Find the function The following limits repres...

Question

Find the function The following limits represent the slope of a curve y = f(x) at the point (a,f(a)). Determine a possible function f and number a; then calculate the limit.

(lim h🠂0) (2+h)⁴-16 / h

Accepted Answer

Welcome back, everyone. In this problem, we want to find a possible function F of X and a number C such that the following limit represents the slope of the curve Y equals F of X at CFFC. Then we want to evaluate the limit, that is the limit as H approaches 0 or 3 + H all cubed minus 27 divided by H. Now what do we know that's going to help us? Well, recall, OK, that given Y equals FFX, OK. At a point, see. FFC, OK. Then the derivative of F of X at that point, so F of C, is going to be equal to the limit as X approaches C of F of X minus F of C, all divided by X minus C. OK? Or we can also say that the derivative F C is going to be equal to the limit. As H approaches 0 of F of C plus H minus F of C. OK, divided by age. Which fits our limit fits our scenario. So now the question is, how can we find FF X and C by or FFX and C from our limit? Well, we can compare the general form to what we have, OK? And by comparison, that tells us that F of C plus H minus F of C are divided by H. Is going to be equal to 3 + H cubed minus 27 divided by H. Now notice both terms here. H is the denominator, and if we cancel through for H, divide through our entire equation by H, then we should get, uh, let me leave this in black here, we should get F of C plus H minus F of C to be equal to 3 + HRQed minus 27. Now, where are we supposed to go from here to find F of X and C? Well, if we notice in our, on the left hand side, FFC is gonna be related to FF C plus H. In other words, we should be able to express one of these terms as the other to show that it's the substitute, OK? That is, whatever value we get for C here must be the same value in C+ H. Now, not here, we have 3 + H all cubed. So can we express 3 in terms of a cube for 27? Well, we know that 27 equals 3 cubed. So we could rewrite this as 3 + H cubed minus 3 cubed, OK? And now notice here that by comparison. 3, OK, we get here FFC. To be equal to C cubed because the alter function is a cube since H is added under the cube. So here we notice C plus H, and then we have 3 + H cub and now we notice here that we have F of C and 3 cubed. So what does this tell us? Well, from this we can conclude. That F of X is equal to X cubed, OK, and C equals 3, OK? Because F of C equals C cubed. Let's put a note here. F of C equals c cubed, which satisfies f of C+ H being equal to C + H R cued. So F of X is X cubed and C equals 3. And now we know we're right because if we test it, then F of 3 is equal to 3 cubed, which equals 27, OK? Now that we know here then that F or the derivative of F of 3 is gonna be equal to the limit as. That's right, uh. The derivative of FF 3 is going to be equal to the limit as it approaches 0 of. 3 plus H cubed minus 27 divided by H then we should now be able to solve for our limit. Now notice here it is approaching 0, and if we substitute it directly into the expression that we now have, the denominator will be 0 and it leads to an indeterminate form. So let's see if we can express our numerator in another way for us to be able to factor out H. Let's expand the 3 + HRQ. No notice here, OK, that. When we do that, OK. Then we're going to get the limit as it approaches 0 of 27. Plus 27 H plus 9 H squared plus HQ minus 27. OK, all of that is being divided by H. Now 27 minus 27 equals 0, and from these three terms, notice they are of a common factor of H. So we could rewrite this as the limit as H approaches 0 of H multiplied by 27 + 9 H plus H squared, OK, all divided by H. Now we can cancel H from from the numerator and denominator. And we will be left with the limit as H approaches 0 of 27 + 9 H + 8 squared and now we can substitute H equals 0 into our, into our expression. So that's gonna be 27 + 9. OK, multiplied by 0 plus 0 squared, which is gonna be equal to 27. OK. So the derivative of F of 3, sorry, F3, OK, is going to be equal to 27. So that is going to be our limit. F of X is X cubed, OK, and C equals 3. Thanks a lot for watching everyone. I hope this video helped.

Find the function The following limits represent the slope of a curve y = f(x) at the point (a,f(a)). Determine a possible function f and number a; then calculate the limit.
(lim h🠂0) (2+h)⁴-16 / h

Key Concepts

Limits

Derivative

Function Evaluation

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