UT Austin M408D Credit by Exam Practice Test 1
July 16, 2025
Instructions:
1. This practice test consists of 14 questions. The first 11 questions are multiple-
choice, and the last 3 questions are free-response style (presented here with answer
choices for practice).
2. No calculators are allowed.
3. Choose the best answer for multiple-choice questions. For free-response style ques-
tions, imagine you would show all your work to receive full credit.
Multiple-Choice Questions
1. Evaluate the integral:
R
x2 ex dx
(a) x2 ex − 2xex + 2ex + C
(b) x2 ex − xex + ex + C
(c) x2 ex + 2xex + 2ex + C
(d) ex (x2 − 2x) + C
2. Evaluate the integral: sin3 x cos2 x dx
R
(a) 1
3
cos3 x − 15 cos5 x + C
(b) − 13 cos3 x + 15 cos5 x + C
(c) 1
3
sin3 x cos3 x + C
(d) sin4 x 13 cos3 x + C
1
4
3. Evaluate the integral: 1
R
x2 −1
dx
(a) ln |x2 − 1| + C
(b) 1
2
ln x−1
x+1
+C
(c) 1
2
ln x+1
x−1
+C
(d) arctan(x) + C
1
4. Evaluate the improper integral:
R∞ 1
1 x2
dx
(a) 0
(b) 1
(c) 2
(d) Diverges
5. Determine whether the series converges or diverges: 2 n
P∞
n=0 3
(a) Converges to 2
(b) Converges to 3
(c) Converges to 1
3
(d) Diverges
6. Determine whether the series converges or diverges:
P∞ n
n=1 n3 +1
(a) Converges by Comparison Test with a p-series (p > 1)
(b) Diverges by Comparison Test with a p-series (p ≤ 1)
(c) Diverges by Ratio Test
(d) Converges by Alternating Series Test
7. Find the interval of convergence for the power series: (x−3)n
P∞
n=1 n
(a) (2, 4)
(b) [2, 4)
(c) (2, 4]
(d) [2, 4]
8. Given vectors a = h1, −2, 3i and b = h4, 0, −1i, find the dot product a · b.
(a) h4, 0, −3i
(b) 1
(c) 7
(d) −3
9. Find the symmetric equations of the line passing through the point (2, −1, 5) and
parallel to the vector v = h3, 0, −2i.
(a) x−2
3
= y+1
0
=z−5
−2
(with the understanding that y+1
0
means y = −1)
(b) x+2
3
= y − 1 = z+5
−2
(c) x−3
2
= y
0
= 5z+2
(d) x = 2 + 3t, y = −1, z = 5 − 2t
10. Identify the surface represented by the equation x2 + y 2 + z 2 = 9.
(a) Ellipsoid
2
(b) Sphere
(c) Paraboloid
(d) Cone
11. Find the first partial derivatives of f (x, y, z) = x2 yz + ln(xy).
(a) ∂f
∂x
= 2xyz + x1 , ∂f
∂y
= x2 z + y1 , ∂f
∂z
= x2 y
(b) ∂f
∂x
= 2x + y + z + 1
, ∂f = x2 + z + xy
xy ∂y
1
, ∂f
∂z
= x2 y
(c) ∂f
∂x
= 2xyz, ∂f
∂y
2 ∂f 2
= x z, ∂z = x y + z 1
(d) ∂f
∂x
= 2xy + y
, ∂f = x2 + xy
xy ∂y
, ∂z = x2 y
x ∂f
Free-Response Style Questions
(These are presented with multiple-choice options for practice, but on an actual
free-response section, you would show all your steps.)
12. Find the directional derivative of f (x, y) = x2 y at the point (1, 2) in the direction
of the vector v = h3, 4i.
(a) 8
(b) 12
(c) 14
(d) 20
13. Set up the double integral to find the area of the region bounded by y = x2 and
y = x.
(a) 0 x2 dy dx
R1Rx
(b) 0 x dy dx
R 1 R x2
R 1 R √y
(c) 0 y dx dy
(d) 0 (x − x2 ) dx
R1
√
14. Convert the Cartesian point (1, 1, 2) to cylindrical coordinates (r, θ, z).
√
(a) (2, π/4, 2)
√ √
(b) ( 2, π/4, 2)
√ √
(c) ( 2, π/2, 2)
√
(d) (1, π/4, 2)
