章鱼世界高等数学-微积分第1章(英文讲稿)

2020年11月26日发(作者：谭以新)
高等数学- Calculus微积分（双语讲稿）Chapter 1 Functions and Models
Calculus (Fifth Edition)
高等数学- Calculus微积分（双语讲稿）
Chapter 1 Functions and Models
1.1 Four ways to represent a function
1.1.1 ☆Definition(1-1) function: A function f is a rule that assigns to each element x in a set A
exactly one element, called f(x), in a set B. see Fig.2 and Fig.3
Conceptions: domain; range (See fig. 6 p13); independent variable; dependent variable.
Four possible ways to represent a function: 1)Verbally语言描述(by a description in words); 2)
Numerically数据表述 (by a table of values); 3) Visually 视觉图形描述(by a graph);
4)Algebraically 代数描述(by an explicit formula).

1.1.2 A question about a Curve represent a function and can’t represent a function
The way ( The vertical line test ) : A curve in the xy-plane is the graph of a function of x if and
only if no vertical line intersects the curve more than once. See Fig.17 p 17

1.1.3 ☆Piecewise defined functions (分段定义的函数)
Example7 (P18)
1－x if x ≤1
f(x)＝﹛
x
2
if x＞1
Evaluate f(0)，f(1)，f(2) and sketch the graph.
Solution：
1.1.4 About absolute value (分段定义的函数)
⑴∣x∣≥0； ⑵∣x∣≤0

Example8 (P19)
Sketch the graph of the absolute value function f(x)＝∣x∣.
Solution：

1.1.5☆☆ Symmetry ，(对称) Even functions and Odd functions (偶函数和奇函数)
⑴ Symmetry See Fig.23 and Fig.24
⑵ ①Even functions: If a function f satisfies f(－x)＝f(x) for every number x in its domain，then f
2
22
is call an even function. Example f(x)＝x is even function because: f(－x)＝ (－x)＝x＝ f(x)
②Odd functions: If a function f satisfies f(－x)＝－f(x) for every number x in its domain，then
f is call an odd function. Example f(x)＝x
3
is even function because: f(－x)＝(－x)
3
＝－x
3
＝－ f(x)
③Neither even nor odd functions:

1.1.6☆☆ Increasing and decreasing function (增函数和减函数)
⑴Definition(1-2) increasing and decreasing function: ① A function f is called increasing on an
interval I if f(x
1
)＜f(x
2
) whenever x
1
＜x
2
in I. ①A function f is called decreasing on an interval I if
f(x
1
)＞f(x
2
) whenever x
1
＜x
2
in I.
See Fig.26. and Fig.27. p21

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高等数学- Calculus微积分（双语讲稿）Chapter 1 Functions and Models
1.2 Mathematical models: a catalog of essential functions p25
1.2.1 A mathematical model p25
A mathematical model is a mathematical description of a real-world phenomenon such as the size of
a population, the demand for a product, the speed of a falling object, the concentration of a product
in a chemical reaction, the life expectancy of a person at birth, or the cost of emission reduction.
1.2.2 Linear models and Linear function P26
1.2.3 Polynomial P27
A function f is called a polynomial if
P(x) ＝ a
n
x< br>n
＋a
n
－
1
x
n
－
1
＋ …＋a
2
x
2
＋a
1
x＋a
0

Where n is a nonnegative integer and the numbers a
0
，a
1
，a
2
，…，a
n< br>－
1
， a
n
are constants called the
coefficients of the polynomial. The domain of any polynomial is R＝(－∞,＋∞).if the leading
coefficient a
n
≠0, then the degree of the polynomial is n. For example, the function
P(x) ＝ 5x
6
＋2x
5
－x
4
＋3
x
－9
⑴ Quadratic function example: P(x) ＝ 5x
2
＋2x－3 二次函数（方程）
⑵ Cubic function example: P(x) ＝ 6x
3
＋3x
2
－1 三次函数（方程）
1.2.4 Power functions 幂函数 P30
A function of the form f(x) ＝x
a
，Where a is a constant, is called a power function. We consider
several cases：
⑴ a＝n where n is a positive integer ， (n＝ 1，2，3，…，)
⑵ a＝1/n where n is a positive integer， (n＝ 1，2，3，…，) The function f(x) ＝ x
1/n

⑶ a＝
n
－1
the graph of the reciprocal function f(x) ＝ x
－1
反比函数
1.2.5 Rational function 有理函数P 32
A rational function f is a ratio of two polynomials:
f(x) ＝P(x) /Q(x)
1.2.6 Algebraic function 代数函数P32
A function f is called algebraic function if it can be constructed using algebraic operations ( such as
addition，subtraction，mult iplication，division，and taking roots) starting with polynomials. Any
rational function is automatically an algebraic function. Examples: P 32
1.2.7 Trigonometric functions 三角函数P33
⑴ f(x) ＝sin x
⑵ f(x) ＝cos x

⑶ f(x) ＝tan x＝sin x / cos x
1.2.8 Exponential function 指数函数P34
The exponential functions are the functions the form f(x) ＝ a
x
Where the base a is a positive
constant.
1.2.9 Transcendental functions 超越函数P35
These are functions that are not a algebraic. The set of transcendental functions includes the
trigonometric，inverse trigonometric，exponential，and logarithmic functions，but it also includes a
vast number of other functions that have never been named. In Chapter 11 we will study
transcendental functions that are defined as sums of infinite series.

1.2 Exercises P 35-38



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高等数学- Calculus微积分（双语讲稿）Chapter 1 Functions and Models
1.3 New functions from old functions
1.3.1 Transformations of functions P38
⑴ Vertical and Horizontal shifts (See Fig.1 p39)
①
y
＝f(x)＋c，（c＞0）shift the graph of
y
＝f(x) a distance c units upward.
②
y
＝f(x)－c，（c＞0）shift the graph of
y
＝f(x) a distance c units downward.
③
y
＝f(x＋c)，（c＞0）shift the graph of
y
＝f(x) a distance c units to the left.
④
y
＝f(x－c)，（c＞0）shift the graph of
y
＝f(x) a distance c units to the right.

⑵ Vertical and Horizontal Stretching and Reflecting (See Fig.2 p39)
①
y
＝c f(x)， （c＞1）stretch the graph of
y
＝f(x) vertically by a factor of c
②
y
＝(1/c) f(x)，（c＞1）compress the graph of
y
＝f(x) vertically by a factor of c
③
y
＝f(x/c)， （c＞1）stretch the graph of
y
＝f(x) horizontally by a factor of c.
④
y
＝f(c x)， （c＞1）compress the graph of
y
＝f(x) horizontally by a factor of c.
⑤
y
＝－f(x)， reflect the graph of
y
＝f(x) about the x-axis
⑥
y
＝f(－x)， reflect the graph of
y
＝f(x) about the y-axis
Examples1: (See Fig.3 p39)
y
＝f( x) ＝cosx，
y
＝f( x) ＝2cosx，
y
＝f( x) ＝（1/2）cosx，
y
＝f( x) ＝cos（x/2），
y
＝f( x) ＝cos2x
Examples2: (See Fig.4 p40)
Given the graph
y
＝f( x) ＝( x)
1/2
，use transformations to graph
y
＝f( x) ＝( x)
1/2
－2，
y
＝f( x)
＝(x－2)
1/2
，
y
＝f( x) ＝－( x)
1/2
，
y
＝f( x) ＝2 ( x)
1/2
，
y
＝f( x) ＝(－x)
1/2


1.3.2 Combinations of functions (代数组合函数)P42
Algebra of functions: Two functions (or more) f and g through the way such as add, subtract,
multiply and divide to combined a new function called Combination of function.
☆Definition(1-2) Combination function : Let f and g be functions with domains A and B. The
functions f ± g，f g and f / g are defined as follows: （特别注意符号(f ± g)( x) 定义的含义）
① (f ± g)( x)＝f(x) ± g( x)， domain ＝ A ∩ B
② (f g)( x)＝f(x) g( x)， domain ＝ A ∩ B
③ (f / g)( x)＝f(x) /g( x)， domain ＝ A ∩ B and g( x)≠0
Example 6 If

f( x) ＝( x)
1/2
，and
g
( x)＝( 4－x
2
)
1/2
，find functions
y
＝f(x)＋g( x)，
y
＝f(x)－g( x)，
y
＝f(x)g( x)，and
y
＝f(x) / g( x)
Solution: The domain of f( x) ＝( x)
1/2
is [0，＋∞ )，The domain of g( x) ＝( 4－x
2
)
1/2
is interval
[－2，2]，The intersection of the domains of f(x) and g( x) is
[0，＋∞ )∩[－2，2]＝[0，2]
Thus，according to the definitions, we have
(f＋g)( x)＝( x)
1/2
＋( 4－x
2
)
1/2
， domain [0，2]
(f－g)( x)＝( x)
1/2
－( 4－x
2
)
1/2
， domain [0，2]
(f g)( x)＝f(x) g( x) ＝( x)
1/2
( 4－x
2
)
1/2
＝ ( 4 x－x
3
)
1/2
domain [0，2]
(f / g)( x)＝f(x)/g( x)＝( x)
1/2
/( 4－x
2
)
1/2
＝[ x/( 4－x
2
)]
1/2
domain [0，2)

1.3.3 ☆☆Composition of functions (复合函数)P45
☆Definition(1-3) Composition function : Given two functions f and g the composite function f⊙g
(also called the composition of f and g ) is defined by
(f⊙g)( x)＝f( g( x)) （特别注意符号(f⊙g)( x) 定义的含义）

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高等数学- Calculus微积分（双语讲稿）Chapter 1 Functions and Models
The domain of f⊙g is the set of all x in the domain of g such that g(x) is in the domain of f . In
other words, (f⊙g)(x) is defined whenever both g(x) and f (g (x)) are defined. See Fig.13 p 44

Example7 If f (g) ＝( g)
1/2
and
g
(x)＝( 4－x
3
)
1/2
find composite functions f⊙g and g⊙f
Solution We have
(f⊙g)(x) ＝f (g (x) ) ＝( g)
1/2
＝(( 4－x
3
)
1/2
)
1/2

(g⊙f)(x) ＝ g (f (x) ) ＝( 4－x
3
)
1/2
＝[ 4－((x)
1/2
)
3
]
1/2
＝[4－(x)
3/2
]
1/2


Example8 If f (x) ＝( x)
1/2
and
g
(x)＝( 2－x)
1/2
find composite functions
① f⊙g ② g⊙f

③ f⊙f ④ g⊙g
Solution We have
① f⊙g＝(f⊙g)(x)＝f (g (x) )＝f(( 2－x)
1/2
)＝(( 2－x)
1/2
)
1/2
＝( 2－x)
1/4

The domain of (f⊙g)(x) is 2－x≥0 that is
x
≤2 {
x
︳
x
≤2 }＝(－∞，2]
② g⊙f＝(g⊙f)(x) ＝g (f (x) )＝g (( x)
1/2
) ＝( 2－( x)
1/2
)
1/2

The domain of (g⊙f)(x) is x≥0 and 2－( x)
1/2
x
≥0 ，that is ( x)
1/2
≤2 ，or x ≤ 4 ，so the
domain of g⊙f is the closed interval [0，4]
③ f⊙f＝(f⊙f)(x)＝f (f(x) )＝f((x)
1/2
)＝((x )
1/2
)
1/2
＝(x)
1/4

The domain of (f⊙f)(x) is [0，∞)
④ g⊙g＝(g⊙g)(x) ＝g (g(x) )＝g (( 2－x)
1/2
) ＝( 2－( 2－x)
1/2
)
1/2

The domain of (g⊙g)(x) is x－2≥0 and 2－( 2－x)
1/2

≥0 ，that is x ≤2 and x ≥－2，
so the domain of g⊙g is the closed interval [－2，2]
Notice: g⊙f⊙
h
＝f (g(
h
(x)))

Example9

Example10 Given F (x)＝cos
2
( x＋9)，find functions f，g，and h such that F (x)＝f⊙g⊙h
Solution Since F (x)＝[cos ( x＋9)]
2
，that is h (x)＝x＋9 g(x)＝cosx f (x)＝x
2


Exercise P 45-48

1.4 Graphing calculators and computers P48

1.5 Exponential functions
⑴An exponential function is a function of the form
f (x)＝a
x
See Fig.3 P56 and Fig.4

Exponential functions increasing and decreasing (单调性讨论)

⑵ Lows of exponents If a and b are positive numbers and x and y are any real numbers. Then
1) a
x+y
＝ a
x
a
y
2) a
x
－
y
＝ a
x
/ a
y
3) (a
x
)
y
＝ a
xy
4) (ab)
x+y
＝ a
x
b
x

⑶ about the number e f (x)＝e
x
See Fig. 14，15 P61
Some of the formulas of calculus will be greatly simplified if we choose the base a .
Exercises P 62-63


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