Finding Functions from Derivatives

In Exercis...

Question

Finding Functions from Derivatives

In Exercises 37–40, find the function with the given derivative whose graph passes through the point P.

g'(x) = 1 / x² + 2x, P(−1, 1)

Accepted Answer

Welcome back, everyone. In this problem, we want to determine the function that passes through the point P29 and has the following derivative of 2 divided by X + 6 X. Use the second corollary of the mean value theorem. Now how is the 2nd corollary going to help us determine the function that passes through the point? Well, recall, OK, that by the color of the mean value theorem, it says that if the derivative of one function F X equals the derivative of another function G of X at each point X, OK. In an open interval AB. Then, OK, there exists a constant sea. OK. There exists a constant sea. Such that FFX Will be equal to G of X plus C. For all. Values of X in the open interval, OK? That is, F minus G will be a constant function on AB on our open interval AB. So now, if we can figure out a function that has the same derivative of F, we can put it into our equation for F X E equals GX + C to thus help us figure out. What the value of C would be and then the value of our function F of X. Now let's first, let's use the first idea here that the derivative of F of X is equal to the derivative of G of X. So this would imply that the derivative of G of X is going to be equal to 2 divided by X2 + 6 X. Now if we were to write this as 2 multiplied by X to the negative 2 plus 6 X, can we figure out what GFX would be? In other words, what could we differentiate to get 22 X to the negative 2 + 6X? Well this must mean that the GX. Is going to be equal to -2 divided by X plus 3 x 2 because the derivative of -2 divided by X equals 2 x negative2 and the derivative of 6 X is 3 x 2. Now, since we have our function GFX, OK, then that means, let me put this in red. At the point where. So at the point P, which is 29, OK. This means that X equals 2 and F of X would be equal to 9, OK. Then, This would imply that F of X, which is 9, is going to be equal to -2 divided by 2, which is X, OK, plus 3 X squad, so that's 3 multiplied by 22 plus C. And at this point then we can use this to help us solve for a C because no. And this means that 9 will be equal to -1 + 12 plus C, OK, which means that C is gonna be equal to 9 minus 11, OK? Because -1 + 12 equals 11 and then we subtract 11 from both sides and thus we'll get C to be equal to -2. Therefore, if C equals -2, OK, then that means that F of X is going to be equal to G X 2 divided by X + 3 X2, OK, plus C R minus 2. This is the function that passes through the point P29 and has a derivative of 2 divided by X2 + 6 X. Thanks for watching everyone. I hope this video helped.

Finding Functions from Derivatives

In Exercises 37–40, find the function with the given derivative whose graph passes through the point P.

g'(x) = 1 / x² + 2x, P(−1, 1)

Key Concepts

Antiderivatives

Initial Conditions

Integration Techniques

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