• 沒有找到結果。

# 2008 W -C L Calculus(I)

N/A
N/A
Protected

Share "2008 W -C L Calculus(I)"

Copied!
20
0
0

(1)

### Calculus (I)

WEN-CHING LIEN

Department of Mathematics National Cheng Kung University

2008

(2)

### Ch3-8: Approximation And Local Linearity

(1) The tangent line approximation

Assume that y =f(x)is differentiable at x =a ; then L(x) =f(a) +f(a)(x−a)

is the linearization of f at x =a.

Remark: f(x) ≈f(a) +f(a)(x −a)

Ex:(plynomial)

(3)

### Ch3-8: Approximation And Local Linearity

(1) The tangent line approximation

Assume that y =f(x)is differentiable at x =a ; then L(x) =f(a) +f(a)(x−a)

is the linearization of f at x =a.

Remark: f(x) ≈f(a) +f(a)(x −a)

Ex:(plynomial)

(4)

### Ch3-8: Approximation And Local Linearity

(1) The tangent line approximation

Assume that y =f(x)is differentiable at x =a ; then L(x) =f(a) +f(a)(x−a)

is the linearization of f at x =a.

Remark: f(x) ≈f(a) +f(a)(x −a)

Ex:(plynomial)

(5)

### Ch3-8: Approximation And Local Linearity

(1) The tangent line approximation

Assume that y =f(x)is differentiable at x =a ; then L(x) =f(a) +f(a)(x−a)

is the linearization of f at x =a.

Remark: f(x) ≈f(a) +f(a)(x −a)

Ex:(plynomial)

(6)

### Ch3-8: Approximation And Local Linearity

(1) The tangent line approximation

Assume that y =f(x)is differentiable at x =a ; then L(x) =f(a) +f(a)(x−a)

is the linearization of f at x =a.

Remark: f(x) ≈f(a) +f(a)(x −a)

Ex:(plynomial)

(7)

(2)Examples

1 Find the linear approximation of f(x) =sin x at x =0

2 Find the linear approximation of f(x) =√

x at x =64 and use it to find an approximate value of√

65

3 Suppose that the per capita growth rate of a

population is 3%, that is, if N(t)denote the population size at time t , then

1 N

dN

dt =0.03

Suppose that the population size at time t =4 is equal to 100. Use a linear approximation to compute the population size at time t =4.1.

(8)

(2)Examples

1 Find the linear approximation of f(x) =sin x at x =0

2 Find the linear approximation of f(x) =√

x at x =64 and use it to find an approximate value of√

65

3 Suppose that the per capita growth rate of a

population is 3%, that is, if N(t)denote the population size at time t , then

1 N

dN

dt =0.03

Suppose that the population size at time t =4 is equal to 100. Use a linear approximation to compute the population size at time t =4.1.

(9)

(2)Examples

1 Find the linear approximation of f(x) =sin x at x =0

2 Find the linear approximation of f(x) =√

x at x =64 and use it to find an approximate value of√

65

3 Suppose that the per capita growth rate of a

population is 3%, that is, if N(t)denote the population size at time t , then

1 N

dN

dt =0.03

Suppose that the population size at time t =4 is equal to 100. Use a linear approximation to compute the population size at time t =4.1.

(10)

(2)Examples

1 Find the linear approximation of f(x) =sin x at x =0

2 Find the linear approximation of f(x) =√

x at x =64 and use it to find an approximate value of√

65

3 Suppose that the per capita growth rate of a

population is 3%, that is, if N(t)denote the population size at time t , then

1 N

dN

dt =0.03

Suppose that the population size at time t =4 is equal to 100. Use a linear approximation to compute the population size at time t =4.1.

(11)

(2)Examples

1 Find the linear approximation of f(x) =sin x at x =0

2 Find the linear approximation of f(x) =√

x at x =64 and use it to find an approximate value of√

65

3 Suppose that the per capita growth rate of a

population is 3%, that is, if N(t)denote the population size at time t , then

1 N

dN

dt =0.03

Suppose that the population size at time t =4 is equal to 100. Use a linear approximation to compute the population size at time t =4.1.

(12)

(2)Examples

1 Find the linear approximation of f(x) =sin x at x =0

2 Find the linear approximation of f(x) =√

x at x =64 and use it to find an approximate value of√

65

3 Suppose that the per capita growth rate of a

population is 3%, that is, if N(t)denote the population size at time t , then

1 N

dN

dt =0.03

Suppose that the population size at time t =4 is equal to 100. Use a linear approximation to compute the population size at time t =4.1.

(13)

### Newton-Raphson Method

Given a function f(x): differentiable onR, Q:

we want to estimate a root of the equation f(x) =0 in the following way–

(1) Start with a given point (x1,f(x1)) and find the tangent line at (x1,f(x1)):

yf(x1) =f(x1)(x −x1)

(2) Solve x2: x2 =x1f(x1) f(x1)

(14)

### Newton-Raphson Method

Given a function f(x): differentiable onR, Q:

we want to estimate a root of the equation f(x) =0 in the following way–

(1) Start with a given point (x1,f(x1)) and find the tangent line at (x1,f(x1)):

yf(x1) =f(x1)(x −x1)

(2) Solve x2: x2 =x1f(x1) f(x1)

(15)

### Newton-Raphson Method

Given a function f(x): differentiable onR, Q:

we want to estimate a root of the equation f(x) =0 in the following way–

(1) Start with a given point (x1,f(x1)) and find the tangent line at (x1,f(x1)):

yf(x1) =f(x1)(x −x1)

(2) Solve x2: x2 =x1f(x1) f(x1)

(16)

### Newton-Raphson Method

Given a function f(x): differentiable onR, Q:

we want to estimate a root of the equation f(x) =0 in the following way–

(1) Start with a given point (x1,f(x1)) and find the tangent line at (x1,f(x1)):

yf(x1) =f(x1)(x −x1)

(2) Solve x2: x2 =x1f(x1) f(x1)

(17)

(3) Replace(x1,f(x1))by(x2,f(x2))in step1 and find

xn+1=xnf(xn) f(xn)

(4) We can form a sequence {x1,x2,x3,· · · ,xn,· · · }

Example: Estimate√ 3.

(18)

(3) Replace(x1,f(x1))by(x2,f(x2))in step1 and find

xn+1=xnf(xn) f(xn)

(4) We can form a sequence {x1,x2,x3,· · · ,xn,· · · }

Example: Estimate√ 3.

(19)

(3) Replace(x1,f(x1))by(x2,f(x2))in step1 and find

xn+1=xnf(xn) f(xn)

(4) We can form a sequence {x1,x2,x3,· · · ,xn,· · · }

Example: Estimate√ 3.

(20)

### Thank you.

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

W EN -C HING L IEN Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung