University of California, Riverside
Department of Mathematics
Midterm Exam
Mathematics 9B - First Year of Calculus
Sample 2 (Rev. 090119)
Instructions: This exam has a total of 120 points. You have 50 minutes.
You must show all your work to receive full credit. You may use any result
done in class. The points attached to each problem are indicated beside the
problem. You are not allowed books, notes, or calculators. Answers should
be written as
√
2 as opposed to 1.4142135....
1. This problem has three parts:
(a) (8 points) State both parts of the fundamental theorem of calculus.
(b) (8 points) Compute d
dx
∫ cos(x)
2
sin tdt
(c) (8 points) Evaluate
∫ pi/4
0
sec2 xdx
2. Evaluate
(a) (12 points) ∫ 2
1
(
2t +
3
t2
)(
4t2 − 5
t
)
dt
(b) (12 points) ∫ 2
0
(
x3 + x
)√
x4 + 2x2 + 4dx
3. Microtutorial (24 points) The population density of a plant species
is f(x) individual per square meter, where x is the distance from the
river, with f(x) ≥ 0 for x ≤ 100 and f(x) = 0 for x ≥ 100. Construct
a definite integral to calculate the number of plants along a section of
the river of length L.
1
4. (24 points) Evaluate the integral:∫
e−2x sin (2x) dx
5. (24 points) Evaluate the integral:∫
tan4 x dx