Finding Functions from Derivatives

Suppose th...

Question

Finding Functions from Derivatives

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

b. f(1) = 0

Accepted Answer

Welcome back, everyone. Consider the derivative of FFX equals 4X cubed for all X. Use the second corollary of the mean value theorem to find F of 3 if FF 1 equals -2. A says it's 24, B 78, C 81, and the D 84. Now what is this second corollary that's supposed to help us figure out FR 3? Well, recall that if the derivative of FX is equal to the derivative of geoX at each point X. In an open interval A, B, OK. Then, There exists. A constant sea. Such that FFX Is equal to G of X + C? OK. For all X inside the open interval AB, OK? That is, F minus G is a constant function and AB. So now, if we can use this idea that their derivatives are the same and get a function for G of X, then we can use that to help us find FFX given that we have FF1 equals -2. Now, since they have the same derivative, OK, then if we take F of X equals GFX plus C. Then at X equals 1, this means that F of 1 will be equal to G 1 plus C. But what is G of 1? But if I put this over in a box here, if we think about it, since the derivative of FF X is the derivative of GF X, which is 4X cubed, let's ask ourselves what can we differentiate to get 4X cubed? Well, our function G of X would be equal to X 4th because the derivative of X 4th is 4XQed by the power rule. We would, we would bring down the power, OK? And then m um subtract one from the power to get 4X cubed. So since G of X equals X 4th and F of 1 equals -2, this means 2 will be equal to 14th plus C. 1 to the fourth is 1, so C will be equal to -2 minus 1, which equals -3. And since that's the case, this implies then that FF X is going to be equal to G X X 4 minus, sorry, plus C, the constant, which is minus 3. FF X equals X 4th minus 3. Therefore, at X equals 3, then F is going to be equal to 34 minus 3. That's 81 minus 3, which equals 78. Thus, using our second corollary F of 3 equals 78. If we look back at our answer choices, that means B 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

b. f(1) = 0

Key Concepts

Antiderivatives

Initial Conditions

Definite Evaluation

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