### Calculus (I)

W^{EN}-C^{HING} L^{IEN}

Department of Mathematics National Cheng Kung University

2008

### 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)

### 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)

### 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)

### 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)

### 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)

(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.

(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.

(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.

(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.

(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.

(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.

### 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)):

*y* −*f*(x1) =*f*^{′}(x1)(x −*x*1)

*(2) Solve x*2*: x*2 =*x*1− *f*(x1)
*f*^{′}(x1)

### 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)):

*y* −*f*(x1) =*f*^{′}(x1)(x −*x*1)

*(2) Solve x*2*: x*2 =*x*1− *f*(x1)
*f*^{′}(x1)

### 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)):

*y* −*f*(x1) =*f*^{′}(x1)(x −*x*1)

*(2) Solve x*2*: x*2 =*x*1− *f*(x1)
*f*^{′}(x1)

### 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)):

*y* −*f*(x1) =*f*^{′}(x1)(x −*x*1)

*(2) Solve x*2*: x*2 =*x*1− *f*(x1)
*f*^{′}(x1)

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

*x**n+1*=*x**n*− *f*(x*n*)
*f*^{′}(x*n*)

(4) We can form a sequence {*x*1,*x*2,*x*3,· · · ,*x**n*,· · · }

Example: Estimate√ 3.

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

*x**n+1*=*x**n*− *f*(x*n*)
*f*^{′}(x*n*)

(4) We can form a sequence {*x*1,*x*2,*x*3,· · · ,*x**n*,· · · }

Example: Estimate√ 3.

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

*x**n+1*=*x**n*− *f*(x*n*)
*f*^{′}(x*n*)

(4) We can form a sequence {*x*1,*x*2,*x*3,· · · ,*x**n*,· · · }

Example: Estimate√ 3.