16
INTEGRATION IN
VECTOR FIELDS
OVERVIEW In this chapter we extend the theory of integration to curves and surfaces in
space. The resulting theory ofline and surface integrals gives powerful mathematical tools
for science and engineering. Line integrals a
1
FUNCTIONS
OVERVIEW Functions are fundamental to the study of calculus. In this chapter we review
what functions are and how they are pictured as graphs, how they are combined and transformed, and ways they can be classified. We review the trigonometric
10
INFINITE SEQUENCES
AND SERIES
OVERVIEW Everyone knows how to add two numbers together, or even several. But how
do you add infinitely many numbers together? In this chapter we answer this question,
which is part of the theory of infinite sequences and
8
TECHNIQUES OF
INTEGRATION
OVERVIEW The Fundamental Theorem tells us how to evaluate a definite integral once we
have an antiderivative for the integrand function. Table 8.1 summarizes the forms of antiderivatives for many of the functions we have studie
4
ApPLICATIONS OF
DERIVATIVES
OVERVIEW In this chapter we use derivatives to find extreme values of functions, to
determine and analyze the shapes of graphs, and to find numerically where a function
equals zero. We also introduce the idea of recovering a
5
INTEGRATION
OVERVIEW A great achievement of classical geometry was obtaining formulas for the
areas and volumes of triangles, spheres, and cones. In this chapter we develop a method to
calculate the areas and volumes of very general shapes. This method,
2
LIMITS AND CONTINUITY
OVERVIEW Mathematicians of the seventeenth century were keenly interested in the study
of motion for objects on or near the earth and the motion of planets and stars. This study
involved both the speed of the object and its directi
CONTENTS
1
Preface
ix
I Functions
1
1.1
1.2
1.3
1.4
Functions and Their Graphs
I
CombiJring Functions; Shifting and Scaling Graphs
Trigonometric Functions 22
Graphing with Calcolators and Computers
30
QuEsTIONS TO GUIDE YOUR REVIEW
PRACTICE EXERCISES
34
3
MATH2822 Mathematical Methods for Actuarial Science II
Assignment 4
Due date: 8 April, 2015
1. In the following exercises, sketch the region of integration and write an equivalent integral
with the order of integration reversed.
4
(a)
4y
0
3/2
f (x, y) d
MATH2822 Mathematical Methods for Actuarial Science II
Assignment 1
Due date: 6 February, 2015
1. Show that for every real number k, the plane
(x 2y + z + 3) + k(2x y z + 1) = 0
contains the line of intersection of the planes
x 2y + z + 3 = 0 and 2x y z +
12
VECTORS AND THE
GEOMETRY OF SPACE
OVERVIEW To apply calculus in many real-world situations and in higher mathematics,
we need a mathematical description of three-dimensional space. In this chapter we introduce three-dimensional coordinate systems and v
9
FIRST-ORDER
DIFFERENTIAL EQUATIONS
OVERVIEW In Section 4.7 we introduced differential equations of the form dy/dx = f(x),
where f is given and y is an unknown function of x. When f is continuous over some interval, we found the general solution y(x) by
3
DIFFERENTIATION
OVERVIEW In the beginning of Chapter 2 we discussed how to determine the slope of a
curve at a point and how to measure the rate at which a function changes. Now that we
have studied limits, we can define these ideas precisely and see th
13
VECTOR-VALUED
FUNCTIONS AND MOTION
IN SPACE
OVERVIEW Now that we have learned about vectors and the geometry of space, we can
combine these ideas with our earlier study of functions. In this chapter we introduce the
calculus of vector-valued functions.
ApPENDICES
Real Numbers and the Real Line
A.1
This section reviews real numbers, inequalities, intervals, and absolute values.
Real Numbers
Much of calculus is based on properties of the real number system. Real numbers are
numbers that can be expressed a
SECOND-ORDER
DIFFERENTIAL EQUATIONS
OVERVIEW In this chapter we extend oUI study of differential equations to those of second
order. Second-order differential equations arise in many applications in the sciences and
engineering. For instance, they can be
7
TRANSCENDENTAL
FUNCTIONS
OVERVIEW Functions can be classified into two broad complementary groups called
algebraic functions and transcendental functions (see Section 1.1). Except for the trigonometric functions, our studies so far have concentrated on
15
MULTIPLE INTEGRALS
OVERVIEW In this chapter we consider the integral of a function of two variables f(x, y)
over a region in the plane and the integral of a function of three variables f(x, y, z) over
a region in space. These multiple integrals are def
14
PARTIAL DERIVATIVES
OVERVIEW Many functions depend on more than one independent variable. For instance,
the volume of a right circular cylinder is a function V = 7Tr 2h of its radius and its height,
so it is a function V(r, h) of two variables rand h.
11
PARAMETRIC EQUATIONS
AND POLAR COORDINATES
OVERVIEW In this chapter we study new ways to define curves in the plane. Instead of
thinking of a curve as the graph of a function or equation, we consider a more general way
of thinking of a curve as the pat
6
ApPLICATIONS OF
DEFINITE INTEGRALS
OVERVIEW In Chapter 5 we saw that a continuous function over a closed interval has a
definite integral, which is the limit of any Riemann sum for the function. We proved that
we could evaluate definite integrals using
MATH1813 Mathematical Methods for Actuarial Science
MATH2822 Mathematical Methods for Actuarial Science II
Assignment 3
Due date: 17 March, 2015
1. Show that the only possible maxima and minima of z on the surface z = x3 + y 3 9xy + 27
occur at (0, 0) and
MATH2822 Mathematical Methods for Actuarial Science II
Assignment 3
Due date: 17 March, 2015
1. Show that the only possible maxima and minima of z on the surface z = x3 + y 3 9xy + 27
occur at (0, 0) and (3, 3). Show that neither a maximum nor a minimum o
MATH2822 Mathematical Methods for Actuarial Science II
Brief Solution to Assignment 1
1. Show that for every real number k, the plane
(x 2y + z + 3) + k(2x y z + 1) = 0
contains the line of intersection of the planes
x 2y + z + 3 = 0 and 2x y z + 1 = 0.
S
2014-15 Second Semester
MATH2822 Mathematical Methods for Actuarial Science II
Assignment 2
Solution outline
1. Since
1 = lim
x0
sin x
sin x
= lim
= 1,
+ |x|
x0
|x|
sin x
does not exist. Therefore neither does
x0 |x|
continuous at the origin.
lim
lim
2. (
2014-15 Second Semester
MATH2822 Mathematical Methods for Actuarial Science II
Assignment 1
Solution outline
1. Let P (x0 , y0 , z0 ) be a point in the intersection of the planes x 2y + z + 3 = 0 and
2x y z + 1 = 0. Then x0 2y0 + z0 + 3 + k(2x0 y0 z0 + 1)
MATH2822 Mathematical Methods for Actuarial Science II
Supplementary Notes 1
Example 1: Let
Show that
2
2
xy(x y )
f (x, y) =
x2 + y 2
0
if (x, y) = (0, 0),
if (x, y) = (0, 0).
2f
2f
(0, 0) =
(0, 0).
yx
xy
Solution: Note that
f
f
(0, k)
(0, 0)
f
x
x
MATH2822 Mathematical Methods for Actuarial Science II
Supplementary Notes 2
Example: (Directional derivative may not be given by dot product)
Let
2x|y|
if (x, y) = (0, 0),
f (x, y) =
x2 + y 2
0
if (x, y) = (0, 0).
1. The directional derivative of f e
MATH2822 Mathematical Methods for Actuarial Science II
Class Exercise 2
1. By considering dierent paths of approach, show that the following functions have no limit
as (x, y) (0, 0).
(a) f (x, y) =
x
(b) f (x, y) =
x2 + y 2
x4
x4 + y 2
Solution.
(a)
x
=