Answers to Review Sheet for Final Exam

(d) d
dy
y + 1
y + 7
y + 7
Solution: 3
y + 1
1
(y + 7) 2
41. Calculate the following limits (make sure to justify your answers).
ln x
(a) lim √
x→1 x + 1
Solution: Just plug in x = 1 to get 0/2 = 0.
(b) lim
x→∞ x 2 e −x
Solution: Use L’Hopital’s rule to get
lim
x→∞
x2 = lim
ex x→∞
2x
= lim
ex x→∞
2
= 0.
ex (c) lim x ln x
x→0 +
Solution: It’s of the form 0 · ∞, so we rewrite as a fraction:
42. Evaluate the following integrals.
(a)
2
0
4 − y 2 dy
lim x ln x = lim
x→0 + x→0 +
ln x
1/x
= lim
x→0 +
1/x
= lim
−1/x2 x→0 +(−x)
= 0.
Solution: You’re supposed to recognize this as the area of a quarter-circle of radius 2,
which is 1
4 (π · 22 ) = π.
√
t + t + 1
(b)
t2 dt
√
t + t + 1
Solution:
t2
t−1 −3/2 −2
dt = + t + t dt = ln |t| − 2t −1/2 − t −1 + C
2 θ
(c) 2 + 1 dθ
1
Solution:
θ 2
θ=2 2
1
2 2
It’s + θ = + 2 − + 1 =
ln 2 θ=1 ln 2 ln 2 2
+ 1.
ln 2
43. Calculate
2
1
5
1
[3f(x) + 4g(x)] dx, given that
f(x) dx = −1,
5
2
f(x) dx = 1,
10
5
−1
g(x) dx = −2, and
1
−1
g(x) dx = 3.