Practice Test
Note: Solutions will be posted mid-day Tuesday, 28 April. If you want more questions, I
can direct you to additional problems.
Part 1: Integration.
1. Compute the following indefinite integrals:
Z
(a) x2 + 3x5 + 7 dx
Z
(b) e7x + x3 − 1 dx
Z
1
1
(c)
+
dx
x x2
Z
(d) sin x + cos x dx
Z
2x + 5x2 + 1
dx
(e)
x
Z
(f) sin 5xdx
2. Compute the following definite integrals:
Z e
1
dx
(a)
x
1
Z π
(b)
sin(2x) dx
Z
0
4
(c)
2x2 +
√
x dx
1
3. Compute the following integrals:
Z
(a) (2x + 1)(2x2 + 2x + 5)77 dx Edited: 4/28: 8:50
Z
3
(b) 3x2 · ex dx
4. Compute the following integrals:
Z ∞
8e−8x dx
Z1 ∞
1
√ dx
x
1
5. Compute the following integrals:
Z
xex dx
Z
x2 ex dx
Z
(8x + 10) ln(5x) dx
6. Compute the following integrals:
Z x
3x3 y 2 + 5xex dy
Z2 π
cos(xy) dx
0
Z Z
x + y dxdy
Part 2: Applications of Integrals
7. Compute the area between the x-axis and the curve for the following function and interval:
f (x) = 2x3 + 4x2 on the interval [−3, 0]
8. State the formula for finding the volume of a solid of revolution.
9. Find the volume of the of resulting solid of revolution formed by rotating the following functions
around the x-axis:
(a) f (x) =
√
4x3 + 2x + 1 from x = 0 to x = 3
(b) g(x) = e3x over the interval [−2, 1]. Round your answer to two decimal places.
10. A delivery van is traveling at 80 km per hour. Give a formula for the position of the van
at time t.
11. Nanako’s pasta company orders 600 cases of imported pasta sauce from Italy every 30 days.
The number of cases on hand t days after the shipment arrives is given by:
√
N (t) = 600 − 20 30t
Find the average daily inventory.
12. Yukiko throws a nerf ball off of the top of a 150 foot apartment complex at her friend Chie.
The ball is tossed with an initial velocity of 10 miles per hour. Give a formula for the position of
the ball at time t. When does the ball hit Chie?
13. Chell needs to compute the area between the function sin x over the interval [0, 2π]. She
does so and gets an answer of 0. Her boss, Mr. Johnson, interrupts her calculations and tells her
he thinks the answer should be 4. Who is correct? What was the error?
Multivariable Calculus
14. What surface does the function x2 + y 2 + z 2 = 9 represent?
15. What surface does the function x + y = 6 represent in 3-dimensions?
16. Let f (x, y) = 2x + 4y − 27xy. Compute fx (x, y), fy (x, y), fxx (x, y), fxy (x, y), and fyy (x, y).
17. Let g(x, y) =
6
. Compute gxx (x, y).
x2 + y 2
18. Let f (x, y, z) = ln |8xy + 5yz − x3 |. Compute fx , fy , fz and fyz
19. The total revenue (in hundreds of dollars) from the sale of x spas and y solar heaters is
approximated by
15 + 169x + 182y − 5x2 − 7y 2 − 7xy
Find the number of each that should be sold to produce the maximum revenue. Find the maximum
revenue.
20. Find where the function has any relative extrema/saddle points for f (x) = 3x2 +2y 3 −18xy+42
21. State the process of finding the maximum or minimum of a function subject to a constraint
(Lagrange Multipliers)
22. Use the method of Lagrange Multipliers to find the Maximum of f (x, y) = 12xy − x2 − 3y 2
subject to x + y = 16. Edited 4/28 at 9:58am
Differential Equations (DE)
23. State what it means for a DE to be “first-order”.
24. Solve the following differential equations (all methods present):
dy
= 3e5x
dx
dy
(b)
= 2xy
dx
dy
y
(c)
= for x > 0
dx
x
(a)
(d) y 0 + y = 2ex
dy
+ 2xy = 4x
dx
dy
(f) x ·
+ 2xy − x2 = 0
dx
(e)
25. Solve the following Initial Value Problem (IVP)
dy
+ 3x2 = 2x;
dx
y(0) = 5
26. A tank holds 100 gallons of water that initially contains 20 pounds of dissolved salt. A
salt solution is flowing into the tank at the rate of 2 gallons per minute. The tank is mixed and
drained at the same rate. The solution entering the tank has a concentration of 3 pounds of salt
per gallon.
(a)Find an expression for the amount of salt in the tank at time t.
(b) What will be the concentration of salt after 30 minutes? 60 minutes? What value is the concentration approaching as time goes by?
Probability
27. What conditions do a function need to satisfy to be a Probability Density Function (PDF)?
28. Which of the following are PDFs?
x2
; [−2, 2]
16
3
(b) f (x) = x2 ; [3, 5]
98
(a) p(x) =
(c) f (x) = 8e−8x on [0, ∞)
29. Find the value of k that makes the following a PDF: f (x) = kx3 over [2, 4].
30. For the function found in the previous problem, find P (2 ≤ x ≤ 3).
31. Find the cumulative distribution function for the PDF f (x) =
Edited 4/28: 10:16.
3 1/2
x
on the interval [1,4]
14
3 2
x on [1, 3], compute the following: Expected Value, Variance,
32. Given the PDF f (x) =
26
Standard Deviation, Median.
33. The rainfall (in inches) in a certain region varies from 32 to 44. Assume the probability
of any given rainfall is equally likely. Give a PDF to represent this scenario.
Sequences and Series
34. Which of the following represent geometric sequences:
(a) {1, 5, 25, 125, 625, . . .}
(b) {1, 6, 11, 16, . . .}
(c) {e, e2 , e3 , e4 , . . .}
1
2
(d)
, 2, 4π, 8π , . . .
π
(e) {1, i, −1, −i, 1, i, . . .} where i is the imaginary number.
35. For the geometric sequence {1, 2, 4, 8, 16, . . .}, compute r, an , and Sn .
36. Aperture Labs purchases a new machine for testing. They initially spend $100, 000 on the
machine. Each year it loses 25% of its value. Find the value of the machine after 5 years.
37. State what an annuity is.
38. Find the amount of the annuity where R = $1800 at 8% interest compounded quarterly
for 12 years.
39. Tell whether each geometric series converges. If it converges, find the sum of the series:
(a) 1+1+1+1+. . .
(b) 1 + 0.8 + 0.64 + 0.512 + . . .
(c) 3+6+12+24+. . .
40. State how we compute the Taylor Series for a function f (x) (we have a formula).
41. Find the Taylor Series for sin x and cos x. The ambitious may find the first 4 or 5 terms
of the Taylor Series for tan x.
42. Find the Taylor Series for f (x) =
−3
4−x
43. Use L’Hospital’s Rule to find the following limits:
ln(x − 1)
x→2 x − 2
√
x−2
(b) lim
x→4 x − 4
ln x
(c) lim 77
x→∞ x
(a) lim