Answer:
(a)[tex]L(x)=48x-64[/tex]
(b)[tex]P_{2}(x)=48x-64+3(x-4)^{2}[/tex]
(c)Using Linear Approximation,L(4.2)=137.6
Using Quadratic Approximation,[tex]P_{2}(4.2)=137.72[/tex]
Step-by-step explanation:
Given: [tex]f(x) = 16x^{3/2}, a = 4[/tex]
(a)Linear Approximation, [tex]L(x)=f(a)+f'(a)(x-a)[/tex]
[tex]f(4) = 16*4^{3/2}=128[/tex]
[tex]f'(x) = 24\sqrt{x}[/tex]
[tex]f'(4) = 24\sqrt{4}=24*2=48[/tex]
[tex]L(x)=128+48(x-4)[/tex]
[tex]L(x)=128+48x-192\\L(x)=48x-64[/tex]
(b)Quadratic Approximation, [tex]P_{2}(x)=f(a)+f'(a)(x-a)+\frac{1}{2}f''(a)(x-a)^{2}[/tex]
[tex]f''(x)=12x^{-1/2}\\f''(4)=12*4^{-1/2}=6[/tex]
[tex]P_{2}(x)=48x-64+3(x-4)^{2}[/tex]
(c)To approximate:
L(x)=48x-64
L(4.2)=48(4.2)-64=137.6
Also, Using Quadratic Approximation
[tex]P_{2}(x)=48x-64+3(x-4)^{2}\\P_{2}(4.2)=48(4.2)-64+3(4.2-4)^{2}=137.72[/tex]
