2008 W -C L Calculus(I)

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Calculus (I)

WEN-CHING LIEN

Department of Mathematics National Cheng Kung University

2008

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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)

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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)

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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)

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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)

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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)

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(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.

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(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.

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(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.

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(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.

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(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.

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(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.

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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)

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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)

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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)

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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)

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(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.

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(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.

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(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.

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Thank you.

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