Copyright

Given that f(3)= 5 and f '(x)= \frac{x}{x^3 + 3}, find the linear approximation of f(x) at x= 3.

Question:

Given that f(3)= 5 and {eq}f '(x)= \frac{x}{x^3 + 3}, {/eq} find the linear approximation of f(x) at x= 3.

Linearization of the Function:

We have to find the linear approximation of f(x) at x= 3. We have given the value of f(3) and with help of f'(x) we will find the value of f'(3), then plug the values into the formula of the linear approximation to get the desired results.

Answer and Explanation:

Plug the value of x=3 into f'(x) and we have $$\begin{align*} f'\left( x \right) &= \frac{x}{{{x^3} + 3}}\\ f'\left( 3 \right) &= \frac{3}{{{{\left( 3 \right)}^3} + 3}}\\ f'\left( 3 \right) &= \frac{3}{{30}}\\ f'\left( 3 \right) &= \frac{1}{{10}}. \end{align*} $$

The linear approximation of f(x) at a= 3 is $$\begin{align*} y &= f\left( a \right) + f'\left( a \right)\left( {x - a} \right)\\ y &= f\left( 3 \right) + f'\left( 3 \right)\left( {x - 3} \right)\\ y &= 5 + \frac{1}{{10}}\left( {x - 3} \right)\\ y &= \frac{x}{{10}} + 5 - \frac{3}{{10}}\\ y &= \frac{1}{{10}}\left( {x + 47} \right). \end{align*} $$


Learn more about this topic:

Loading...
How to Estimate Function Values Using Linearization

from Math 104: Calculus

Chapter 10 / Lesson 2
5K

Related to this Question

Explore our homework questions and answers library