Question

a.Find a linear approximation of 4√14 . b. Justify visually whether your approximation is an over- or underestimate.

Answer 1

15

Explanation:

Let f(x) be the function

`\bf f(x)=√(x)`

A linear approximation of f is the Taylor polynomial of degree one:

`\bf f(x)\approx f(a)+f'(a)(x-a)`

Taking a = 16, and given that

`\bf f'(x)=\displaystyle(1)/(2√(x))`

we get

`\bf f(x)\approx f(16)+f'(16)(x-16)=√(16)+\displaystyle(1)/(2√(16))(x-16)=4+\displaystyle(x-16)/(8)`

so

`\bf √(14)=f(14)\approx 4+\displaystyle(14-16)/(8)=3.75\Rightarrow\\\\\Rightarrow 4√(14)\approx 4(3.75)=15`

Since 16 > 14, we can deduce that this is an overestimate.