Finding Functions from Derivatives

Suppose th...

Question

Finding Functions from Derivatives

Suppose that f'(x) = 2x for all x. Find f(2) if

a. f(0) = 0

Accepted Answer

Welcome back, everyone. In this problem, consider the derivative of FF X equals 4XQ for all X. Use the second corollary of the mean value theorem to find F of 3 if F of 0 equals 0. A says it's 27, B 32, C 81, and the D says it's 84. Now what is the second corollary and how is it going to help us to find FF 3? Well, recall, OK, that the second corollary of the mean value theorem states that if the derivative of FX is equal to the derivative of GX at each point X. In an open interval AB, OK. Then There exists a constant C. OK, there exists a constant C. Such that, OK. Purge that. F of X will be equal to G of X plus C, OK? G of X plus C for all X that's in the open interval AB, OK? In other words, F minus G is going to be a constant function on the open interval AB. Now, in that case, then noticing our problem, we're given the derivative of FF X. So if we can figure out the value of FFX, then we should be able to use corollary to to help us figure out what FF 3 is going to be. Now, given the derivative of FF X equals 4 X cubed, let's ask ourselves what can we differentiate to get 4X cubed? Well, that means. That if this is true, OK, then. GFX, the derivative of G of X will also be equal to 4 x cubed, which would imply that G of X is equal to x 4th because when we differentiate X 4th by the power rule, we bring down the power 4 and then we subtract 1 from the power to make the power 3. So that means G of X will be equal to X to the 4th. And now since they both have the same derivative like we said from correllary 2 of the mean value theorem. Since FFX equals GX plus C. Then at X equals 0, OK. Then F of 0 would be equal to G 0 plus C. Like we found earlier, G of X is X4, and in our problem statement, we know that F of 0 equals 0. So 0 must be equal to 0 4th plus C. and thus that means then that C has to be equal to 0. Now C equals 0, OK. OK, then that means FFX is going to be equal to GFX plus 0, or FF X will just be equal to geofix, which is equal to X to the 4th. Therefore, at X equals 3, then F of 3 is going to be equal to 34, which equals 81. Therefore, F of 3 equals 81, which means that C is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Finding Functions from Derivatives

Suppose that f'(x) = 2x for all x. Find f(2) if

a. f(0) = 0

Key Concepts

Antiderivatives

Initial Conditions

Evaluating Functions

Watch next