18.01 Practice Exam 3
Problem 1.
a) (10) Derive the trigonometric formula
cos 2x = 1 2 sin2 x and use it to evaluate
sin2 xdx.
e
b) (10) Dierentiate x ln x, and use your answer to evalute
ln x dx.
1
Problem 2. (15) K-mart is selling at half-price its left
Solutions to 18.01 Exam 3
Problem 1(20 points) A particle moves along the positive xaxis with velocity 5 units/second. How
fast is the particle moving away from the point (0, 3) (which is on the yaxis) when the particle is
7 units away from (0, 3)?
Soluti
5.1 The Idea of the Integral
CHAPTER 5
5.1
(page 181)
INTEGRALS
The Idea of the Integral
(page 181)
Problems 1-3 review sums and differences from Section 1.2. This chapter goes forward t o integrals and
derivatives.
1. If fo, f l , f2, f3, f4 = 0,2,6,12,2
6.1 An Overview
(page 234)
CHAPTER 6
6.1
EXPONENTIALS AND LOGARITHMS
An Overview
(page 234)
The laws of logarithms which are highlighted on pages 229 and 230 apply just as well to 'natural logs." Thus
In yz = In y In z and b = eln b . Also important :
+
b
Exam 3 Review
18.01 Fall 2006
Exam 3 Review
Integration
1. Evaluate de nite integrals. Substitution, rst fundamental theorem of calculus (FTC 1), (and
hints?)
2. FTC 2:
If F (x) =
Z
d
dx
x
Z
x
f (t) dt = f (t)
a
f (t) dt, nd the graph of F , estimate F ,
18.01 Practice Questions for Exam 4 Fall 2006
Problem 1.
Problem 2.
x4
dx.
(x + 1)(x2 + 4)
Evaluate
Evaluate
2
0
(x2
dx
by making the substitution x = 2 tan u.
+ 4)2
Problem 3.
1
1
2
0
1
b) Let F (x) =
x2n2 ex dx.
0
1
e
0
2
x2n ex dx to
a) Derive a reduct
18.01 Exam 5
Problem 1(25 points) Use integration by parts to compute the antiderivative,
2
x3 ex dx.
2
2
Solution to Problem 1 The derivative of ex is 2xex . Thus, set
2
u = x2 ,
dv = xex dx
2
du = 2xdx v = ex /2.
Then, by integration by parts,
udv = uv
Exam 4 Review
18.01 Fall 2006
Exam 4 Review
1. Trig substitution and trig integrals.
2. Partial fractions.
3. Integration by parts.
4. Arc length and surface area of revolution
5. Polar coordinates
6. Area in polar coordinates.
Questions from the Students
18.01 Exam 4
dx
Problem 1. (15 points) Evaluate
x( x + 1)
Problem 2. (15 points) Evaluate
(ln x) x dx
2
2
Problem 3. (20 points) Use a trigonometric substitution to evaluate
(Be careful evaluating the limits)
1
0
dx
(4 + x 3 )3/ 3
Problem 4. a. (10 poin
18.01 Practice Questions for Exam 3 Fall 2006
1
1. Evaluate
a)
0
x dx
1 + 3x2
/2
cos3 x sin 2x dx
b)
/3
1
2. Evaluate
x dx directly from its denition as the limit of a sum.
0
n
Use upper sums (circumscribed rectangles). You can use the formula
i=
1
1
n(n
18.01 Exam 4
Problem 1(25 points) A solid is formed by revolving about the xaxis the region bounded by the
xaxis, the line x = 0, the line x = a, and the curve,
x
y = b sin
.
a
Find the volume of the solid.
You may use the halfangle formulas,
2
cos (/2)