2
Instantaneous Rate of Change:
The Derivative
Suppose that y is a function of x, say y = f (x). It is often necessary to know how sensitive
the value of y is to small changes in x.
EXAMPLE 2.1 Take, for example, y = f (x) = 625 x2 (the upper semicircle
324
this equation into two functions, f (x, y ) = 4 x2 y 2 and f (x, y ) = 4 x2 y 2 ,
representing the upper and lower hemispheres. Each of these is an example of a function
with a restricted domain: only certain values of x and y make sense (namely, thos
358
Chapter 15 Multiple Integration
d
y3
15
y
y2
y1
c
Multiple Integration
x
a
Consider a surface f (x, y ); you might temporarily think of this as representing physical
topographya hilly landscape, perhaps. What is the average height of the surface (or
a
324
this equation into two functions, f (x, y ) = 4 x2 y 2 and f (x, y ) = 4 x2 y 2 ,
representing the upper and lower hemispheres. Each of these is an example of a function
with a restricted domain: only certain values of x and y make sense (namely, thos
Calculus
Early Transcendentals
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B
Useful Formulas
Algebra
Remember that the common algebraic operations have precedences relative to each other:
for example, mulitplication and division take precedence over addition and subtraction, but
are tied with each other. In the case of ties, wor
452
Appendix A Selected Answers
1.3.6. cfw_x | x 0
2.3.7. 3
1.3.7. cfw_x | h r x h + r
1.3.8. cfw_x | x 1
A
1.3.9. cfw_x | 1/3 < x < 1/3
1.3.10. cfw_x | x 0 and x = 1
1.3.11. cfw_x | x 0 and x = 1
Selected Answers
2.3.8. 172
2.3.9. 0
2.3.10. 2
2.3.11. do
452
Appendix A Selected Answers
2.3.7. 3
1.3.6. cfw_x | x 0
1.3.7. cfw_x | h r x h + r
1.3.8. cfw_x | x 1
A
1.3.9. cfw_x | 1/3 < x < 1/3
1.3.10. cfw_x | x 0 and x = 1
1.3.11. cfw_x | x 0 and x = 1
Selected Answers
2.3.8. 172
2.3.9. 0
2.3.10. 2
2.3.11. do
426
Chapter 17 Dierential Equations
We start by considering equations in which only the rst derivative of the function appears.
17
DEFINITION 17.1 A rst order dierential equation is an equation of the form
F (t, y, y) = 0. A solution of a rst orde
426
Chapter 17 Dierential Equations
We start by considering equations in which only the rst derivative of the function appears.
17
DEFINITION 17.1 A rst order dierential equation is an equation of the form
F (t, y, y) = 0. A solution of a rst orde
17
Dierential Equations
Many physical phenomena can be modeled using the language of calculus. For example,
observational evidence suggests that the temperature of a cup of tea (or some other liquid)
in a room of constant temperature will cool over time a
16
Vector Calculus
This chapter is concerned with applying calculus in the context of vector elds. A
two-dimensional vector eld is a function f that maps each point (x, y ) in R2 to a twodimensional vector u, v , and similarly a three-dimensional vector
358
Chapter 15 Multiple Integration
d
y3
15
y
y2
y1
c
Multiple Integration
x
a
Figure 15.1
Consider a surface f (x, y ); you might temporarily think of this as representing physical
topographya hilly landscape, perhaps. What is the average height of the
15
Multiple Integration
Consider a surface f (x, y ); you might temporarily think of this as representing physical
topographya hilly landscape, perhaps. What is the average height of the surface (or
average altitude of the landscape) over some region?
A
14
Partial Dierentiation
In single-variable calculus we were concerned with functions that map the real numbers R
to R, sometimes called real functions of one variable, meaning the input is a single real
number and the output is likewise a single re
306
Chapter 13 Vector Functions
13
Vector Functions
Figure 13.1
We have already seen that a convenient way to describe a line in three dimensions is to
provide a vector that points to every point on the line as a parameter t varies, like
1, 2, 3 + t 1,
306
Chapter 13 Vector Functions
13
Vector Functions
Figure 13.1
We have already seen that a convenient way to describe a line in three dimensions is to
provide a vector that points to every point on the line as a parameter t varies, like
1, 2, 3 + t 1,
13
Vector Functions
We have already seen that a convenient way to describe a line in three dimensions is to
provide a vector that points to every point on the line as a parameter t varies, like
1, 2, 3 + t 1, 2, 2 = 1 + t, 2 2t, 3 + 2t .
Except that thi
276
Chapter 12 Three Dimensions
You must then imagine that the z axis is perpendicular to the other two. Just as we have
investigated functions of the form y = f (x) in two dimensions, we will investigate three
dimensions largely by considering functions;
276
Chapter 12 Three Dimensions
You must then imagine that the z axis is perpendicular to the other two. Just as we have
investigated functions of the form y = f (x) in two dimensions, we will investigate three
dimensions largely by considering functions;
12
Three Dimensions
So far we have been investigating functions of the form y = f (x), with one independent and
one dependent variable. Such functions can be represented in two dimensions, using two
numerical axes that allow us to identify every point i
234
Chapter 11 Sequences and Series
and then
1
= 1 0 = 1.
2i
There is one place that you have long accepted this notion of innite sum without really
thinking of it as a sum:
lim 1
i
11
0.3333 =
3
Sequences and Series
3
3
3
1
3
+
+
+
+ = ,
10 100 1000 100
234
Chapter 11 Sequences and Series
and then
1
= 1 0 = 1.
2i
There is one place that you have long accepted this notion of innite sum without really
thinking of it as a sum:
lim 1
i
11
0.3333 =
3
Sequences and Series
3
3
3
3
1
+
+
+
+ = ,
10 100 1000 100
11
Sequences and Series
Consider the following sum:
111
1
1
+
+ i +
2 4 8 16
2
The dots at the end indicate that the sum goes on forever. Does this make sense? Can
we assign a numerical value to an innite sum? While at rst it may seem dicult or
impossible
218
Just as we describe curves in the plane using equations involving x and y , so can we
describe curves using equations involving r and . Most common are equations of the form
r = f ( ).
10
EXAMPLE 10.1 Graph the curve given by r = 2. All points with r
218
Just as we describe curves in the plane using equations involving x and y , so can we
describe curves using equations involving r and . Most common are equations of the form
r = f ( ).
10
EXAMPLE 10.1 Graph the curve given by r = 2. All points with r
10
Polar Coordinates,
Parametric Equations
Coordinate systems are tools that let us use algebraic methods to understand geometry.
While the rectangular (also called Cartesian) coordinates that we have been using are
the most common, some problems are ea