MATH 18.01, FALL 2016 - PROBLEM SET # 3A: SOLUTIONS
Professor: Jared Speck
Due: by 1:45pm on 10-7-16
(in the boxes outside of Room 4-174 during the day; write your name and recitation
instructor on your homework)
Part I (15 points)
Notation: The problems

MATH 18.01, FALL 2016 - PROBLEM SET # 3B: PART II SOLUTIONS
1. (Sept. 29; max-min problems; 2 + 10 + 5 = 17 points) 4.3: 30extra+ab. You may assume that the
paper has a vertical length L, where L 2a. Before doing the two problems stated in the text, first

MATH 18.01, FALL 2016 - PROBLEM SET # 4
Professor: Jared Speck
Due: by Friday 1:45pm on 10-14-16
(in the boxes outside of Room 4-174; write your name, recitation instructor, and
recitation meeting days/time on your homework)
18.01 Supplementary Notes (inc

MATH 18.01, FALL 2016 - PROBLEM SET # 1
Professor: Jared Speck
Due: by 1:45pm on 9-16-16
(in the boxes outside of Room 4-174 during the day; write your name and recitation
instructor on your homework)
18.01 Supplementary Notes (including Exercises and Sol

MATH 18.01, FALL 2016 - PROBLEM SET # 3B
Professor: Jared Speck
Due: by Friday 1:45pm on 10-07-16 (turn in along with Problem Set #
3A)
(in the boxes outside of Room 4-174; write your name, recitation instructor, and
recitation meeting days/time on your h

MATH 18.01, FALL 2016 - PROBLEM SET # 2
Professor: Jared Speck
Due: by 4:00pm on Thursday 9-22-16
(in the boxes outside of Room 4-174 during the day; write your name and recitation
instructor on your homework)
18.01 Supplementary Notes (including Exercise

MATH 18.01, FALL 2016 - PROBLEM SET # 5
Professor: Jared Speck
Due: by Friday 1:45pm on 10-28-16
(in the boxes outside of Room 4-174; write your name, recitation instructor, and
recitation meeting days/time on your homework)
18.01 Supplementary Notes (inc

MATH 18.01, FALL 2016 - SOLUTIONS FOR THE PROBLEM SET # 2
Part II
Problem 1. (now; slope, basic curve sketching; 1 + 2 = 3 points)
a) Find an even function E(x) and an odd function O(x) such that the function f (x) = x4 /(x + 1)
can be decomposed as f (x)

MATH 18.01, FALL 2016 - PROBLEM SET # 3A
Professor: Jared Speck
Due: by 1:45pm on 10-7-16
(in the boxes outside of Room 4-174 during the day; write your name and recitation
instructor on your homework)
18.01 Supplementary Notes (including Exercises and So

MATH 18.01, FALL 2016 - PROBLEM SET # 4
Part II (30 points)
Problem 1. (Oct. 6; Newtons method; 1 + 2 + 2 = 5 points) This problem will show you why it is
important to choose a good starting point x0 when you are applying Newtons method. Consider the
func

MATH 18.01, FALL 2016 - PROBLEM SET #1 SOLUTIONS
(PART II)
Problem 1
(a): We can rewrite E(x) + O(x) = 0 as O(x) = E(x). Therefore for all x we
have,
O(x) = E(x) = E(x) = O(x) = O(x).
The identity O(x) = O(x) implies O(x) = 0 for all x. As a consequence,

Unit 4. Applications of integration
4A. Areas between curves.
4A-1 a)
1
Z
1/2
3
1
(3x 1 2x2 )dx = (3/2)x2 x (2/3)x3 1/2 = 1/24
b) x = ax = x = a or x = 0. There are two enclosed pieces (a < x < 0 and
0 < x < a) with the same area by symmetry. Thus the tot

Unit 7. Infinite Series
7A: Basic Definitions
7A-1
n
X
X
1
1
4
1
=
=
.
=
n
4
4
1
(1/4)
3
0
0
a) Sum the geometric series:
b) 1 1 + 1 1 + . . . + (1)n + . . . diverges, since the partial sums sn are successively
1, 0, 1, 0, . . . , and therefore do not ap

Unit 3. Integration
3A. Differentials, indefinite integration
3A-1 a) 7x6 dx. (d(sin 1) = 0 because sin 1 is a constant.)
b) (1/2)x1/2 dx
c) (10x9 8)dx
d) (3e3x sin x + e3x cos x)dx
e) (1/2 x)dx + (1/2 y)dy = 0 implies
dy =
y
1/2 xdx
1 x
1
dx
dx = 1
=

2. Applications of Differentiation
2A. Approximation
d
b
b
a + bx =
f (x) a + x by formula.
dx
2 a
2 a + bx
r
bx
bx
By algebra:
a + bx = a 1 +
a(1 + ), same as above.
a
2a
2A-1
2A-2 D(
1 b
1
b
1/a
1
b
1
f (x) 2 x; OR:
)=
=
(1 x).
2
a + bx
(a + bx)
a

SOLUTIONS TO 18.01 REVIEW PROBLEMS
Unit 1: Differentiation
R1-0.
a)
nRT
V +1
b)
(x + 1) cos x sin x
(x + 1)2
b)
R1-1
a)
m0 v
v 2 /c2 )3/2
c)
c0 (2k + 1)
2
sin x cos x
1
c) x1/3 sec2 x + x2/3 tan x
x
3
(3cos2 x2 + 1)( sin x2 + 1)x
sin x2 + 1 x
f)
e)
x2 +

SOLUTIONS TO 18.01 EXERCISES
Unit 1. Differentiation
1A. Graphing
1A-1,2 a) y = (x 1)2 2
b) y = 3(x2 + 2x) + 2 = 3(x + 1)2 1
2
2
1
1
-2
1a
1A-3 a) f (x) =
1
-2
-1
1b
2a
2b
x3 3x
(x)3 3x
=
= f (x), so it is odd.
4
1 (x)
1 x4
b) (sin(x)2 = (sin x)2 , so it

18.01 REVIEW PROBLEMS AND SOLUTIONS
Unit I: Differentiation
R1-0 Evaluate the derivatives. Assume all letters represent constants, except for the
independent and dependent variables occurring in the derivative.
m0
dp
=?
b) m = p
,
a) pV = nRT,
dV
1 v 2 /c

INT.
IMPROPER INTEGRALS
In deciding whether an improper integral converges or diverges, it is often awkward or
impossible to try to decide this by actually carrying out the integration, i.e., finding an
antiderivative explicitly. For example both of these

X. EXPONENTIALS
AND LOGARITHMS
1. The Exponential and Logarithm Functions.
We have so far worked with the algebraic functions those involving polynomials and
root extractions and with the trigonometric functions. We now have to add to our list the
exponen

FT. SECOND FUNDAMENTAL THEOREM
1. The Two Fundamental Theorems of Calculus
The Fundamental Theorem of Calculus really consists of two closely related theorems,
usually called nowadays (not very imaginatively) the First and Second Fundamental Theorems. Of

A. APPROXIMATIONS
In science and engineering, you often approximate complicated functions by simpler ones
which are easier to calculate with, and which show the relations between the variables more
clearly. Of course, the approximation must be close enoug

PI. PROPERTIES OF INTEGRALS
For ease in using the definite integral, it is important to know its properties. Your book
lists the following1 (on the right, we give a name to the property):
a
Z
(1)
Zb a
(2)
f (x) dx =
b
Z
f (x) dx
integrating backwards
a
f

G. GRAPHING FUNCTIONS
To get a quick insight int o how the graph of a function looks, it is very helpful to know
how certain simple operations on the graph are related to the way the function expression
looks. We consider these here.
1. Right-left transla

F. HEAVISIDES COVER-UP METHOD
The eponymous method was introduced by Oliver Heaviside as a fast way to do a decomposition into partial fractions. In 18.01 we need the partial fractions decomposition in order
to integrate rational functions (i.e., quotient

AV. AVERAGE VALUE
What was the average temperature on July 4 in Boston?
T
The temperature is a continuous function f (x), whose graph over the
24-hour period might look as shown. How should we define the average
value of such a function over the time inte

C. CONTINUITY AND
DISCONTINUITY
1. One-sided limits
We begin by expanding the notion of limit to include what are called one-sided limits,
where x approaches a only from one side the right or the left. The terminology and
notation is:.
right-hand limit
li

MVT. MEAN-VALUE THEOREM
There are two forms in which the Mean-value Theorem can appear;1 you should get
familiar with both of them. Assuming for simplicity that f (x) is differentiable on an interval
whose endpoints are a and b, or a and x, the theorem sa