Question

How do you estimate $\vartriangle f$ using the linear approximation and use a calculator to compute both the error and the percentage error given $f(x) = \sqrt {18 + x} $$a = 7$ and $\vartriangle x = - 0.5$?

Answer 1

Hint: In this question we have been given with the error in the function and the function is to be calculated at $f(7)$ therefore, we will use the formula of linear approximation by first finding out the derivative of the function and then find the percentage error from the value of the error.

Formula used: $f(a + \vartriangle x) = f(a) + f'(a) \times \vartriangle x$

Complete step-by-step solution:
We have the given function as:$f(x) = \sqrt {18 + x} $
We have the error for the function as:$\vartriangle x = - 0.5$
Now the formula for linear approximation is: $f(a + \vartriangle x) = f(a) + f'(a) \times \vartriangle x$
Therefore, we will first find out $f'(a)$ by taking the derivative of the function.
we have: $f(x) = \sqrt {18 + x} $
The derivative of the term can be written as:
$ \Rightarrow f'(x) = \dfrac{{d(\sqrt {18 + x} )}}{{dx}}$
Now we know that $\dfrac{d}{{dx}}\sqrt x = \dfrac{1}{{2\sqrt x }}$ therefore, on using this formula, we get:
$ \Rightarrow f'(x) = \dfrac{1}{{2\sqrt {18 + x} }}$
Now we can write $f(a)$ and $f'(a)$as:
$ \Rightarrow f(a) = \sqrt {18 + 7} $
Which can be simplified as:
$ \Rightarrow f(a) = \sqrt {25} $
On taking the square root, we get:
$ \Rightarrow f(a) = 5$
Now $f'(a) = \dfrac{1}{{2\sqrt {18 + 7} }}$
On simplifying we get:
$ \Rightarrow f'(a) = \dfrac{1}{{2\sqrt {25} }}$
On taking the square root, we get:
$ \Rightarrow f'(a) = \dfrac{1}{{2 \times 5}}$
On multiplying the terms in the denominator, we get:
$ \Rightarrow f'(a) = \dfrac{1}{{10}}$
Now we have:
$ \Rightarrow f(7 - 0.5) = 5 + \dfrac{1}{{10}} \times 0.5$
On simplifying the expression by using a calculator, we get:
$ \Rightarrow f(7 - 0.5) = 5.05$
Now $\vartriangle f = f(7 - 0.5) - f(7)$
On substituting, we get:
$ \Rightarrow \vartriangle f = 5 - 5.05$
On subtracting, we get:
$\vartriangle f = - 0.05$, which is the total error.
Now to find the percentage of error, we will use the formula of percentage which is: $ \Rightarrow per = \left( {\dfrac{{part}}{{whole}} \times 100} \right)\% $
Now on substituting the value of $part = 0.05$ and $whole = 5$, which is the actual value of the function, we get:
$ \Rightarrow per = \left( {\dfrac{{0.05}}{5} \times 100} \right)\% $
On using the calculator, we get the error as:
$ \Rightarrow per = 1\% $
Therefore, the total percentage error is $1\% $.

Note: It is to be remembered that the term error in mathematics is the difference between the true value and the approximate value of a term.
The most common types of errors in mathematics are round-off error and the truncation error.