basic integrals 13

Question

Find f such that f′(x)=10x−7​, f(9)=0.

Answer 1

This requires an integral. Since there are no endpoints for integration, we will use indefinite integrals, so don't forget a + C in the work! This is important for incorporating the given information into the final answer.

f(x) = ∫10x - 7 dx = 10(x² / 2) - 7x + C = 5x² - 7x + C

We know that f(9) = 0:

f(x) = 5x² - 7x + C

f(9) = 0 = 5(9)² - 7(9) + C

0 = 405 - 63 + C

0 = 342 + C

C = -342

So f(x) = 5x² - 7x - 342.