Name:
Math 1220
Final Exam Review
This review is a fairly comprehensive picture of what we have covered this semester, and all1 of these
problems are of an appropriate level2 for the final exam.
1. Compute the following derivatives:
(a) Dx ln(sin(x))
(b) Dx 2cos(x)
(c) Dx x4x
2. Compute the following integrals. Note that I have deliberately not told you which techniques to use; if
you are pressed for time, you can still study in a useful way by simply figuring out which technique to
use for each, without computing the integral.
Z
Z
Z π
6
2x2 + x − 4
cos x
(a)
axn dx
(s)
dx
(j)
(sin x)2
dx
x3 − x2 − 2x
0
Z
Z
Z
2
cos x
(b)
ex dy
(t)
dx
(k)
tan x dx
sin4 x − 16
Z
Z∞
Z
cos x
1
x
(c)
dx
(u)
dx
(l)
xe dx
1.000001
1 + sin x
x
0
Z
Z∞
Z
1
(d)
ex sin ex dx
(v)
dx
(m)
ln x dx
2 + 2x + 10
x
−∞
Z
Z∞
Z √
1
2
3
(e)
dx
2
(w)
xe−x dx
(n)
x 1 − x dx
x ln(3x)
1
Z 2√
Z3
Z π
2
1
(f)
4 − x2 dx
dx
(x)
sin6 x dx
(o)
3
−2
−1 x
0
Z
Z
Z3
1
x
6
(g)
dx
(p)
tan
x
dx
2
(y)
dx
√
1 − 4x
9 − x2
0
Z1
Z
2x
Z π
2
(h)
dx
cos x
(q)
sin2 3x cos2 3x dx
4
(z)
dx
0 1+x
1
− sin x
Z
0
Z
x
6
4e
x
(i)
dx
√
(r)
dx
2x
1−e
(x − 2)2 (1 − x)5
3. Compute the following limits.
x2
sin x−x
limx→0+ cosx x
(a) limx→0+
(b)
1
(c) limx→∞ x x
(d) limx→∞
x3
ex
(e) limx→0+ x ln x
(f) limn→∞ sinn n
4. Determine if each of the following series converge or diverge.
1
k=1 k ln k
(c)
P∞
−k 2
k=1 k 10
(d)
P∞
(a)
P∞
(b)
P∞
n
n=1 n2 +2n+1
(e)
P∞
7n
n=1 n!
(f)
P∞
sin n
n=1 n2
n
1
n=1 n 2
1 The last two problems are actually quite challenging, since you have not seen many problems of that sort before, so they would
be considered challenge problems on the exam as well.
2 That’s not to say that they are all the same difficulty level, of course.
Final Exam Review
Math 1220
5. Determine if the following series converge or diverge. If they converge, evaluate the sum (you may
have to be very clever).
(a)
P∞
k=1 3
k 5−k
(b)
P∞
1
k=1 k 2 −1
(c)
2k
k=0 k!
P∞
6. Determine if each series diverges, converges conditionally, or converges absolutely.
(a)
P∞
k=1 (−0.25)
k
(b)
P∞
n k
k=1 (−1) k−1
7. If you want to use Sn to approximate the sum
the error to be less than 0.001?
Pn
k=1
8. Determine the convergence set for the power series
(c)
P∞
1
k=1 k sin k
(−1)k
k ,
what do you need to choose for n in order for
Pn
(x−1)n
n .
k=1
9. Find the terms of the Taylor series for each of the following functions through (x − a)5 . If you wish,
1
in order to do these
you may use the Taylor series representations of sin(x), cos(x), ln(x), ex , and 1−x
problems faster.
(a) f (x) =
1
;
(1−x)2
a=0
(b) f (x) = sin(x) cos(x); a = 0
(c) f (x) = ex ; a = 1
10. Solve the differential equation
dy y
− = 3x3
dx x
if y = 3 when x = 1.
11. Given the differential equation
dy
dx
= 1 − y 2 , compute:
(a) limx→∞ y if y = 0 when x = 0
(b) limx→∞ y if y = 10 when x = 0
(You may want to plot a slope field.)
12. Use Euler’s method to approximate a solution to the differential equation
y 0 = −y
over the interval [0, 1] with h = 0.2.
13. (Challenge) Represent 0.099 as a fraction.
14. (Challenge) Given f (x) = sin(x):
(a) Find the Taylor polynomial of order 5 based at a = 0.
(b) Find a formula for R5 (x) (i.e. the remainder).
(c) Use your formula for R5 (x) to find an upper bound on the value of sin
π
4