Calculus of One and Several Variables and Linear Algebra
MA 1 abc

Fall 2014
Practice problems Final Exam
1. Derive identities for sin 3 and cos 3 by calculating ei3 = (ei )3 .
2. Show that if z and w are complex numbers then zw = zw.
3. If z = a + ib and w = c + id are complex numbers with a, b, c, d real, derive a
formula f
Calculus of One and Several Variables and Linear Algebra
MA 1 abc

Fall 2014
MATH 1A HW6 SOLUTIONS
1. Section 4.3 Question 2 [10 pts total]
Problem. Prove that
X
1
n
log
n
log(log
n)
n=2015
diverges.
Solution. Using the integral test explained in cranks, the series in the problem converges or diverges if
1
x
log
x
log(log
x)
201
Calculus of One and Several Variables and Linear Algebra
MA 1 abc

Fall 2014
Solution to HW 5, Ma 1a Fall 2016
Section 3.8 Exercise 1: Let f be a twice continuously differentiable function
on the real line with f 0 (x) > 1 for every value of x and f 00 (x) 1 for every
value of x. Let x0 be a real number with f (x0 ) < 12 and f
Calculus of One and Several Variables and Linear Algebra
MA 1 abc

Fall 2014
Solutions for Problem set 4
3.1.1 Fix > 0. At each point a of the interval [c, d], let a > 0 be the least upper
bound of the set of for which x a < implies that f (x) f (a) < . (We have that
a > 0, since f is continuous at a. Moreover, we have that if
Calculus of One and Several Variables and Linear Algebra
MA 1 abc

Fall 2014
Problem Set 3, Due October 17,2016 2:00 A.M.
From the official course notes Calculus for Cranks:
From Section 2.1 Exercises 2
From Section 2.2 Exercises 2,6
From Section 2.3 Exercises 1,3
1
Calculus of One and Several Variables and Linear Algebra
MA 1 abc

Fall 2014
Problem Set 4, Due October 24,2016 2:00 A.M.
From the official course notes Calculus for Cranks:
From Section 3.1 Exercise 1
From Section 3.2 Exercise 1
From Section 3.3 Exercises 1,2
From Section 3.4 Exercise 1
1
Calculus of One and Several Variables and Linear Algebra
MA 1 abc

Fall 2014
Problem Set 2, Due October 10,2016 2:00 A.M.
From the official course notes Calculus for Cranks:
From Section 1.3 Exercises 6,7
From Section 1.4 Exercises 2,3,5
1
Calculus of One and Several Variables and Linear Algebra
MA 1 abc

Fall 2014
Problem Set 1, Due October 3,2016 2:00 A.M.
From the official course notes Calculus for Cranks:
From Section 1.2 Exercises 4,5,6,7
From Section 1.1 Exercise 7 (but feel free to use your results and ideas from the section
1.2 problems for this problem.)
1
Calculus of One and Several Variables and Linear Algebra
MA 1 abc

Fall 2014
Problem Set 7, Due November 21,2016 2:00 A.M.
From the official course notes Calculus for Cranks:
From Section 4.5 Exercises 1,2
From Section 5.1 Exercises 2
From Section 5.3 Exercises 1,2
1
Calculus of One and Several Variables and Linear Algebra
MA 1 abc

Fall 2014
Problem Set 5, Due November 14,2016 2:00 A.M.
From the official course notes Calculus for Cranks:
From Section 4.3 Exercises 2,3,4
From Section 4.4 Exercises 1,2
1
Calculus of One and Several Variables and Linear Algebra
MA 1 abc

Fall 2014
Problem Set 5, Due November 7,2016 2:00 A.M.
From the official course notes Calculus for Cranks:
From Section 3.8 Exercise 1
From Section 4.1 Exercise 1,4,5,6
1
Calculus of One and Several Variables and Linear Algebra
MA 1 abc

Fall 2014
Problem Set 8, Due November 28,2016 2:00 A.M. Blanket extension to
November 30, 2016 4:00 P.M.
From the official course notes Calculus for Cranks:
From Section 6.1 Exercises 1,2,3
From Section 6.2 Exercises 1,2
1
Calculus of One and Several Variables and Linear Algebra
MA 1 abc

Fall 2014
Problem Set 6, Due November 17,2014
From the official course notes Calculus for Cranks:
From Section 4.1 Exercise 1
From Section 4.2 Exercise 1
From Section 4.3 Exercise 1
1
Calculus of One and Several Variables and Linear Algebra
MA 1 abc

Fall 2014
Review Problems: Math 1a Midterm 2014
These arent problems that will be on the midterm. These arent problems you have
to turn in. These are problems that might help you if you study them.
1. Use induction to prove for r 6= 1
n
X
jrj =
j=1
rn+1 r
nrn+1
).
Calculus of One and Several Variables and Linear Algebra
MA 1 abc

Fall 2014
Problem Set 7, Due November 12,2014
From the official course notes Calculus for Cranks:
From Section 4.4 Exercises 1,2
From Section 4.5 Exercises 1,2
1
Calculus of One and Several Variables and Linear Algebra
MA 1 abc

Fall 2014
Problem Set 5, Due November 10,2014
From the official course notes Calculus for Cranks:
From Section 3.5 Exercise 1,2
From Section 3.6 Exercise 1
From Section 3.7 Exercises 1,2
1
Calculus of One and Several Variables and Linear Algebra
MA 1 abc

Fall 2014
Problem Set 8, Due December 3,2014
From the official course notes Calculus for Cranks:
From Section 5.1 Exercises 1,2
From Section 5.2 Exercises 1,2
From Section 5.3 Exercise 1
1
Calculus of One and Several Variables and Linear Algebra
MA 1 abc

Fall 2014
Problem Set 3, Due October 27,2014
From the official course notes Calculus for Cranks:
From Section 3.1 Exercise 1
From Section 3.2 Exercise 1
From Section 3.3 Exercises 1,2,3
1
Calculus of One and Several Variables and Linear Algebra
MA 1 abc

Fall 2014
Problem Set 3, Due October 20,2014
From the official course notes Calculus for Cranks:
From Section 2.2 Exercises 3,4
From Section 2.3 Exercises 1,2,3
1
Calculus of One and Several Variables and Linear Algebra
MA 1 abc

Fall 2015
1
Problem 3.3.1
Since f is O(1), there are > 0 and a constant C so that if h < then f (h) < C. Since the denition of
limit as h 0 only depends on values of h smaller than , we have
lim
h0
f (h)g(h)
Cg(h)
lim
h0
h
h
g(h)
= C lim
h0 h
=0
since g is o(h).
Calculus of One and Several Variables and Linear Algebra
MA 1 abc

Fall 2015
1
Problem 2.1.5
Use the squeeze theorem to calculate
(
)
3
4+ 2 .
n
lim n
n
)
3
4+ 2 .
n
(
Solution. Let
an = n
We have
(
2+
3
4n
)2
4+
3
n
3
3
2+
2+ .
4n
n
Hence if we dene
(
)
3
3
bn = n 2 +
2 =
4n
4
then an bn for all n. Similarly, we have
(
)2
3
1
3
Calculus of One and Several Variables and Linear Algebra
MA 1 abc

Fall 2015
Math 1a, 2015 problem set 7, due Monday,November 23 10:00 A.M.
Do problems 1 and 2 from section 5.1 of Cranks , problems 1 from section 5.2 of
Cranks, problems 1 and 2 from section 5.3 of Cranks. (Be sure and refresh Cranks before
you do this. Chapter 5 m