微積分 (二) ★★ 期末考試會考進度：

微積分 (二)

★★ 期末考試會考進度：

1. 數列、極限；級數、級數和(Sequence and its limit, series and its sum) 2. 正項級數收斂或發散之檢驗法(Test for the convergence or divergence of

the series with positive terms)

3. 交錯級數、冪級數、泰勒多項式(Alternative series, power series Taylor polynomial)

4. 馬克勞林級數、泰勒級數、泰勒定理(Maclaurin series, Taylor series, Taylor theorem)

5. 多變數函數、偏導函數(Multi variables, Partial derivative) 6. 梯度、方向導數及切平面方程式。

7. 雙變數的極值(Relative extrema for two variables) 8. 拉格朗治乘子(Lagrange multipliers)

9. 重積分(Multiple integration)

九十七學年度微積分(二)期末會考題型

★共出十題

1. 無窮級數和(參考：附件-期末會考參考題庫之第一題)

2. 無窮級數收斂發散之判斷(參考：附件-期末會考參考題庫之第二題) 3. 泰勒及馬克勞林級數 (參考：附件-期末會考參考題庫之第三題) 4. 偏微分 (參考：附件-期末會考參考題庫之第四題)

5. 梯度、方向導數及切平面方程式(參考：附件-期末會考參考題庫之第五

題)

6. 偏微分---隱函數 (參考：附件-期末會考參考題庫之第六題) 7. 極値與鞍點 (參考：附件-期末會考參考題庫之第七題)

8. 極値--拉格朗治乘子法(參考：附件-期末會考參考題庫之第八題) 9. 二重積分 (參考：附件-期末會考參考題庫之第九題)

10. 二重積分---積分次序交換(參考：附件-期末會考參考題庫之第十題)

九十七學年度微積分(二)期中會考參考題庫

一、 無窮級數和

1. Find the sum (if it exists) (a)

∑

^∞

= −

2

5 1

2

k k

k

(b)

∑

^∞

=

⋅ −

+

⋅

1

1

4 3 2 2 3

k

k k k

2. Find the sum (if it exists) (a)

∑

^∞

= +

1

3 1

2

k k

k

(b)

∑

^∞

=

⋅ −

−

⋅

1

1

6 3 2 4 3

k

k k k

3. Find the sum (if it exists)

∑

^∞

=₁(2 −1)(2 +1) 6

k

k k

4. Find the sum (if it exists)

∑

^∞

=₁(3 −1)(3 +2) 9

k

k k

5. Find the sum (if it exists)

∑

^∞

=₁ ( +2) 1

k

k k

6. Find the sum (if it exists)

4

3 2 1 3 )1 (

25 2 5 1 2 )1 ( 13

11 10 1 9 4 )5 (

L

L L

+ + +

+ + + +

+ + +

c

b a

7. Find the sum (if it exists)

2 1 3 1 2 1 1 ) ( 125

18 25

6 5 2 3 ) 2

(

a

− + − +L

b

+ + + +L

二、無窮級數收斂發散之判斷

8. Determine convergence or divergence of each series。

(a)

∑

^∞

= +

+

16 1

10

k

k

k

(b)

∑

^∞

=

+

1 3

1 2

k k

k

9. Determine convergence or divergence of each series。

(a)

∑

^∞

= −

+

1

3 1

4 3 2

n

n

n

(b)

∑

^∞

= −

+

1

2) 3

3 (4

k

k

k k

10. Determine convergence or divergence of each series。

(a)

∑

^∞

=1 ! 5

k k

k (b)

∑

^∞

=13 5

1

k

k

11. Determine convergence or divergence of each series。

(a)

∑

^∞

= − −

1 2 1

) 1 1 (

n

n

n

(b)

∑

^∞

= +

−

1 5 4

1 6

2 3

n n

n

12. Given

∑

^∞

=

− +

0 3

) 2 ) (

1 (

n

n n

n x

, find the radius and interval of convergence

13. Find the series’ radius and interval of convergence (a)

∑

^∞

=1 n

n

n

x (b)

∑

^∞

=1 2 n

n

n x

三、泰勒及馬克勞林級數

14. Find the 4th Taylor polynomial for

f

(

x

)=ln

x

, at

x

=1, (b) Using the result of (a) to approximate ln(1.2), (c) estimate its error.

15. (a) Find Maclaurin series for

x e

^x，(b) find the third Maclaurin polynomial for

x2

e

⁻ (c) Using the result of (b) to find the approximate value of

∫

f

( )=

1 − 0

2dx

e ^x

16. Assume that 1 ( 1) , as 1

1 1

0

2 − = − <

+

−

+ =

∑

^∞

=

x x

x

x x

_k

k

L k Find (a) Maclaurin

series for tan⁻¹

x ∑

^∞

= − +

0 2 1

) 1 1 ( ) (

k

k

b k

17. Assume that 1 ( 1) , as 1

1 1

0

2 − = − <

+

−

+ =

∑

^∞

=

x x

x

x x

_k

k

L k Find (a) Maclaurin

series for ln(1+

x

)

∑

^∞

=

− + 1

1 1 ) 1 ( ) (

k

k

b k

18. We know that

∑

^∞

=

+

− +

=

0

1 2

)!

1 2 ) ( 1 ( sin

k

k k

k

x x , try to find Maclaurin series for cos3x

四、偏微分

19. f(x,y)= x²y² −3x+2y，thenfind f_x(−2，3，) f_y(−2，3)

20.

x y

y xy x

g

( , )= − ，find the value of

2) (2,

and 2)

(2, ∂ −

∂

−

∂

∂

y g x

g

.

21. Find

y g x

g

∂

∂

∂

∂ and for

g

(

x

,

y

)=ln(

x

³ +

y

²) .

22. Find

y z x z

∂

∂

∂

∂ , for

z

=

e

^xsin

xy

23. If

f

(

x

,

y

)=

x

³

e

^y −6

y

ln

x

, find all the second partial derivatives of

f

(

x

,

y

) 24. . If

g

(

x

,

y

)=cos(

x

−2

y

), find all the second partial derivatives of

g

(

x

,

y

)

五、梯度、方向導數及切平面方程式

25. Find ∇

f

at the given point.

f

(

x

)=

x

² +

y

² −2

z

²+

z

ln

x

, (1,1,1)

26. Find ∇

f

at the given point.

f

(

x

)=

x

²+

y

²+

z

² +ln(

xyz

), (−1 ,2,−2)

27. Find the derivative of 1 ,2, 1) in the

direction of

j k

( point at the , 3 2 )

, ,

(

x y z

=

x

² +

y

² −

z

²

P

− −

f

i

u

v=2v+2v−2r

28. Find the derivative of

f

(

x

,

y

,

z

)=

xy

+

yz

+

zx

, at thepoint

P

(1 ,−1,2) in the direction of

u

v=3

i

v+6v

j

−2

k

r

29. Find the tangent plane and normal line to the surface

f

(

x

,

y

,

z

)=

xy

−

yz

−

zx

−7=0 )

1 3, , 1 ( point

at the

P

−

30. Find the tangent plane and normal line to the surface 4 0 )

1

3

2 ^{2 2}

2−

xy

−

y

+

z

− =

x

,

1 , 2 ( point at the

P

− 六、偏微分---隱函數

31. If

x

³ −2cos

xy

+

y

² +7=0，find

dx dy

32. If 4

x

²

y

³ +7

x

³

y

⁴ −

x

⁴ +3

y

² −17=0，find

dx dy

33.

y

z x

z z xy z

y

x

∂

∂

∂

= ∂

+ 3 find and

If ^{2 3 2} ，

34. If

y z x xz z

z y y

x

∂

∂

∂

= ∂

−

−

+ ^{2 2} 6 0, find ,

2

七、極値與鞍點

35. Given. 2 4 , find the relative extrema and saddle

point, if it exists.

2 )

,

(

x y

=

x

² −

xy

+

y

² +

x

+

y

−

f

36. Given. 3 4 , find the relative extrema and saddle

point, if it exists

2 2

) ,

(

x y

=

xy

−

x

² −

y

² +

x

+

f

37. Given 9 7, find the relative extrema and saddle

point, if it exists.

3 3 )

,

(x y =x³ +y³ − x² − y² − x+ f

38. Given f(x,y)=x³ −y³−2xy+6, find the relative extrema and saddle point, if it

exists.

39. Given f(x,y)=4xy−x⁴ −y⁴, find the relative extrema and saddle point, if it exists 八、極値--拉格朗治乘子法

40. Find all points on the circle

x

² + y² =18 at which the product

xy

is a maximum.

41. Using Lagrange multipliers to find all points on the 8 at which the product

4

x

² + y² =

xy

is a minimum

42. Use Lagrange multipliers to minimize ^{2 2} with the constraint

that 4

4 2

) , ,

(

x y z x y z

f

= + +

2

x

−

y

+

z

=

43. Using Lagrange multipliers to maximize

f

(

x

,

y

,

z

)=

xy

+

yz

+

xz

with the constraint that

x

+

y

+

z

=1

九、二重積分

44. Find ² x²ydydx

0

4 1 6

∫ ∫

45. Evaluate ³ (2x y)dxdy

1

3

0 +

∫ ∫

46. Evaluate

∫∫

R

xy

cos

ydA

,

R

: −1≤

x

≤1, 0≤

y

≤π

47. Evaluate

∫∫

R ₊

dA x

xy

2 1

3

,

R

: 0≤

x

≤1, 0≤

y

≤2

48. If

R

={(

x

,

y

)

y

³ ≤

x

≤

y

²,0≤

y

≤1}，

f

(

x

,

y

)=2

x

+3

y

，find

∫∫

R

f

(

x

,

y

)

dA

49. If ,2 }

4 0 1

) ,

{(

x y x x y x

R

= ≤ ≤ ≤ ≤ ，

f

(

x

,

y

)=12

x

²

y

，find

∫∫

R

f

(

x

,

y

)

dA

十、二重積分---積分次序交換

50. Evaluate

dydx y

y

∫ ∫

0^π x^π

sin

51. Evaluate

e dxdy

y x

∫ ∫

0^{2 −}y 4 2

52. Evaluate dydx y xe

x y

∫ ∫

0^{2 −} ₋ 4 0

2 2

4 53. Evaluate

∫ ∫

0⁸ ₊

2

3 4

^y

x

1

dxdy