5E-03(pp 126-135)
1/17/06
1:49 PM
Page 126
CHAPTER 3
By measuring slopes at points on the sine curve, we get strong visual evidence that the derivative of the sine function is the cosine function.
D erivatives
5E-03(pp 126-135)
1/17/06
1:49 PM
Page 127
In
5E-12(pp 736-745)
1/18/06
10:08 AM
Page 736
CHAPTER 12
Bessel functions, which are used to model the vibrations of drumheads and cymbals, are dened as sums of innite series in Section 12.8. Notice how closely the computergenerated models (which involve Be
5E-04(pp 222-231)
1/17/06
2:40 PM
Page 222
CHAPTER 4
Scientists have tried to explain how rainbows are formed since the time of Aristotle. In the project on page 232, you will be able to use the principles of differential calculus to explain the formation
5E-08(pp 510-519)
1/17/06
5:19 PM
Page 510
CHAPTER 8
The techniques of this chapter enable us to nd the height of a rocket a minute after liftoff and to compute the escape velocity of the rocket.
T echniques of Integration
5E-08(pp 510-519)
1/17/06
5:19 P
5E-17(pp 1090-1099)
1/19/06
2:27 PM
Page 1090
CHAPTER 17
The collection of wind velocity vectors at any given time during a tornado is an example of a vector eld.
V ector Calculus
5E-17(pp 1090-1099)
1/19/06
2:27 PM
Page 1091
In this chapter we study the
5E-15(pp 0922-0931)
1/18/06
2:46 PM
Page 922
CHAPTER 15
We can visualize a function of two variables by its graph (a surface) or by its level curves (a contour map).
P artial Derivatives
5E-15(pp 0922-0931)
1/18/06
2:46 PM
Page 923
So far we have dealt wi
5E-06(pp 374-383)
1/17/06
4:15 PM
Page 374
CHAPTER 6
The volume of a sphere is the limit of sums of volumes of approximating cylinders.
A pplications of Integration
5E-06(pp 374-383)
1/17/06
4:15 PM
Page 375
In this chapter we explore some of the applicat
5E-07(pp 412-421)
1/17/06
4:37 PM
Page 412
CHAPTER 7
The graphs of the functions of The graphs of the functions this chapter appear as reecof this chapter appear tions of each otherthe natural as reections of each exponential and logarithmic otherthe natu
Math408D Homework 1 Soultions
Mike Harmon September 13, 2008
1
Section 7.8
ln(ln(x) x 0 = 0 . So, use LHospitals rule.
18. limx
ln(ln(x) = lim x x x lim
d (ln(ln(x) dx d (x) dx
= lim 30. limx0
cos(mx)cos(nx) x2
1 ln(x)
1 x
x
1
=
0 =0 1
=
0 0
LHospitals R
Rahman, Tarique Homework 10 Due: Nov 21 2007, 6:00 pm Inst: Vandenbout This print-out should have 25 questions. Multiple-choice questions may continue on the next column or page nd all choices before answering. The due time is Central time. 001 (part 1 of
5E-18(pp 1176-1185)
1/19/06
3:42 PM
Page 1176
CHAPTER 18
The charge in an electric circuit is governed by the differential equations that we solve in Section 18.3.
S econd-Order Differential Equations
5E-18(pp 1176-1185)
1/19/06
3:42 PM
Page 1177
The basi
5E-16(pp 1016-1025)
1/18/06
4:13 PM
Page 1016
CHAPTER 16
If we approximate a solid by rectangular columns and let the number of columns increase, the limit of sums of volumes of columns is the volume of the solid.
M ultiple Integrals
5E-16(pp 1016-1025)
1
5E-14(pp 884-893)
1/18/06
11:47 AM
Page 884
CHAPTER 14
The calculus of vectorvalued functions is used in Section 14.4 to prove Keplers laws. These describe the motion of the planets about the Sun and also apply to the orbit of a satellite about the Earth,
5E-13(pp 828-837)
1/18/06
11:09 AM
Page 828
CHAPTER 13
Wind velocity is a vector because it has both magnitude and direction. Pictured are velocity vectors indicating the wind pattern over San Francisco Bay at 12:00 P.M. on June 11, 2002.
V ectors and the
5E-11(pp 686-695)
1/18/06
9:31 AM
Page 686
CHAPTER 11
Parametric curves are used to represent letters and other symbols on laser printers. See the Laboratory Project on page 705.
P arametric Equations and Polar Coordinates
5E-11(pp 686-695)
1/18/06
9:31 A
5E-10(pp 622-631)
1/18/06
9:18 AM
Page 622
CHAPTER 10
By analyzing pairs of differential equations we gain insight into population cycles of predators and prey, such as the Canada lynx and snowshoe hare.
W 150
100
R 3000
R
50 2000
W
W
120
80 0 1000 2000 3
5E-09(pp 582-591)
1/17/06
6:20 PM
Page 582
CHAPTER 9
Integration enables us to calculate the force exerted by water on a dam.
Further Applications of Integration
5E-09(pp 582-591)
1/17/06
6:20 PM
Page 583
We looked at some applications of integrals in Cha
5E-05(pp 314-323)
1/17/06
3:37 PM
Page 314
CHAPTER 5
To compute an area we approximate a region by rectangles and let the number of rectangles become large. The precise area is the limit of these sums of areas of rectangles.
I ntegrals
5E-05(pp 314-323)
1
5E-02(pp 064-073)
1/17/06
1:24 PM
Page 64
CHAPTER 2
The idea of a limit is illustrated by secant lines approaching a tangent line.
L imits and Rates of Change
5E-02(pp 064-073)
1/17/06
1:25 PM
Page 65
In A Preview of Calculus (page 2) we saw how the idea
5E-FM.qk
1/19/06
11:09 AM
Page 1
CA L C U L U S
5E-Preview (pp 02-09)
1/17/06
11:44 AM
Page 2
By the time you nish this course, you will be able to explain the formation and location of rainbows, compute the force exerted by water on a dam, analyze the po