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
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:
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:
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, p
Calculus
Early Transcendentals
This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike License. To
view a copy of this license, visit http:/creativecommons.org/licenses/b
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,
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
Se
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
Se
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 equatio
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 equatio
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)
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
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 hill
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
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 si
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
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
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
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 investi
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 investi
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
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
Seq
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
Seq
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 in
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
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
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 ha