逢甲大學 111 學年度第 2 學期 工三、工四 微積分教學進度詳表
課本：Thomas’s Calculus Early Transcendentals (13th ed.)
作者：George B. Thomas, Maurice D. Weir, Joel R. Hass, Christopher Heil, Antonio Behn.
CHAPTER 8 TECHNIQUES OF INTEGRATION
§8.1
Using Basic Integration Formulas
授課內容：Table 8.1(P471)之編號 1~8、10、12、14、18 及 19(令 a  1 )。
§8.2
Integration by Parts
授課內容：Product Rule in Integral Form, Integration by Parts Formula;
Evaluating Definite Integrals by Parts, Integration by Parts Formula for Definite
Integrals;
Examples 1, 2, 3, 4, 6, 7(彈性)
習題演練：4, 6, 8, 11(變數改為 x), 23, 30, 47
自行練習：3, 5, 7, 9, 29(提示:令 u  ln x  x  eu and dx  eu du ), 71
§8.3
Trigonometric Integrals
 sin
授課內容：Integrals of the Form
Integrals of the Form
m
x cos n x dx ; Integrals of Powers of tan x and sec x ;
 sin mx sin nx dx ,  sin mx cos nx dx
and
 cos mx cos nx dx ;
Examples 1, 2, 補充  cos2 xdx , 3(改為  sin 2 x cos 2 xdx ), 5, 6, 7, 8
習題演練：3, 9, 11, 45, 56
自行練習：5, 7, 13, 51, 54
§8.4
Trigonometric Substitutions
授課內容：Trigonometric Substitutions(P489);
Examples 1, 補充 
習題演練: 1, 3, 5, 7(改成

1
1
dx , 3, 4(改為 
dx, x  2 )
2 3/2
2
(4  x )
x 4
1  t 2 dt ), 13
1
自行練習: 2, 6, 8, 14
§8.5
Integration of Rational Functions by Partial Fractions
Examples 1, 2(方法自行決定)
習題演練：1, 3
自行練習: 2, 4
§4.5
ˆ
s Rule
Indeterminate Forms and LHopital
0
ˆ
s Rule );
, Theorem 6( LHopital
0

Indeterminate Forms
, 0 and    ;

授課內容：Indeterminate Form
Indeterminate Powers
00 ,0 and 1 .
Examples 1(a)(b), 2, 4(b)(c), 5, 6, 7, 8
習題演練：8, 24, 41, 44, 53, 61
自行練習：13, 15, 25, 52, 59
§8.8
Improper Integrals
授課內容：Infinite Limits of Integration, Def: Improper Integrals of Type I;
Integrands with Vertical Asymptotes, Def: Improper Integrals of Type II;
Test for Convergence and Divergence, 彈性:Theorem 2, Theorem 3

Examples 補充  e x dx , 1(彈性), 2, 3, 4, 5.
0
習題演練：1, 4, 5, 7, 25.
自行練習：3, 6, 8, 10, 24.
CHAPTER 10
INFINITE SEQUENCES AND SERIES
本章重點：Taylor Polynomials, Taylor Series Representation for a Function, Geometric Series, the
Radius of Convergence(彈性)
§10.1
Sequences
2
授課內容：Representing Sequences; Convergence and Divergence, Definitions(P588, 589);
Calculating Limits of Sequences, Theorem1~2; (彈性進度: Theorem 3~4)
(收斂及發散的定義簡略帶過，讓學生熟悉基本的計算及數列的斂散性即可)
主要重點放在以下數列的結果:
0,
x|  | 1


l i mx  
1,
x 1
n
d i v e r g e n t , o t h e r w i s e

n
Examples 1, 2, 3(d), 4(a)(c), 5, 6, 7(彈性), 9(d)
習題演練: 4, 40
自行練習: 6, 42
§10.2
Infinite Series
授課內容：Infinite Series, Definitions(P599); Geometric Series; Combining Series, Theorem8

 ar
n 1
n 1
說明

a

,
| r | 1

  1 r
divergent, | r | 1
n
 xn  lim  x k  lim(1  x  x 2 
n=0
n 
k =0
(a) Pn ( x )  1  x  x 
2
n 
 xn ) 
1  x n1
1
x =

1 x
1 x
n
1
, x <1
1 x
當 n  且 x 1.
或
1
 Pn ( x ) 誤差趨近 0
1 x
1
1
 Pn ( x) 
 (1  x  x 2 
1 x
1 x
(b) 說明

此時定義
 xn )
1
1  x n1 x n1


 0 當 n  且 x 1.
1 x 1 x
1 x
1
 1  x  x2 
1 x
 xn 
Examples 1, 2, 4, 5(彈性), 9(a)
3
, x  1 (可稍微引入冪級數的概念)
習題演練: 11, 14
自行練習: 7, 9, 12
§10.3
The Integral Test (本節重點: p -series)
授課內容：說明 p -series

1
n
n 1
p
與瑕積分


1
1
dx 之間收斂發散的關係(畫圖說明即可)
xp
Example 3
習題演練: 1
自行練習: 2
§10.5
The Ratio and Root Tests (不講 Root Test 及 Absolute Convergence，簡表省略此標題)

1
授課內容：Theorem13-The Ratio Test; 說明  與
n 1 n

1
n
n 1
2
分別為 p -series 發散及收斂，
但無法使用 the Ratio Test 得知此結果
Examples 2(a)(b), 4(a)(b) 使用 the Ratio Test (列為考題)
習題演練: 1, 5
自行練習: 2, 3
§10.7
Power Series
授課內容：Def: Power Series (P631);
Theorem 18 (省略 absolutely 的字眼);
Def: Radius of Convergence (p634~635) (理解就好);
Theorem 21; Theorem 22
Examples 1, 3(c)(d), 4, 5, 6
習題演練：55
自行練習：56
§10.8
Taylor and Maclaurin Series
授課內容：Def: Taylor Polynomial of Order n (P642),
4
泰勒多項式(理解函數的非線性近似)
f (a)
f (a)
Pn ( x)  f (a) 
( x  a) 
( x  a) 2 
1!
2!
f ( n ) (a )

( x  a) n (可配合 GGB)
n!
Ex 求 Pn ( x) (1) f ( x)  1  x  x 2 a  0,1
(理解 n 次多項式與其 n 次泰勒多項式相等)
(2) f ( x)  e x ， a  0 .
(3) f ( x) 
1
，a 0.
1 x
(4) f ( x)  sin x ， a  0 .
(5) f  x   ln x ， a  1.
Def: Taylor and Maclaurin Series (P641)
計算上例(2)~(5)的 Taylor Series)
習題演練：1, 3, 23
自行練習：2, 4, 24
§10.9
Convergence of Taylor Series
授課內容：介紹以下幂級數跟函數的關係;Theorem 23 (P645, 彈性)
x x2
(1) e  1   
1! 2!
xn
 
n!
x
x3
(2) sin x  x  
3!
, x
( 1)n x 2 n1


(2n  1)!
.
, x .
(3)
1
 1  x  x2 
1 x
(4)
1
1
1/ 2

=
x 2  [( x  2)] 1  [( x  2) / 2]

 xn 
1 ( x  2) ( x  2)2



2
22
23
,
x 1
 ( 1)n
( x  2)n

2n1
補充: 利用已知的泰勒級數求函數的級數表示式
5
, 0  x  4. (P632)
1. Find Taylor series at 0 of the following functions.
(1) f ( x)  sin x 2
(2) f ( x)  xe x
2. Find the Taylor series representation of f  x  
1
at x  2 .
1 x
習題演練：1, 10, 12
自行練習：2, 11
CHAPTER 14
§14.1
PARTIAL DERIVATIVES
Functions of Several Variables
授課內容：Def: Functions of n Variables;
Functions of Two Variables;
Graphs, Level Curves, and Contours of Functions of Two Variables, Def: Level
Curves;
Functions of Three Variables, Def: Level Surfaces
Examples 1, 2, 3.
(參考教具網址 : http://140.134.140.60/calculus-ggb/)
習題演練：3(a), 6(find just the domain), 14, 16.
自行練習：1(c), 5(find just the domain), 13, 15.
§14.3
Partial Derivatives
授課內容：Partial Derivatives of a Function of Two Variables, Def: Partial Derivatives of a
Function of Two Variables with respect to x, Def: Partial Derivatives of a Function of
Two Variables with respect to y;
Calculations;
Functions of More than Two Variables;
Second-Order Partial Derivatives;
The Mixed Derivative Theorem, Theorem 2;
Partial Derivatives of Still Higher Order
Examples 1, 2, 5, 6, 9, 10, 11.
習題演練：2, 7, 19, 30, 31, 42, 51
自行練習：1, 6, 21, 29, 32, 44, 53.
§14.4
The Chain Rule
授課內容：Functions of Two Variables, Theorem 5;
Functions of Three Variables, Theorem 6;
6
Functions Defined on Surfaces, Theorem 7;
Implicit Differentiation Revisited, Theorem 8, Equations (2);
Functions of Many Variables
Examples 1, 2, 3, 4, 5, 6.
習題演練：2, 3, 5, 9, 17, 28, 29, 35.
自行練習：1, 6, 18, 27, 31, 34.
§14.5
Directional Derivatives and Gradient Vector
授課內容：Directional Derivatives in the Plane : Def: Directional Derivative of Two Variables;
Calculation and Gradients : Def: Gradient of a Function of Two Variables; Theorem 9;
Properties of the Directional Derivative;
Gradients and Tangent to the Level curves : Tangent Line to a Level curve;
Functions of Three Variables : Def: Gradient of a Function of Three Variables; Def:
Directional Derivative of Three Variables
Examples 1, 2, 3, 4, 6
習題演練：2, 4, 9, 12, 20, 22, 29(a)(b)
自行練習：1, 3, 7, 11, 15, 19, 21
§14.6
Tangent Planes and Differentials
授課內容：Tangent Planes and Normal Lines, Def: Tangent Plane and Normal Line, Equations
(1)~(3);
How to Linearize a Function of Two Variables, Def: Linearization and Standard Linear
Approximation;
Differentials, Def: Total Differential;
Functions of More Than Two Variables;
Examples 1, 2, 5, 8(Find just the linearization).
補充範例 1. Let f ( x, y)  2 x 2  3 y 3 .
(a) Find the Linearization L( x, y ) of f at the point (2, 1) .
(b) Use (a) to approximate the value of f (2.01, 0.98) .
(c) Find df
(d) Estimate f (2.01, 0.98)  f (2, 1) by (c).
補充範例 2. Use linear approximation to approximate the value of
(2.95) 2  7(0.98) 2 .
習題演練：2, 4, 10, 12, 26, 39.
自行練習：1, 3, 9, 29, 40.
7
§14.7
Extreme Values and Saddle Points
授課內容：Derivative Tests for Local Extreme Values, Def: Local Extrema of a Function of Two
Variables, Theorem 10, Def: Critical Point, Def: Saddle Point, Theorem 11
Examples 1, 2, 3, 4.
習題演練：2, 4, 20.
自行練習：1, 3
CHAPTER 15
§15.1
MULTIPLE INTEGRALS
Double and Iterated Integrals over Rectangles
授課內容：Double Integrals;
Double Integrals as Volumes;
Fubini’s Theorem for Calculating Double Integrals, Theorem 1
Examples 1, 2.
補充範例: If R  ( x, y ) 1  x  1, 2  y  2 , evaluate

1  x 2 dA by
R
interpreting it as the volume of a solid.
習題演練：2, 6, 10, 17, 19, 24.
自行練習：1, 4, 5, 11, 15, 23.
§15.2
Double Integrals over General Regions
授課內容：Double Integrals over Bounded, Nonrectangular Regions;
Volumes, Theorem 2;
Finding Limits of Integration, Using Vertical Cross-sections, Using Horizontal Crosssections;
Properties of Double Integrals;
Examples 1, 2(題目改成 
1
0

1
y
sin x 2 dx dy ), 3, 4(Only write out the iterated
integral)。
補充範例: Evaluate
2
4
0
2y
 
2
e x dxdy
習題演練：2, 10, 13, 19, 23, 26 , 35, 49, 54, 57.
自行練習：20, 47, 58.
§15.3
Area by Double Integration*
(*彈性進度)
8
授課內容：Areas of Bounded Regions in the Plane, Def: Area by Double Integration;
Average Value, Def: Average Value of a Function.
Examples 1, 3.
習題演練：1.
自行練習：2.
§15.4
Double Integrals in Polar Form
授課內容：Integrals in Polar Coordinates; Changing Catesian Integrals into Polar Integrals;
(簡單介紹極座標，只考慮圓,半圓或¼圓的區域描述；極座標積分變換公式)
Examples 3, 4,
[彈性補充範例:
1
 
1 x 2
1  1 x
2
1  x 2  y 2 dydx (單位球體積) ,


0
e x dx ]
習題演練：10
自行練習：9
§15.8* Substitutions in Multiple Integrals*
(*彈性進度)
授課內容：Substitutions in Double Integrals, Def: The Jacobian, Theorem 3;
Examples 2, 4.
習題演練：1, 7.
自行練習：3, 6.
9
2