Name:
MTH 8: Test 1 Review
March 3, 2017
Please write neatly and clearly. On all problems, you must show your work and identify
your final answer. Partial credit will be given. You may use a calculator, but you may not use a
computer, mobile device, or any notes.
1. Find the area of the region bounded by the curves
y = x,
y = 5x − x2 .
2. Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region
bounded by the given curves about the specified line.
a. y = 0,
b. y = 5,
y = sin(x),
0 ≤ x ≤ π; about y = −2.
2
y = x − 5x + 9; about x = −1.
3. Consider the unbounded region enclosed by the curves
x = 1,
y = 0,
y=
1
.
x
a. Determine the area of this region, or show that it is infinite.
b. Consider the unbounded solid obtained by rotating this region about the x-axis. Determine the
volume of this region, or show that it is infinite.
4. Consider the differential equation y 00 + y = 0.
a. Determine whether the function y = sin(x) − 2 cos(x) is a solution.
b. Determine whether the function y = x2 is a solution.
5. To the right is a graph of the slope field for the differential equation y 0 = .1y(10 − y) .
a. Draw the solution curve for the solution with
y(0) = 3
b. If y(0) = 14, estimate y(9).
6.
a. Solve the initial value problem
dy
= .03y,
dt
b. What will y(5) be?
y(0) = 100.
7. Bismuth-210 is a radioactive substance. If x = x(t) is mass of Bismuth-210 in a sample remaining after
t days, then x satisfies the differential equation
dx
= kx.
dt
Suppose a sample originally contains 1000 mg, and it contains 870 mg after 1 day.
a. Solve the above differential equation to determine the mass of Bismuth-210 remaining after t days.
b. How much Bismuth-210 remains after 9 days?
c. Determine the half-life of Bismuth-210.
8. You decide to retire. Your retirement account earns interest, compounded continuously, at an annual
rate of 4%. The balance is currently $1, 500, 000, and you now withdraw money continuously at a rate
of $120, 000 per year.
a. Write an initial value problem describing the balance after t years. (This means to write a differential
equation, along with a value of the function at a specific point.)
b. Solve the initial value problem.
9. Calculate the derivative of the following functions.
a. f (x) = x ln x − x
b. f (x) = ln(ax + 4), where a is a constant
c. f (x) = 5 ln(x2 + 4x)
1
2
d. f (x) = 5e− 2 x
√
4x − 1
e. f (x) = 3 ln
x+2
10. Calculate the following integrals.
Z
p
a.
x2 4 + x3 dx
Z
b.
e−5x dx
Z
c.
x sin(2x) dx
Z
d.
sin(kx) dx, where k is a constant
Z
e.
e4x dx
Z
2
f.
xe(x ) dx
Z
g.
x2 e4x dx
Z
h.
cos4 (x) sin(x) dx
Z
5
dx
i.
2x + 1
Z
sin(x)
j.
dx
1 + cos(x)
Z
2
k.
dx
x2 − 5x + 6