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首页Thomas' Calculus 13e: 理解导向的现代微积分入门
Thomas' Calculus 13e: 理解导向的现代微积分入门
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"Thomas' Calculus 13e 是一本权威的微积分教材,由 George B. Thomas Jr. 原著,经 Maurice D. Weir、Joel Hass 和 Christopher Heil 协力修订,特别适合数学、工程学和自然科学专业的学生进行传统的三学期或四季度课程学习。该版本强调概念理解,致力于培养学生的计算基础,同时适应了现代教育趋势和今日学生的需求。 本书的核心在于其精确的解释、精心挑选的例子、高质量的插图和经过时间考验的习题集,这些构成了教材的基础。作者们不断更新内容,以反映学生准备方式的变化以及微积分在快速发展的世界中的实际应用。例如,它可能包括极限理论、导数和积分的基本概念,多元函数的导数和极值,微分方程,级数,以及泰勒级数和幂级数等内容,这些都是微积分的重要组成部分。 在编排上,Thomas' Calculus 13e 注重逻辑连贯性和实用性,通过实例和问题驱动的学习方法,帮助学生建立起对微积分概念的深刻理解和应用能力。此外,教材还可能包含许多实际问题的解决方案,如物理学、工程学、经济学中的问题,以增强学生将理论知识与实际生活和职业需求相结合的能力。 编辑团队由资深的专业人士组成,确保了教材的质量和一致性。从封面设计到软件开发,每个环节都体现出对教育质量的严谨追求。无论是初学者还是进阶学习者,都能从中受益,获得全面而系统的微积分知识,为他们的学术研究和未来职业生涯打下坚实的基础。"
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2 Chapter 1: Functions
Often a function is given by a formula that describes how to calculate the output value
from the input variable. For instance, the equation
A = pr
2
is a rule that calculates the
area A of a circle from its radius r (so r, interpreted as a length, can only be positive in this
formula). When we define a function
y = ƒ(x)
with a formula and the domain is not stated
explicitly or restricted by context, the domain is assumed to be the largest set of real
x-values for which the formula gives real y-values, which is called the natural domain. If
we want to restrict the domain in some way, we must say so. The domain of y = x
2
is the
entire set of real numbers. To restrict the domain of the function to, say, positive values of
x, we would write “y = x
2
, x 7 0.”
Changing the domain to which we apply a formula usually changes the range as well.
The range of y = x
2
is [0,
q
). The range of y = x
2
, x Ú 2, is the set of all numbers
obtained by squaring numbers greater than or equal to 2. In set notation (see Appendix 1),
the range is 5x
2
" x Ú 26 or 5y "y Ú 46 or 34,
q
).
When the range of a function is a set of real numbers, the function is said to be real-
valued. The domains and ranges of most real-valued functions of a real variable we con-
sider are intervals or combinations of intervals. The intervals may be open, closed, or half
open, and may be finite or infinite. Sometimes the range of a function is not easy to find.
A function ƒ is like a machine that produces an output value ƒ(x) in its range whenever we
feed it an input value x from its domain (Figure 1.1). The function keys on a calculator give an
example of a function as a machine. For instance, the
2x key on a calculator gives an output
value (the square root) whenever you enter a nonnegative number x and press the
2x key.
A function can also be pictured as an arrow diagram (Figure 1.2). Each arrow associates
an element of the domain D with a unique or single element in the set Y. In Figure 1.2, the
arrows indicate that ƒ(a) is associated with a, ƒ(x) is associated with x, and so on. Notice that
a function can have the same value at two different input elements in the domain (as occurs
with ƒ(a) in Figure 1.2), but each input element x is assigned a single output value ƒ(x).
EXAMPLE 1 Let’s verify the natural domains and associated ranges of some simple
functions. The domains in each case are the values of x for which the formula makes sense.
Input
(domain)
Output
(range)
xf(x)
f
FIGURE 1.1 A diagram showing a
function as a kind of machine.
x
a
f(a)
f(x)
D = domain set Y = set containing
the range
FIGURE 1.2 A function from a set D
to a set Y assigns a unique element of Y
to each element in D.
Function Domain (x) Range ( y)
y = x
2
(-
q
,
q
) 30,
q
)
y = 1>x (-
q
,0)∪ (0,
q
)(-
q
,0)∪ (0,
q
)
y = 2x
30,
q
)
30,
q
)
y = 24 - x (-
q
,4430,
q
)
y = 21 - x
2
3-1, 14
30, 14
Solution The formula y = x
2
gives a real y-value for any real number x, so the domain
is (-
q
,
q
). The range of y = x
2
is
30,
q
)
because the square of any real number is non-
negative and every nonnegative number y is the square of its own square root, y =
1
2y
2
2
for y Ú 0.
The formula y = 1>x gives a real y-value for every x except x = 0. For consistency
in the rules of arithmetic, we cannot divide any number by zero. The range of y = 1>x, the
set of reciprocals of all nonzero real numbers, is the set of all nonzero real numbers, since
y = 1>(1>y). That is, for y ≠ 0 the number x = 1>y is the input assigned to the output
value y.
The formula y = 2x gives a real y-value only if x Ú 0. The range of y = 2x is
30,
q
) because every nonnegative number is some number’s square root (namely, it is the
square root of its own square).
In y = 24 - x , the quantity 4 - x cannot be negative. That is, 4 - x Ú 0, or
x … 4. The formula gives real y-values for all x … 4. The range of 24 - x is
30,
q
),
the set of all nonnegative numbers.
![](https://csdnimg.cn/release/download_crawler_static/10655129/bg11.jpg)
1.1Functions and Their Graphs 3
The formula y = 21 - x
2
gives a real y-value for every x in the closed interval from
-1 to 1. Outside this domain,
1 - x
2
is negative and its square root is not a real number.
The values of 1 - x
2
vary from 0 to 1 on the given domain, and the square roots of these
values do the same. The range of
21 - x
2
is 30, 14.
Graphs of Functions
If ƒ is a function with domain D, its graph consists of the points in the Cartesian plane
whose coordinates are the input-output pairs for ƒ. In set notation, the graph is
5
(x,ƒ(x))
"
x
∊
D
6
.
The graph of the function
ƒ(x) = x + 2
is the set of points with coordinates (x, y) for
which y = x + 2. Its graph is the straight line sketched in Figure 1.3.
The graph of a function ƒ is a useful picture of its behavior. If (x, y) is a point on the
graph, then y = ƒ(x) is the height of the graph above (or below) the point x. The height
may be positive or negative, depending on the sign of ƒ(x) (Figure 1.4).
x
y
−20
2
y = x + 2
FIGURE 1.3 The graph of ƒ(x) = x + 2
is the set of points (x, y) for which y has the
value
x + 2.
y
x
012
x
f(x)
(x, y)
f(1)
f(2)
FIGURE 1.4 If (x, y) lies on the graph of
ƒ, then the value
y = ƒ(x) is the height of
the graph above the point x (or below x if
ƒ(x) is negative).
012−1−2
1
2
3
4
(−2, 4)
(−1, 1) (1, 1)
(2, 4)
3
2
9
4
,
x
y
y = x
2
ab
FIGURE 1.5 Graph of the function
in Example 2.
EXAMPLE 2 Graph the function y = x
2
over the interval 3-2, 24.
Solution Make a table of xy-pairs that satisfy the equation y = x
2
. Plot the points (x, y)
whose coordinates appear in the table, and draw a smooth curve (labeled with its equation)
through the plotted points (see Figure 1.5).
How do we know that the graph of y = x
2
doesn’t look like one of these curves?
xy= x
2
-24
-11
00
11
3
2
9
4
24
y = x
2
?
x
y
y = x
2
?
x
y
![](https://csdnimg.cn/release/download_crawler_static/10655129/bg12.jpg)
4 Chapter 1: Functions
To find out, we could plot more points. But how would we then connect them? The basic
question still remains: How do we know for sure what the graph looks like between the
points we plot? Calculus answers this question, as we will see in Chapter 4. Meanwhile,
we will have to settle for plotting points and connecting them as best we can.
Representing a Function Numerically
We have seen how a function may be represented algebraically by a formula (the area
function) and visually by a graph (Example 2). Another way to represent a function is
numerically, through a table of values. Numerical representations are often used by engi-
neers and experimental scientists. From an appropriate table of values, a graph of the func-
tion can be obtained using the method illustrated in Example 2, possibly with the aid of a
computer. The graph consisting of only the points in the table is called a scatterplot.
EXAMPLE 3 Musical notes are pressure waves in the air. The data associated with
Figure 1.6 give recorded pressure displacement versus time in seconds of a musical note
produced by a tuning fork. The table provides a representation of the pressure function
over time. If we first make a scatterplot and then connect approximately the data points
(t, p) from the table, we obtain the graph shown in the figure.
−0.6
−0.4
−0.2
0.2
0.4
0.6
0.8
1.0
t (sec)
p (pressure)
0.001 0.002 0.004 0.0060.003 0.005
Data
FIGURE 1.6 A smooth curve through the plotted points
gives a graph of the pressure function represented by the
accompanying tabled data (Example 3).
Time Pressure Time Pressure
0.00091 -0.080 0.00362 0.217
0.00108 0.200 0.00379 0.480
0.00125 0.480 0.00398 0.681
0.00144 0.693 0.00416 0.810
0.00162 0.816 0.00435 0.827
0.00180 0.844 0.00453 0.749
0.00198 0.771 0.00471 0.581
0.00216 0.603 0.00489 0.346
0.00234 0.368 0.00507 0.077
0.00253 0.099 0.00525 -0.164
0.00271 -0.141 0.00543 -0.320
0.00289 -0.309 0.00562 -0.354
0.00307 -0.348 0.00579 -0.248
0.00325 -0.248 0.00598 -0.035
0.00344 -0.041
The Vertical Line Test for a Function
Not every curve in the coordinate plane can be the graph of a function. A function ƒ can
have only one value ƒ(x) for each x in its domain, so no vertical line can intersect the
graph of a function more than once. If a is in the domain of the function ƒ, then the vertical
line x = a will intersect the graph of ƒ at the single point (a,ƒ(a)).
A circle cannot be the graph of a function, since some vertical lines intersect the circle
twice. The circle graphed in Figure 1.7a, however, does contain the graphs of functions of
x, such as the upper semicircle defined by the function ƒ(x) =
21 - x
2
and the lower
semicircle defined by the function g(x) =-
21 - x
2
(Figures 1.7b and 1.7c).
![](https://csdnimg.cn/release/download_crawler_static/10655129/bg13.jpg)
1.1Functions and Their Graphs 5
Piecewise-Defined Functions
Sometimes a function is described in pieces by using different formulas on different parts
of its domain. One example is the absolute value function
0
x
0
= e
x, x Ú 0
-x, x 6 0,
First formula
Second formula
whose graph is given in Figure 1.8. The right-hand side of the equation means that the
function equals x if x Ú 0, and equals -x if x 6 0. Piecewise-defined functions often
arise when real-world data are modeled. Here are some other examples.
EXAMPLE 4 The function
ƒ(x) = c
-x, x 6 0
x
2
,0… x … 1
1, x 7 1
First formula
Second formula
Third formula
is defined on the entire real line but has values given by different formulas, depending on
the position of x. The values of ƒ are given by y =-x when x 6 0, y = x
2
when
0 … x … 1, and y = 1 when x 7 1. The function, however, is just one function whose
domain is the entire set of real numbers (Figure 1.9).
EXAMPLE 5 The function whose value at any number x is the greatest integer less
than or equal to x is called the greatest integer function or the integer floor function. It
is denoted
:
x
;
. Figure 1.10 shows the graph. Observe that
:
2.4
;
= 2,
:
1.9
;
= 1,
:
0
;
= 0,
:
-1.2
;
=-2,
:
2
;
= 2,
:
0.2
;
= 0,
:
-0.3
;
=-1,
:
-2
;
=-2.
EXAMPLE 6 The function whose value at any number x is the smallest integer
greater than or equal to x is called the least integer function or the integer ceiling func-
tion. It is denoted
<
x
=
. Figure 1.11 shows the graph. For positive values of x, this function
might represent, for example, the cost of parking x hours in a parking lot that charges $1
for each hour or part of an hour.
−110
x
y
(a) x
2
+ y
2
= 1
−110
x
y
−11
0
x
y
(b) y =
"1 −
x
2
(c) y =−"1 −
x
2
FIGURE 1.7 (a) The circle is not the graph of a function; it fails the vertical line test. (b) The
upper semicircle is the graph of a function ƒ(x)
= 2
1
-
x
2
. (c) The lower semicircle is the graph
of a function g
(x) =-
21 - x
2
.
−2 −10 1 2
1
2
x
y
y =−x
y = x
2
y = 1
y = f(x)
FIGURE 1.9 To graph the
function
y = ƒ(x) shown here,
we apply different formulas to
different parts of its domain
(Example 4).
x
y =
0x 0
y =
x
y =
−x
y
−3 −2 −10 1 2 3
1
2
3
FIGURE 1.8 The absolute value
function has domain
(-
q
,
q
) and
range
30,
q
).
1
−2
2
3
−2 −1123
y = x
y = :x;
x
y
FIGURE 1.10 The graph of the
greatest integer function y
=
:
x
;
lies on or below the line
y = x, so
it provides an integer floor for x
(Example 5).
![](https://csdnimg.cn/release/download_crawler_static/10655129/bg14.jpg)
6 Chapter 1: Functions
Increasing and Decreasing Functions
If the graph of a function climbs or rises as you move from left to right, we say that the
function is increasing. If the graph descends or falls as you move from left to right, the
function is decreasing.
x
y
1−1−223
−2
−1
1
2
3
y = x
y = <x=
FIGURE 1.11 The graph
of the least integer function
y
=
<
x
=
lies on or above the line
y = x, so it provides an integer
ceiling for x (Example 6).
(a)
(b)
0
x
y
y = x
2
(x, y)
(−x, y)
0
x
y
y = x
3
(x, y)
(−x, −y)
FIGURE 1.12 (a) The graph of y = x
2
(an even function) is symmetric about the
y-axis. (b) The graph of y
= x
3
(an odd
function) is symmetric about the origin.
DEFINITIONS Let ƒ be a function defined on an interval I and let x
1
and x
2
be
any two points in I.
1. If ƒ(x
2
) 7 ƒ(x
1
) whenever x
1
6 x
2
, then ƒ is said to be increasing on I.
2. If ƒ(x
2
) 6 ƒ(x
1
) whenever x
1
6 x
2
, then ƒ is said to be decreasing on I.
It is important to realize that the definitions of increasing and decreasing functions
must be satisfied for every pair of points x
1
and x
2
in I with x
1
6 x
2
. Because we use the
inequality 6 to compare the function values, instead of
…,
it is sometimes said that ƒ is
strictly increasing or decreasing on I. The interval I may be finite (also called bounded) or
infinite (unbounded) and by definition never consists of a single point (Appendix 1).
EXAMPLE 7 The function graphed in Figure 1.9 is decreasing on (-
q
,04 and increas-
ing on 30, 14. The function is neither increasing nor decreasing on the interval
31,
q
)
because of the strict inequalities used to compare the function values in the definitions.
Even Functions and Odd Functions: Symmetry
The graphs of even and odd functions have characteristic symmetry properties.
DEFINITIONS A function
y = ƒ(x) is an
even function of x if ƒ(-x) = ƒ(x),
odd function of x if ƒ(-x) =-ƒ(x),
for every x in the function’s domain.
The names even and odd come from powers of x. If y is an even power of x, as in
y = x
2
or y = x
4
, it is an even function of x because (
-
x)
2
=
x
2
and (-x)
4
= x
4
. If y is an
odd power of x, as in y = x or y = x
3
, it is an odd function of x because (-x)
1
=-x and
(-x)
3
=-x
3
.
The graph of an even function is symmetric about the y-axis. Since ƒ(-x) = ƒ(x), a
point (x, y) lies on the graph if and only if the point (-x, y) lies on the graph (Figure 1.12a).
A reflection across the y-axis leaves the graph unchanged.
The graph of an odd function is symmetric about the origin. Since ƒ(-x) =-ƒ(x), a
point (x, y) lies on the graph if and only if the point (-x, -y) lies on the graph (Figure 1.12b).
Equivalently, a graph is symmetric about the origin if a rotation of 180° about the origin leaves the
graph unchanged. Notice that the definitions imply that both x and -x must be in the domain of ƒ.
EXAMPLE 8 Here are several functions illustrating the definition.
ƒ(x) = x
2
Even function: (
-
x)
2
=
x
2
for all x; symmetry about y-axis.
ƒ(x) = x
2
+ 1 Even function: (-x)
2
+ 1 = x
2
+ 1 for all x; symmetry about
y-axis (Figure 1.13a).
ƒ(x) = x Odd function: (-x) =-x for all x; symmetry about the origin.
ƒ(x) = x + 1 Not odd: ƒ(-x) =-x + 1, but -ƒ(x) =-x - 1. The two are not
equal.
Not even: (-x) + 1 ≠ x + 1 for all x ≠ 0 (Figure 1.13b).
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