What is the value of f(6)?

Question

Let f be a differentiable function such that f(2)=1 and f'(x) = sin(x^2 - 5)

Answer 1

Is this a linear approximation problem? Are you being asked to find the exact value of f(6) or an approximate value of f(6)? The latter case is straightforward and pretty simple, but finding the exact value requires integrating sin(x² - 5). This is not so fun.

If you send me a message, I can likely help you in either case.

Thank you for the Comment below. Here is how to find f(6)

By the Fundamental Theorem of Calculus,

∫ f ' (x) dx = f(b) - f(a) where the integral is from a to b.

In this case, we know f ' (x) = sin(x² - 5); we also know f(2) = 1. So, let a = 2 and b = 6. It follows that the above equation becomes

∫ sin(x² - 5) dx = f(6) - 1 where the integral is from 2 to 6

So, the exact value is f(6) = ∫ sin(x² - 5) dx + 1 where the integral is from 2 to 6. We can plug this formula into a graphing calculator to get the solution to three decimal places.

I used my graphing calculator and found ∫ sin(x² - 5) dx rounded to three decimal places equals 0.032. Therefore, f(6) = 0.032 + 1 = 1.032.