18 014 Problem Set 9 Solutions
Total: 24 points
Problem : Integrate
(a)
Z
(b)
(x2
dx
4x + 4)(x2
Z
x4
4x + 5)
.
dx
.
2x2
Solution (4 points) (a) We use the method of partial fractions to write
1
(x
2)2
Practice Exam
Problem 1 Find
lim
h!0
R 1+h
0
Solutions
R 1 t2
2
et dt
e dt
0
.
2)
h(3 + h
Solution First, using that the limit of a product is the product of limits, we get
lim
h!0
Because
1
3+h2
R 1+
K.l
The fundamental theorems of calculus.
Here are the two basic theorems relating integrals and
derivatives. You should know the proofs of these theorems.
First, we need to discuss "onesided" derivat
The trigonometric functions.
For the present, we shall assume the following theorem
1 concerning existence of the sine and cosine functions. Later
on, when we study power series, we shall prove this t
99 dx where here [x] is de ned
R1 3
p
Problem : Compute 99 (2x 198)2 x
to be the largest integer x.
Solution By the properties of the integral, we know that the above is equal
to
Z 4
p
4
(x)2 x dx.
18 014 Problem Set 7 Solutions
Total: 12 points
Problem : Given a function g continuous everywhere such that g(1) = 5 and
R1
Rx
g(t)dt = 2, let f (x) = 1 0 (x t)2 g(t)dt. Prove that
2
0
Z x
Z x
0
f (x
1.1
Theorem. Let m and n pg integers,- let :1) 0. Let
h(y) = WW :9; we.
Then h _i_s_ differentiable, and
h'(y) = dyl.
~Eroof. Step L; We first prove the theorem in the case m = l.
Let f(x) = XI1 f
N.1
Ilnteérationj
_T_h_e substitution gm
Apostol proves only one version of the substitution rule, the one given in Theorem 1
following. Sometimes the converse is needed; we prove this result in Theor
18 014 Problem Set 6 Solutions
Total: 24 points
Problem : A water tank has the shape of a right-circular cone with its vertex
down. Its alititude is 10 feet and the radius of the base is 15 feet. Wate
18 014 Problem Set 8 Solutions
Total: 24 points
Problem : Compute
Z
1
xf 00 (2x)dx
0
00
given that f is continuous for all x, and f (0) = 1, f 0 (0) = 3, f (1) = 5, f 0 (1) = 2,
f (2) = 7, f 0 (2) = 4
H.1
The small sgan theorem and the extremevalue theorem.
There are three fundamental theorems concerning a function
that is continuous on a closed interval [a,b]. The first
is the IntermediateValue
J.l
Exercises on derivatives
1.
3.
Define a new derivative by the formula
3 3
D#f(x) = lim (f(X+h) h- (f(X) _
h+0
Assuming that f and g are continuous, and that D#f(x)
and D#g(x) exist, derive formula
15.1 Vector Fields
CHAPTER 15
15.1
(page 554)
VECTOR CALCULUS
Vector Fields
(page 554)
An ordinary function assigns a value f (x) t o each point x. A vector field assigns a vector F (x, y) t o each
po
13.1 Surfaces and Level Curves
CHAPTER 13
(page 475)
PARTIAL DERIVATIVES
Surfaces and Level Curves
13.1
J
L
The graph of z = f (x, y) is a surface in xyz space. When f is a linear function, the surfac
14.1 Double Integrals
(page 526)
CHAPTER 14
MULTIPLE INTEGRALS
Double Integrals
14.1
(page 526)
sR
The most basic double integral has the form $ JR dA or $ dy dx or $ JR dx dy. It is the integral of 1
12.1ThePositionVector
CHAPTER 12
12.1
(page4521
MOTION ALONG A CURVE
The Position Vector
This section explains the key vectors that describe motion. They are functions o f t . In other words, the time
Practice Exam
Problem 1 Evaluate
R
Solutions
t3
p +t dt
1+t2
Solution The problem can be simpli ed as t3 + t = t(t2 + 1). Then by substitution
with u = t2 + 1 and thus du = 2t dt we have
Z p
Z
1
1
1
2