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:

$f(x)\approx f(a)+f^{\prime }(a)(x-a)$

We want to use $x=4.16$ and $a=4$. It is necessary to compute $f^{\prime }(4)$. The derivative of $f(x)$ is

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

so

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

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