Question

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

Answer 1

Answer:

(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,$P_{2}(4.2)=137.72$

Step-by-step explanation:

Given: $f(x) = 16x^{3/2}, a = 4$

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

$f(4) = 16*4^{3/2}=128$

$f'(x) = 24\sqrt{x}$

$f'(4) = 24\sqrt{4}=24*2=48$

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

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

(b)Quadratic Approximation, $P_{2}(x)=f(a)+f'(a)(x-a)+\frac{1}{2}f''(a)(x-a)^{2}$

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

$P_{2}(x)=48x-64+3(x-4)^{2}$

(c)To approximate:

L(x)=48x-64

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

Also, Using Quadratic Approximation

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