Duration: 100 minutes
Tuesday February 3, 2026
MAT187 Calculus II
Term Test 1
Full Name:
Student number:
Email :
@mail.utoronto.ca
Signature:
Instructions:
1. This test contains a total of 12 pages and a total of 40 marks.
2. DO NOT DETACH ANY PAGES FROM THIS TEST.
3. There are no aids permitted for this test, including calculators.
4. Cellphones, smartwatches, or any other electronic devices are not permitted. They must be turned off and in your
bag under your desk or chair. These devices may not be left in your pockets.
5. Write clearly and concisely in a linear fashion. Organize your work in a reasonably neat and coherent way.
6. Show your work and justify your steps on every question unless otherwise indicated. A correct answer without
explanation will receive no credit unless otherwise noted; an incorrect answer supported by substantially correct
calculations and explanations may receive partial credit.
7. For questions with a boxed area, ensure your answer is completely inside the box.
8. The back side of pages will not be scanned nor graded. Use the back side of pages for rough work only.
9. You must use the methods learned in this course to solve all of the problems.
10. Multiple choice problems containing ⃝ indicate that you are to select exactly one answer.
11. Multiple choice problems containing □ indicate that you may select more than one answer.
12. DO NOT START the test until instructed to do so.
GOOD LUCK!
Multiple Choice: No justification is required. Only your final answer will be graded.
Z
1. Consider the integral
x
p
x2 + 1 dx. The substitution u =
√
x2 + 1 yields which of the following integrals? [1 mark]
Indicate your final answer by filling in exactly one circle below (unfilled
Z
⃝
filled
).
p
u u2 − 1 du
Z
⃝
u du
Z
⃝
u2 du
Z
⃝
2u du
Z
2. Consider the integral
x sec2 x dx. Applying integration by parts yields which of the following expressions? [1 mark]
Indicate your final answer by filling in exactly one circle below (unfilled
filled
).
Z
x2
sec2 x − tan x dx + C
2
Z
x2
⃝
tan x − sec2 x dx + C
2
Z
⃝ x tan x − sec x tan x dx + C
⃝
Z
⃝ x tan x −
tan x dx + C
3. Which of the following expressions is the correct partial fraction decomposition for the rational function
[1 mark]
Indicate your final answer by filling in exactly one circle below (unfilled
⃝
B
Cx + D
A
+
+ 2
x−1 x+1
x +1
⃝
A
B
Cx
Dx
Ex + F
+
+
+
+ 2
2
2
x − 1 x + 1 (x − 1)
(x + 1)
x +1
⃝
A
B
C
D
Ex + F
+
+
+
+ 2
2
2
x − 1 x + 1 (x − 1)
(x + 1)
x +1
⃝
A
B
Cx + D
+
+ 2
2
2
(x − 1)
(x + 1)
x +1
filled
).
x3 + x 2 + x + 2
?
(x2 − 1)2 (x2 + 1)
Multiple Choice: No justification is required. Only your final answer will be graded.
Z
4. Suppose we have an integral
f (x) dx, and after making the substitution x = 2 sec(θ), we arrive at
Z
f (x) dx = 3θ + sin 2θ + C.
Z
Thus,
f (x) dx is given by [1 mark]
Indicate your final answer by filling in exactly one circle below (unfilled
filled
).
x
+ sin(sec−1 x) + C
2
√
x2 − 4
−1 x
+C
⃝ 3 sec
+4
2
x2
√
4 − x2
−1 x
⃝ 3 sec
+4
+C
2
x2
√
x2 − 4
−1 x
⃝ 3 sec
+2
+C
2
x
⃝ 3 sec−1
Z 2
′′
5. Suppose f (x) is a continuous function on (−∞, ∞). Using that
Z 2
(x2 − 4)f ′′ (x) dx is given by [2 mark]
f (x) dx = 10 and f (−2) = f (2) = 4, then
−2
−2
Indicate your final answer by filling in exactly one circle below (unfilled
filled
).
⃝ -12
⃝ -6
⃝ 0
⃝ 6
⃝ 12
6. Suppose f (x) is decreasing and concave up on the interval [a, b]. Based on this information, which of the following
Z b
numerical methods will underestimate
f (x) dx? [2 marks]
a
Indicate all correct answer(s) by filling in the corresponding square(s) below (unfilled □ filled ■).
□ Ln - a left-endpoint Riemann sum with n subintervals.
□ Rn - a right-endpoint Riemann sum with n subintervals.
□ Mn - a midpoint Riemann sum with n subintervals.
□ Tn - the trapezoidal rule with n subintervals
□ Sn - the composite Simpson’s rule with over n subintervals
Multiple Choice: No justification is required. Only your final answer will be graded.
Z 7
7. Suppose we want to approximate
P (x) dx, where P (x) is a cubic polynomial. Which of the following numerical
1
approximations will provide the smallest absolute error? [2 mark]
Indicate your final answer by filling in exactly one circle below (unfilled
⃝
99
X
P
6
6
1+k
100 100
P
6
6
1+k
200 200
P
1 6
6
1 + (k − )
2 50 50
k=0
⃝
200
X
k=1
⃝
50
X
k=1
filled
).
⃝ P (1) + 4P (4) + P (7)
Z 4
8. Which one of the following yields a trapezoidal rule approximation of
f (x) dx? [2 marks]
0
Indicate all correct answer(s) by filling in the corresponding square(s) below (unfilled □ filled ■).
□ 2(f (0) + f (4))
1
1
□ 2 f (0) + f (2) + 2 f (4)
□ f (0) + 2f (2) + f (4)
1
1
□ 2 f (0) + f (1) + f (2) + f (3) + 2 f (4)
□ f (0) + 2f (1) + 2f (2) + 2f (3) + f (4)
Z ∞
9. For what value(s) of p does
0
1
dx converge? [2 mark]
xp
Indicate your final answer by filling in exactly one circle below (unfilled
⃝ p<0
⃝ 0<p<1
⃝ p=1
⃝ p>1
⃝ No such value exist
filled
).
Multiple Choice: No justification is required. Only your final answer will be graded.
Z 2
10. Which of the following shaded regions yields the Simpson’s rule approximation, S4 , of
f (x) dx, where f (x) is
0
given by the dashed line in each plot? [1 mark]
Indicate your final answer by filling in exactly one circle below (unfilled
⃝
⃝
⃝
√
1 − x2 and y =
1
. [2 mark]
2
Indicate your final answer by filling in exactly one circle below (unfilled
⃝
1 π
−
2
6
).
⃝
11. Evaluate the area of the region enclosed by y =
√
π
3
⃝
−
3
4
√
3 π
⃝
−
4
3
√
3 π
−
⃝
2
6
π
⃝
6
filled
filled
).
Short Answer: Unsupported answers will not receive full credit. Organize your work in a reasonably neat and coherent
way. Put your final answer in the boxes provided.
Z a
f (x) dx converges.
12(a) The graphs of f (x), and g(x) on (0, ∞) are pictured below. Suppose first that
0
y
f (x)
g(x)
a
x
For each part below, determine if the given improper integral converges or diverges, or if there is not enough information
to tell.
Z a
(i)
g(x) +
0
1
x
dx. [2 marks]
⃝ converges
Based on the clearly written justification above, the improper integral ⃝ diverges
⃝ could converge or diverge, we cannot tell.
Z a
(ii)
0
f (x)
dx. [2 marks]
1 + g(x)
⃝ converges
Based on the clearly written justification above, the improper integral ⃝ diverges
⃝ could converge or diverge, we cannot tell.
Short Answer: Unsupported answers will not receive full credit. Organize your work in a reasonably neat and coherent
way. Put your final answer in the boxes provided.
Z ∞
f (x) dx converges.
12(b) The graphs of f (x), and g(x) on (0, ∞) are pictured below. Now suppose only that
a
y
f (x)
g(x)
a
x
For each part below, determine if the given improper integral converges or diverges, or if there is not enough information
to tell.
Z ∞
(i)
f (x2 ) dx. [2 marks]
a
⃝ converges
Based on the clearly written justification above, the improper integral ⃝ diverges
⃝ could converge or diverge, we cannot tell.
Z ∞
(ii)
x2 f (x) dx. [2 marks]
a
⃝ converges
Based on the clearly written justification above, the improper integral ⃝ diverges
⃝ could converge or diverge, we cannot tell.
Short Answer: Unsupported answers will not receive full credit. Organize your work in a reasonably neat and coherent
way. Put your final answer in the boxes provided.
Z 2
13. In this problem we consider approximations of the definite integral
xex dx.
0
(a) Suppose we approximate this integral using a right-endpoint Riemann sum, Rn . Determine if our approximation
Z 2
will overestimate or underestimate
xex dx. [3 marks]
0
Based on the clearly written justification above, Rn will
⃝ overestimate
⃝ underestimate
(b) Suppose now we approximate this integral using a midpoint Riemann sum, Mn . Determine if our approximation
Z 2
will overestimate or underestimate
xex dx. [3 marks]
0
Based on the clearly written justification above, Mn will
⃝ overestimate
⃝ underestimate
Short Answer: Unsupported answers will not receive full credit. Organize your work in a reasonably neat and coherent
way. Put your final answer in the boxes provided.
Z 2
13. (continued) In this problem we consider approximations of the definite integral
xex dx.
0
(c) Suppose instead we want to approximate our integral using Simpson’s rule, S2 . Find an upper bound for the absolute
Z 2
error between
xex dx and S2 . [3 marks]
0
Hint: Use the fact that 2 < e < 3.
Z 2
Based on the clearly written justification above,
xex dx − S2 ≤
0
(d) The exact error for Simpson’s rule is given by the equation
Z 2
0
xex dx − Sn = −
f ′′′′ (z)(b − a)5
,
180n4
where z ∈ [0, 2] is some unknown number. Use this equation to determine if S2 will overestimate or underestimate
Z 2
xex dx. [2 marks]
0
⃝ overestimate the integral
Based on the clearly written justification above, S2 will ⃝ underestimate the integral
⃝ possibly overestimate or underestimate, we cannot tell
Short Answer: Unsupported answers will not receive full credit. Organize your work in a reasonably neat and coherent
way. Put your final answer in the boxes provided.
14. Given two particles of equal (and constant) charge q > 0, Coulomb’s law states that the electrostatic force experienced
by one of the particles is given by
q2
F = Ke 2 ,
r
where Ke is a positive constant, and r denotes the distance between the centres of each particle.
Suppose an external agent reduces the distance between the particles from r0 to r1 . The work required for this action is
given by the definite integral
Z r1
q2
W =−
Ke 2 dr.
r
r0
(a) Suppose the particles are initially r0 > 0 units apart. Determine the amount of work required for an external agent
to bring the particles to zero distance apart. If finite, compute the amount of work required, otherwise, justify why
the integral diverges. [2 marks]
Based on the clearly written justification above, the work required
⃝ converges
⃝ diverges
(b) Suppose the particles are initially r0 > 0 units apart. Determine the amount of work required for an external agent
to separate the particles to infinitely far apart. If finite, compute the amount of work required, otherwise, justify
why the integral diverges. [2 marks]
Based on the clearly written justification above, the work required
⃝ converges
⃝ diverges
IF NEEDED, USE THIS PAGE TO CONTINUE OTHER QUESTIONS.
If you wish to have this page marked, make sure to refer to it in your original solution.
IF NEEDED, USE THIS PAGE TO CONTINUE OTHER QUESTIONS.
If you wish to have this page marked, make sure to refer to it in your original solution.