Final Exam Practice
Exam Date: December 16, 2014, 8:00 am
You will have 2 hours to complete the final. The final exam will cover Chapters 6-9, and
Chapter 10.5-10.7. You will be allowed one 8.5” x 11” piece of paper with any notes that you
find useful. I recommend including the flow chart to test for convergent series and derivatives
and integrals of inverse trig functions, hyperbolic functions, and inverse hyperbolic functions.
No calculators will be allowed during the exam.
1. Find
dy
dx
(a) y = sin2 (x3 )
(b) y = tanh−1 (sin(x))
(c) y = x1+x
(d) y = 2x cos(2x ) + xπ
2. Evaluate the following integrals.
Z
(a)
cot(x) dx
Z
(b)
ex sinh(ex ) dx
√
Z
2/2
(c)
Z0
(d)
ln(x) dx
Z
(e)
Z
(f)
Z
(g)
(h)
(i)
(j)
(k)
1
√
dx
1 − x2
x2 sin(x) dx
sin3 (x) cos−4 (x) dx
dx
9 + x2
Z
dx
√
x− x
Z
t+9
dt
t3 + 9t
Z ∞
1
dx
x ln x
e
Z 3
dx
1/3
1 (x − 1)
√
1
3. Evaluate the following limits.
2x − sin x
x→0
x
1/x
(b) lim x
(a) lim
x→∞
(c) lim [ln(x + 1) − ln(x − 1)]
x→∞
ex
x→∞ x2
x2 + 2x + 1
(e) lim
x→3
x2 − 9
(d) lim
4. Determine whether the sequence converges or diverges, and if it converges, find lim an .
n→∞
(a) an = 2−n sin n
4n − 3
(b) an =
2n
5. Indicate whether the given series converges or diverges.
∞
X
1
(a)
n2
n=1
(b)
(c)
(d)
∞
X
n=1
∞
X
n=1
∞
X
2n
1
n1/2
2−n
n=1
6. Determine if the following series converges absolutely, converges conditionally, or diverges.
(a)
(b)
∞
X
(−1)n
x
n=1
∞
X
sin k
(−1)k √
k k
k=1
7. Find the convergent set for the given power series:
(a)
(b)
∞
X
xn
n=1
∞
X
n=1
n2
(x + 1)n
n!
2
8. For f (x) =
√
1+x+1
(a) Find the Taylor polynomial of order 3 centered about a = 0
(b) Approximate f (1.12) using the polynomial from (a).
(c) Find a bound for the error in your approximation (i.e. find R3 )
3 3
9. For the Cartesian coordinates √ , √ , find three different ways to represent this
2 2
point in polar coordinates. (At least one of the points should have a negative r-value).
10. For r = 3 cos θ and r = 1 + cos θ
(a) Find the interaction points.
(b) Find the area of the region outside of the cardioid and inside the circle.
11. Solve
dy
2y
+
= (x + 1)3
dx x + 1
3