Use definition (2) (p. 135) to find the slope of the line tang...

Question

Use definition (2) (p. 135) to find the slope of the line tangent to the graph of f at P.

f(x) = 8 - 2x²; P(0, 8)

Accepted Answer

Hi, everyone, let's take a look at this practice problem dealing with tangent lines. This problem says to use the following limit definition to determine the slope of the line tangent to the graph of F at P, where F of X is equal to 5 minus X2, and P is located at -2.1. And we're given the definition that M tangent is equal to the limit as H approaches 0 of the quantity of F of A plus H minus F of A in quantity, divided by H. We're given 4 possible choices as our answers. For choice A, we have -9. for choice B, we have -3. For choice C, we have 2, and for choice D, we have 4. Now, the first thing we need to do is identify the value for A and A is going to be the X coordinate of our point P. That means that A is equal to -2. That also means that F of A, which we will also need, is equal to F minus 2. Which is equal to 1, which is the Y coordinate for a point P. Now we also need to calculate F of A plus H. Which is equal to F of -2 plus H, since A is equal to -2. And we were given the function F of X in our problem, and so we're going to replace X with the quantity of minus 2 + H. So F of minus 2 plus H is equal to 5 minus the quantity of minus 2 plus H in quantity squared. And so we will we will go ahead and multiply out that square term. This is going to be equal to 5 minus the quantity of 4 minus or H. Plus H squad in quantity. And so, we will go ahead and distribute the minus sign, and collect like terms. So this is going to be equal to. First term we'll have 5 minus 4, which is 1. Then plus or h minus h squared. So now we have all the information that we need to use our definition for the slope of our tangent line. And so that means that M tangent is going to be equal to the limit, as H approaches 0. Of the quantity of F of A plus H, and we just calculated that, that is 1 + 4H minus H2d. And then we'll have minus F of A, which is 1, and that whole quantity is divided by H. So, simplifying the numerator, we see that we have 1 minus 1, and when we add those two together, they will add up to 0, so they cancel out, and the remaining terms have each of them, so we can cancel one factor of each in each of those terms. So this means we will have the limit as H approaches 0, of the quantity of 4 minus H. And here we can evaluate the limit by direct substitution, so setting H equal to 0, that's going to be equal to 4 minus 0, which is equal to 4. And so that is the value of our slope that we're looking for. And if we compare that to our answer choices, this corresponds to answer choice D. So I hope that this explanation has been useful, and I'll see you in the next video.

Use definition (2) (p. 135) to find the slope of the line tangent to the graph of f at P.f(x) = 8 - 2x2; P(0, 8)

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Use definition (2) (p. 135) to find the slope of the line tangent to the graph of f at P.
f(x) = 8 - 2x²; P(0, 8)