Question

How do you use linear approximation to find the value of ${\left( {1.01} \right)^{10}}$ ?

Answer 1

Hint : In this question, we have to find the value of ${\left( {1.01} \right)^{10}}$ using the linear approximation method. In order to apply this method, we need to use the formula $y = f\left( a \right) + f'\left( a \right)\left( {x - a} \right)$ for a function $f\left( x \right)$ which we will take as $f\left( x \right) = {x^{10}}$ and for such a point $a$ which is close to the value we need to find, thus $a = 1$ . We will put the $x = 1.01$ in the formula to obtain the value of $y$ which would be our answer.

Complete step-by-step answer :
(i)
We are asked to find the approximate value of ${\left( {1.01} \right)^{10}}$ through a linear approximation method. In this method we will apply the formula:
$y = f\left( a \right) + f'\left( a \right)\left( {x - a} \right)$
Where, $f\left( x \right)$ is the function we will define by looking at the exponent we have got in our question and $a$ will be the approximated and nearest value of $x$ .
Therefore, $f\left( a \right)$ will be the value of the function $f\left( x \right)$ at the point $a$ and $f'\left( a \right)$ will be the value of the derivative of the function $f\left( x \right)$ at the point $a$ .
As we are asked to calculate ${\left( {1.01} \right)^{10}}$ , we can clearly define our function $f\left( x \right)$ as ${x^{10}}$ . Therefore,
$f\left( x \right) = {x^{10}}$
Since, we need the derivative of the function too, we will differentiate both sides with respect to $x$ . Therefore, we will get:
$f'\left( x \right) = 10{x^9}$
(ii)
In the function $f\left( x \right) = {x^{10}}$ , we want the value of $x$ to be $0.01$ in order to obtain our answer and since the value of $a$ will be the approximated value of $x$ , we will have $a = 1$ .
Therefore, the values of $f\left( a \right)$ and $f'\left( a \right)$ will be:
$f\left( a \right) = f\left( 1 \right) = {1^{10}} = 1$
And,
$f'\left( a \right) = f'\left( 1 \right) = 10 \times {1^9} = 10$
(iii)
Now, since we have all the values to put in the formula, we will put $x = 1.01$ to obtain the value of $y$ which would be our answer. Therefore,
$y = f\left( a \right) + f'\left( a \right)\left( {x - a} \right) \\\ y = 1 + 10\left( {1.01 - 1} \right) \\\ y = 1 + 10\left( {0.01} \right) \\\ y = 1 + 0.1 \\\ y = 1.1 \;$
Since, we get $y = 1.1$ , we can say that through linear approximation method, ${\left( {1.01} \right)^{10}} = 1.1$
So, the correct answer is “1.1”.

Note : Linear approximation is also known as linearization. It is a method to approximate the value of a function at a particular point. It is a quick and simple method that estimates a value otherwise it is very difficult to find. But note that since it is a rough approximation, the further away you get from the point $x = a$ , the less precise the approximation becomes.