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Dimas [21]
4 years ago
15

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

Mathematics
1 answer:
Dafna1 [17]4 years ago
3 0

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

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