• 沒有找到結果。

1. (a) (2 分) 求 R sin(x)dx.

N/A
N/A
Protected

Academic year: 2021

Share "1. (a) (2 分) 求 R sin(x)dx."

Copied!
1
0
0

加載中.... (立即查看全文)

全文

(1)

微乙小考四 (2018/11/29)

1. (a) (2 分) 求 R sin(x)dx.

(b) (2 分) 求 R

01 1+x1 2

dx

(c) (2 分) 若 f(x) = R

1x

tan

10

(2t)dt, 則 f

0

(x) 為何?

sol: (a) R sin xdx = − cos x + C (b) R

1

0 1

1+x2

dx = tan

−1

(x)|

10

= tan

−1

(1) − tan

−1

(0) =

π4

(c) f

0

(x) = tan

10

(2x)

2. (a) (4 分) 證明

654

≤ R

4 0

1

1+x3

dx ≤ 4.

(b) (3 分) 假設 [0, 4] = [0, 2] ∪ [2, 4] 是 [0, 4] 的一個分割. 試求在此分割下, R

04 1+x13

的最小黎曼和.

sol: (a) let f (x) =

1+x1 3

is decreasing on [0,4]

so,

1+413

=

651

≤ f (x) ≤

1+01 3

= 1 R

4

0 1 65

≤ R

4

0

f (x)dx ≤ R

4 0

1dx

4 65

≤ R

4

0 1 1+x3

≤ 4 (b) t

1

= 2, t

2

= 4

2

X

i=1

f (t

i

) = f (2) · 2 + f (4) · 2 = 2 · ( 1

1 + 8 + 1

1 + 64 ) = 148 585 3. (a) (2 分) 陳述微積分基本定理.

(b) (3 分) 試求

dxd

 R

b(x)

a(x)

f (t)dt  . (c) (2 分) 使用 (b) 計算

dxd



R

x3

x2

ln(cos t)dt  . sol: (a) if f is continuous on [a,b]

Let F (x) = R

x

a

f (t)dt then F

0

(x) = f (x) if G

0

(x) = f (x) then R

b

a

f (x)dx = G(b) − G(a) (b)

dxd

( R

b(x)

a(x)

f (t)dt) if f(t) is continuous on [c,d]

dxd

( R

b(x)

a(x)

f (t)dt) =

dxd

( R

b(x)

C

f (t)dt− R

a(x)

C

f (t)dt)

=

dxd

( R

b(x)

C

f (t)dt) −

dxd

( R

a(x)

C

f (t)dt)

=

d(b(x))d

( R

b(x)

C

f (t)dt) ·

d(b(x))dx

d(a(x))d

( R

a(x)

C

f (t)dt) ·

d(a(x))dx

(by F.T.O.C)=f (b(x)) · b

0

(x) − f (a(x)) · a

0

(x)

(c) by(b)

d dx (

Z

x3 x2

ln(cos t)dt) = ln(cos(x

3

)) · 3x

2

− ln(cos(x

2

)) · 2x

1

參考文獻

相關文件

[r]

The Seed project, REEL to REAL (R2R): Learning English and Developing 21st Century Skills through Film-making in Key Stage 2, aims to explore ways to use film-making as a means

反之, 有了 parametric equation, 我們可利用這些在 R n 的 direction vectors, 利 用解聯立方程組的方法求出和這些 direction vectors 垂直的 normal vectors,

而利用 row vectors 的方法, 由於可以化為 reduced echelon form, 而 basis 是由此 reduced echelon form 中的 nonzero vectors 所組成, 所以雖然和來的 spanning

We point out that extending the concepts of r-convex and quasi-convex functions to the setting associated with second-order cone, which be- longs to symmetric cones, is not easy

Hence, we have shown the S-duality at the Poisson level for a D3-brane in R-R and NS-NS backgrounds.... Hence, we have shown the S-duality at the Poisson level for a D3-brane in R-R

We compare the results of analytical and numerical studies of lattice 2D quantum gravity, where the internal quantum metric is described by random (dynamical)

 依序填入該學生社團負責人之相關資訊,並於下方