a and b are two variables that satisfy
R2
0
(−ax2 + b) dx = 0. What is ab ?
3. If f (x) = ex g(x), where g(2) = 1 and g 0 (2) = 2, find f 0 (2).
4. The radius r of a circle is increasing at a rate of 2 meters per minute. Find the rate of
change, in meters2 / minute, of the area when r is 6 meters.
5. Find
lim
x→0
sin(x) − x
.
x cos(x) − x
6. For what positive value k does the equation ln x = kx2 have exactly one solution?
7. Compute
Z
0
π
2
ex (sin x + cos x − 2)
dx.
(cos x − 2)2
8. Let f be a differentiable function such that f 0 (0) = 4 and f (0) = 3. Compute
!x
f ( x1 )
.
lim
x→∞
f (0)
9. Compute
∞
Z
0
10. Using the fact that
(1
1+x11
1+x3
+ x2 ) ln x
ln
dx.
∞
X
1
π2
=
,
n2
6
n=1
compute
Z
1
(ln x) ln(1 − x) dx.
0