Question

A. Find the linear approximating polynomial for the following function centered at the given point a.point a.

b. Find the quadratic approximating polynomial for the following function centered at the given point a.
c. Use the polynomials obtained in parts a. and b. to approximate the given quantity.
f(x) = 16x^3/2 a = 4; approximate 16(4.2^3/2).

Answer 1

(a)
`L(x)=48x-64`

(b)
`P_(2)(x)=48x-64+3(x-4)^(2)`

(c)Using Linear Approximation,L(4.2)=137.6

Using Quadratic Approximation,
`P_(2)(4.2)=137.72`

Explanation:

Given:
`f(x) = 16x^(3/2), a = 4`

(a)Linear Approximation,
`L(x)=f(a)+f'(a)(x-a)`

`f(4) = 16*4^(3/2)=128`

`f'(x) = 24√(x)`

`f'(4) = 24√(4)=24*2=48`

`L(x)=128+48(x-4)`

`L(x)=128+48x-192\\L(x)=48x-64`

(b)Quadratic Approximation,
`P_(2)(x)=f(a)+f'(a)(x-a)+(1)/(2)f''(a)(x-a)^(2)`

`f''(x)=12x^(-1/2)\\f''(4)=12*4^(-1/2)=6`

`P_(2)(x)=48x-64+3(x-4)^(2)`

(c)To approximate:

L(x)=48x-64

L(4.2)=48(4.2)-64=137.6

Also, Using Quadratic Approximation

`P_(2)(x)=48x-64+3(x-4)^(2)\\P_(2)(4.2)=48(4.2)-64+3(4.2-4)^(2)=137.72`