Math 3B Midterm 2 Practice Problems
Problem 1. Compute the following definite integrals.
Z e2
dx
(1)
x(ln(x))2
e
Z 3 √
x x2 + 1 dx
(2)
0
Z
π/3
x sin(x) dx
(3)
0
Z
(4)
4
√
ln( x) dx
1
Problem 2. Compute the following indefinite integrals.
Z
x
√
(1)
dx
x−5
Z
√
(2)
sin( x) dx
Z
(3)
Z
(4)
x3 − 3x2 + x − 6
dx
x4 + 3x2 + 2
x3 arctan(x) dx
Z
(5)
sin(ln(x)) dx
Problem 3. Compute the following limits. Your answer should be a number.
n X
k−1
k−1 1
(1) lim
cos
n→∞
n
n
n
k=1
(2) lim
n→∞
n
X
k=1
sin
kπ
n
exp
Z
Problem 4. Evaluate
kπ
n
π
n
x−1
dx in two ways:
(x − 1)2
(1) use a u-substitution, and
1
D. Penneys
(2) break up the integrand as
term.
x−1
(x−1)2
=
x
(x−1)2
−
1
(x−1)2
and use partial fractions for the first
Compare your answers.
Problem 5. Compute the following improper integrals.
Z ∞
(1)
e−|x| dx
−∞
Z
1
ln |x| dx
(2)
−1
Z
π/2
(3)
0
Z
∞
(4)
−∞
cos(x)
p
dx
sin(x)
1
dx
1 + x2
Problem 6. Determine whether the following improper integrals converge. If they do,
calculate them.
Z 4
1
dx
(1)
4
0 x
Z 2
1
(2)
1/3
0 (x − 1)
Z ∞
dx
(3)
dx
x ln(x)
e
Z ∞
1
dx
(4)
2
−∞ x − 1
Problem 7. Determine for which p > 0 the following integrals converge:
Z 1
1
dx
(1)
p
0 x
Z ∞
1
(2)
dx
xp
1
Z 2
1
(3)
dx
p
0 (x − 1)
Hint: Try different p first, and see if you can guess a pattern. Then try to prove your guess.
Problem 8. Find the general solution to the following differential equations
2
(1)
dy
dx
(2)
dy
dx
= 3x
√
= 3x − 1
(3)
dy
dx
= 1/x
Problem 9. Solve the following initial value problems.
(1)
dy
dx
= cos(2π(x − 3)) where y(3) = 1.
(2)
dy
dx
=
(3)
dy
dx
= 6y − 2y 2 where y(1) = 5.
(4)
dy
dx
=
x+1
y
where y(0) = 2
(5)
dy
dx
=
y
x+1
where y(0) = 1
(6)
dy
dx
= ye−x where y(0) = 1
1
x−2
where y(3) = 0 and y(1) = 1.
Problem 10. Use separation of variables to find all solutions to the following differential
equations. Then find their equilibria and their limiting behavior as t → ∞.
(1)
dy
dt
= −2y
(2)
dy
dt
= y(y − 1)(y − 2)
(3)
dN
dt
= N2 + 1
(4)
dN
dt
= rN (1 − N/K) for constants r, K > 0
3