Finding Functions from Derivatives

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Question

Finding Functions from Derivatives

Suppose that f(0) = 5 and that f'(x) = 2 for all x. Must f(x) = 2x + 5 for all x? Give reasons for your answer.

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. True or false, provided that GF 2 is equal to -3 and G of X is equal to -5 for all X, we can say that G of X is equal to -5 X plus 7 for all X. Awesome. So it appears for this particular problem we're asked to determine whether or not this statement, which is given to us by the problem itself, is true or false. So, with that said, We need to recall first off that the mean value theorem, if a function F has a constant derivative, F of X. So, once again, we're recalling and using the mean value theorem. So the mean value theorem states that if a function F has a constant derivative of F of X is equal to K. On an interval I, then F or function F is a linear function on I or interval. Specifically, F of X is equal to KX plus B, so KX plus B, for some constant value B. So let us note that we are given that G of X is equal to 5 for all X. And Since the derivative of G is constant, since we have this negative 5 value. That means, since we have our constant -5, that means as a result, that G must be in the form of G of X, so we need to take G of X. So G of X is equal to -5 X plus C, where C is some constant value C. So using our initial condition, which is G of 2 is equal to -3, we need to plug in our value of X equals 2 into our function of G of X is equal to, so we need to take G of X. is equal to -5 X plus C, so when you take -5 multiplied by 2, plus C is equal to -3, which will mean That -10 plus C is equal to -3, which will mean that Our constant C is equal to 7. So therefore, that means that G of X equals -5 X plus C satisfies both the constant derivative condition and the initial value G of 2 equals -3. By our statement, by our theorems, so no other function can have. A derivative -5 everywhere, and also pass through the parenthesis 2.3. So therefore, without a doubt, our following statement has to be true. Since it's true because G of X, so let's make a note of this, since G of X is equal to -5 for all X using. G of 2 is equal to -3, which gives us that our constant value C is equal to 7. So therefore we can say safely that G of X is equal to -5 X plus 7 for all values of X. So that means that our final answer has to be true. And that's it, that's our final answer. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Finding Functions from Derivatives

Suppose that f(0) = 5 and that f'(x) = 2 for all x. Must f(x) = 2x + 5 for all x? Give reasons for your answer.

Key Concepts

Derivative

Initial Condition

Linear Function

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