UT Austin M408D Credit by Exam Practice Test
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
x cos(x2 ) dx
(a) 1 2
2
x sin(x2 ) + C
(b) sin(x2 ) + C
(c) 1
2
sin(x2 ) + C
(d) − 12 sin(x2 ) + C
2. Evaluate the integral: arcsin x dx
R
√
(a) x arcsin x − 1 − x2 + C
√
(b) x arcsin x + 1 − x2 + C
(c) √ 1
1−x2
+C
(d) x arcsin x + 12 ln |1 − x2 | + C
3. Evaluate the integral: tan2 x dx
R
(a) ln | sec x| + C
(b) tan x − x + C
(c) 1
3
tan3 x + C
(d) sec2 x − x + C
1
4. Determine whether the series converges or diverges:
P∞
√1
n=1 n
(a) Converges
(b) Diverges
(c) Converges conditionally
(d) Converges absolutely
5. Find the radius of convergence for the power series:
P∞ xn
n=0 n!
(a) R = 0
(b) R = 1
(c) R = ∞
(d) R = e
6. Determine whether the series converges, diverges, or converges conditionally: (−1)n+1
P∞
n=1 n
(a) Diverges
(b) Converges absolutely
(c) Converges conditionally
(d) Neither converges nor diverges
7. Which of the following is the Maclaurin series for sin x?
(a)
P∞ (−1)n x2n+1
n=0 (2n)!
(b)
P∞ (−1)n x2n+1
n=0 (2n+1)!
(c)
P∞ (−1)n x2n
n=0 (2n)!
(d)
P∞ x2n+1
n=0 (2n+1)!
8. Which of the following describes the line given by the parametric equations x =
1 + 3t, y = 2 − t, z = 5?
(a) A line parallel to the xy-plane.
(b) A line passing through the point (1, 2, 5) with direction vector h3, −1, 0i.
(c) A plane perpendicular to the z-axis.
(d) A line with direction vector h1, 2, 5i.
9. Find the equation of the plane passing through the point (1, 2, 1) and with normal
vector n = h1, 2, 3i.
(a) 2x − y + 3z = 1
(b) x + 2y + 3z = 8
(c) x + 2y + 3z = 0
(d) x − 2y + 3z = 0
10. Identify the surface represented by the equation x2 + y 2 − z 2 = −1.
2
(a) Elliptic Cone
(b) Hyperboloid of two sheets
(c) Ellipsoid
(d) Paraboloid
11. Find the first partial derivatives of f (x, y) = x2 y.
(a) ∂f
∂x
= 2xy, ∂f
∂y
= x2
(b) ∂f
∂x
= x2 , ∂f
∂y
= 2xy
(c) ∂f
∂x
= 2x + y, ∂f∂y
= x2
(d) ∂f
∂x
= y, ∂f
∂y
=x
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. What is the gradient of the function f (x, y) = sin(xy)?
(a) hy cos(xy), x cos(xy)i
(b) hcos(xy), cos(xy)i
(c) hcos(xy), − cos(xy)i
(d) h− sin(xy), − sin(xy)i
√
13. Set up the double integral of f (x, y) = x over the region bounded by y = x and
x = 1.
(a) 0 y2 x dx dy
R1R1
R 1 R √x
(b) 0 0 x dy dx
(c) 0 0 x dy dx
R1Rx
(d) 0 x x dy dx
R1R1
14. Which of the following correctly relates the spherical coordinate ρ to Cartesian
coordinates (x, y, z)?
(a) ρ 2 = x2 + y 2 + z 2
(b)
p
ρ = x2 + y 2
(c) ρ=x+y+z
(d) ρ 2 = x2 + y 2
