Science:Math Exam Resources/Courses/MATH110/April 2016/Question 08 (a)/Solution 1

We want to approximate the value

${\frac {1}{4.16}}.$

To this end, we define the function $f(x)={\frac {1}{x}}$.

We will use a linear approximation:

$f(x)\approx f(a)+f'(a)(x-a)$

We want to use $x=4.16$ and $a=4$. It is necessary to compute $f'(4)$. The derivative of $f(x)$ is

$f'(x)=-{\frac {1}{x^{2}}}$

so

$f'(4)=-{\frac {1}{4^{2}}}=-{\frac {1}{16}}$

so

$f(4.16)\approx {\frac {1}{4}}-{\frac {1}{16}}(4.16-4)$

and therefore

$f(4.16)\approx 0.25-{\frac {0.16}{16}}$

and we get the approximation

$f(4.16)\approx 0.24.$

This is believable because we expect to get something just a little less than $0.25$, since we are dividing by a number that is just a little more than 4.

Answer: ${\frac {1}{4.16}}\approx 0.24$