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.
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
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
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
The collection of wind velocity vectors at any given time during a tornado is an example of a vector eld.
V ector Calculus
In this chapter we study the
We can visualize a function of two variables by its graph (a surface) or by its level curves (a contour map).
P artial Derivatives
So far we have dealt wi
The volume of a sphere is the limit of sums of volumes of approximating cylinders.
A pplications of Integration
In this chapter we explore some of the applicat
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
ln(ln(x) x 0 = 0 . So, use LHospitals rule.
ln(ln(x) = lim x x x lim
d (ln(ln(x) dx d (x) dx
= lim 30. limx0
0 =0 1
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
The charge in an electric circuit is governed by the differential equations that we solve in Section 18.3.
S econd-Order Differential Equations
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
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,
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
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
By analyzing pairs of differential equations we gain insight into population cycles of predators and prey, such as the Canada lynx and snowshoe hare.
80 0 1000 2000 3
Integration enables us to calculate the force exerted by water on a dam.
Further Applications of Integration
We looked at some applications of integrals in Cha
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.
The idea of a limit is illustrated by secant lines approaching a tangent line.
L imits and Rates of Change
In A Preview of Calculus (page 2) we saw how the idea
CA L C U L U S
5E-Preview (pp 02-09)
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