Calculus (I)
WEN-CHING LIEN
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 −x1)
(2) Solve x2: x2 =x1− 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 −x1)
(2) Solve x2: x2 =x1− 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 −x1)
(2) Solve x2: x2 =x1− 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 −x1)
(2) Solve x2: x2 =x1− f(x1) f′(x1)
(3) Replace(x1,f(x1))by(x2,f(x2))in step1 and find
xn+1=xn− f(xn) f′(xn)
(4) We can form a sequence {x1,x2,x3,· · · ,xn,· · · }
Example: Estimate√ 3.
(3) Replace(x1,f(x1))by(x2,f(x2))in step1 and find
xn+1=xn− f(xn) f′(xn)
(4) We can form a sequence {x1,x2,x3,· · · ,xn,· · · }
Example: Estimate√ 3.
(3) Replace(x1,f(x1))by(x2,f(x2))in step1 and find
xn+1=xn− f(xn) f′(xn)
(4) We can form a sequence {x1,x2,x3,· · · ,xn,· · · }
Example: Estimate√ 3.