Question

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

Answer 1

Answer:

15

Step-by-step explanation:

Let f(x) be the function

$\bf f(x)=\sqrt{x}$

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

$\bf f'(x)=\displaystyle\frac{1}{2\sqrt{x}}$

we get

$\bf f(x)\approx f(16)+f'(16)(x-16)=\sqrt{16}+\displaystyle\frac{1}{2\sqrt{16}}(x-16)=4+\displaystyle\frac{x-16}{8}$

so

$\bf \sqrt{14}=f(14)\approx 4+\displaystyle\frac{14-16}{8}=3.75\Rightarrow\\\\\Rightarrow 4\sqrt{14}\approx 4(3.75)=15$

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