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≤ 3 for all values of x. How large can f(2) possibly be ?

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中原大學

109

學年度 上學期

下學期 考試命題紙 第二次會考

科目名稱: 微積分 (上)(3 學分) 考試時間: 12 月 9 日第二節

I. 填充題. (45 分)

1. The local maximum value of the function g(x) = x + 2 sin x in the interval [0, 2π] is 2π 3 +

√

3

2. Suppose that f (0) = 4 and f′(x)

≤ 3 for all values of x. How large can f(2) possibly be ?

Ans： 10

3. If it is known that

∫ 9 1

f (x) dx = 13 and

∫ 9 6

f (x) dx = 4, find

∫ 6 1

f (x) dx = 9

4. Express lim

n→∞

∑n i=1

(x3i+xisin xi)∆x as an integral on the interval [0, π]. Ans：

∫ π

0

(x3+ x sin x) dx

5. Let f (x) = x

√

6

− x. The increasing interval of f is (−∞, 4) . The decreasing interval of f is (4, 6)

6. Find lim

x→∞(

√

9x2+ x

− 3x) =

1 6

7. Find lim

x→∞(x

− √

x) = ∞

8. Let f (x) =

x

3

(x + 1)2. The slant asymptote of f is y = x

− 2

II.計算、證明題. (60分)

1. Show that the equation

2x +

cos

x = 0

has exactly one real root.

2. Use the Mean Value Theorem to prove the inequality

|

sin

a −

sin

b | ≤ |a−b|

, for all

a

and

b

.

3. Use the definition of the integral

∫ b a

f (x) dx =

lim

n→∞

∑n

i=1

f (x

i

)∆x

where

∆x = b − a n ,

x

i

= a + i · ∆x

to prove that

∫ b a

x dx = b

2

− a

2

2

.

4. Evaluate the integral by interpreting it in terms of areas

∫ 0

−3

(1 +

√

9 − x

2

) dx

.

(2)

5. Let

f (x) = x x − 1

.

(a) Find the intervals on which

f

is increasing or decreasing.

(b) Find the intervals on which

f

is concave upward or downward.

(c) Sketch the graph of

f

.

6. Find the horizontal and vertical asymptotes of the function

f (x) =

√ 2x

2

+ 1

3x − 5

.

(3)

109學年度第一學期理工電資學院微積分(3學分)第二次會考答案2020.12.9

題號 答案 來源

1 略 3.2

− 習題 19

2 略 3.2

− 習題 31

3 略 3.4

− 習題 32

∗

4 3 + 9

4

π

4.2

− 習題 27

5 (a)f (x)is decreasing on (

−∞, 1), (1, ∞).

(b)f (x)is concave upword on (1,

∞), is concave downword on (−∞, 1). 3.5 − 習題 9

(c)略

6 The vertical asymptotes is x = 5

3

, the horizontal asymptotes is y = ± √

2

3 3.4

− 例題 4

* 為非勾選習題、類似題.

證明、作圖題過程略過.

參考文獻

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