Sample Final Exam
Math 1220-004 Fall 2014
Instructions:
1 This is a closed-book exam.
2 Time and Location: 10:30 a.m-12:30 p.m, AEB 350, 12/15/2014, Monday
3 Materials to be included:
(1) Chapter 6: 6.1-6.4, 6.8-6.9
(2) Chapter 7: 7.1-7.5
(3) Chapter 8: 8.1-8.4
(4) Chapter 9: 9.1-9.8
(5) Chapter 10: Nothing
4 No calculators or any other types of electronic devices are allowed.
5 The ”list of tests” will be provided again, which will be the same list as
was provided during the last midterm exam.
Sample Final Exam
(1) Let f (x) = x5 + 2x3 + 4x + 3
(a) Prove that f has an inverse, which is denoted by f −1 .
(b) Evaluate (f −1 )0 (10).
(2) Differentiate the following expressions with respect to x:
(a)
(2 ln x + x2 )tan (x
3 +1)
(b)
(sin x2 )(2x+cos x)
(3) Prove the following identities.
(a)
sin(arccos x) =
p
1 − x2
cos(arcsin x) =
p
1 − x2
(b)
(4) Prove the following identities.
(a)
(arcsin x)0 = √
1
1 − x2
(b)
(arccos x)0 = − √
1
1 − x2
(5) Using integration by parts, find:
(a)
Z
x2 cosh xdx
(b)
Z
(c)
Z
(d)
Z √
ln x
√ dx
x
1
arctan ( )dy
y
2x ln x3 dx
(6) Find the integrals below:
Z
Z
sin3 xdx and tan3 xdx
(a)
Z
Z
4
(b)
sin xdx and tan4 xdx
Z
(c)
sin2 x cos2 xdx
Z
(d)
tan6 x cot6 xdx
(7) Find the following integrals:
(a)
(b)
Z
x2 + 3x
√
dx
x+4
Z
0
1
√
t
dt
t+1
(8) Find the following integrals by the use of partial fraction decomposition
(a)
Z
1
dx
(x − 1)(x2 + 1)
(b)
Z
x
dx
(x − 1)4
(9) Evaluate the following limits
(a)
lim (csc x −
x→0
1
)
x
(b)
1
lim (cos x − sin x) x
x→0
(c)
lim
x→1−
(d)
lim (
x→1
ln (1 − x)
cot πx
1
x
−
)
x − 1 ln x
(e)
limπ (x tan x −
x→ 2
π
sec x)
2
(10) Evaluate the following improper integrals:
(a)
Z
2
e6x dx
−∞
(b)
Z
+∞
x
dx
x2 + 1
−∞
(c)
Z
π
2
tan x
dx
(ln cos x)2
π
3
Z
∞
(d) For what values of p does
1
Z
1
(e) For what values of p does
0
1
dx converge?
xp
1
dx converge?
xp
(11) Determine convergence or divergence for each of the positive series below.
(1)
∞
X
ke−3k
2
k=1
(2)
∞
X
n−4 n
(
)
n
n=1
(3)
∞
X
n=1
(4)
∞
X
k=1
1
sin2 ( )
n
n
2 + n5n
(12) Determine whether the following series are absolutely convergent, conditionally convergent, or divergent.
(1)
∞
X
(−1)n
n=1
(2)
1
(ln n)4
∞
X
sin n
(−1)n √
n n
n=1
(3)
∞
X
(−1)n
n=1
nn
(2n)!
(13) Determine the convergence set for the following power series.
(1)
∞
X
(−1)n
n=0
(2)
(2x − 3)n
√
4n n
∞
X
(x + 2)n ln n
n=0
n3n
(14) Use the simplest method you can think of to find the first five nonzero
terms of the Taylor series for each of the following. Also indicate the
set of convergence:
(1)
f (x) =
x
at a = 0
1 + x2
f (x) =
1
at a = 1
1 + 2x
(2)
(3)
f (x) = (e−x sin x − 1 + x) at a = 0