Question

If f(3) = 8 and f’(3) = -4, then how do you use the linear approximation to estimate f(3.02) ?

Answer 1

You have to do the question in the following way. First you have to write the equation of the tangent at point 3. Then you get the line in the form of f(3) and f’(3) and x. Then you can substitute the values f(3) = 8 and f’(3) = -4. You have to substitute x as 3.02. After all the substitutions and the calculations, you get the final answer.

Complete step by step answer:
The first step to do in order to solve the question is to write the equation of the tangent at point 3. We can write it in the following way:

$\Rightarrow y-f\left(3\right) = f’\left(3\right) \left(x-3\right)$
Now we have to substitute the values of f(3) and f’(3). The values are given as f(3) = 8 and f’(3) = -4. Therefore, we get :
$\Rightarrow y-8 = -4\left(x-3\right)$
$\Rightarrow y-8 = -4x+12$
Now we have to substitute the value of x as 3.02 . Hence, we get y as:
$\Rightarrow y-8 = -4\left(3.02\right)+12$
$\Rightarrow y-8 = -12.08+12$
$\Rightarrow y=7.92$

Therefore, we get the final answer for the question, if f(3) = 8 and f’(3) = -4, then how do you use the linear approximation to estimate f(3.02), as 7.92

Note: When you get this type of problem, you first have to write the tangent line at the point where you get the values of f(x) and f’(x). Also, be careful while substituting the values, you may get confused between f(x) and f’(x). Also, remember the form of the tangent equation.