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) = 3x² - 4x; P(1, -1)

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 2 X2 + 3 X, and point P is located at 2.14. 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 minus 5, for choice B, we have 7, for choice C, we have 11, and for choice D, we have 16. Now, in order to use our definition, we first need to 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 is equal to F of 2. Which is equal to 14, which is the Y coordinate of our point P. We also need to calculate the quantity of F of A plus H. Which is equal to F of 2 + H since A is equal to 2. And here we can use our function F of X to calculate this quantity. So, in our function F of X, we're we're going to replace X with the quantity of 2 + H. So F of 2 + H is going to be equal to, 2, multiplying the quantity of 2 + H in quantity squared, plus 3 multiplied by the quantity of 2 + H. And so we can multiply out all of these terms, and this is going to be equal to, or 2, multiplying the quantity of 2 plus H in quantity squared. We will get 8 + 8 H. Plus 2 H squared. Then we'll have plus the quantity of 3, multiplying the quantity of 2 plus H, and when we multiply that out, we'll get plus 6 + 3 H. So now we just need to collect like terms, and doing so this is going to be equal to 14 plus 11 H. Plus 2 H2. So now I have all the information that I need to apply our definition that we're given and the problem to calculate our slope. We'll have M tangent. is equal to the limit, as H approaches 0, of the quantity of F of A plus H, which we calculated to be 14 + 11 H. Plus 2 H squared. Then we have minus F of A, but FFA was 14, so we'll have minus 14, and that whole quantity is divided by H. Now, if I look in the numerator, I see that I have a 14 and a -14. And so when I add those together, they cancel out that they add to zero. And the remaining terms have H in them, so I can divide the remaining terms in the numerator by H. So doing so, this is going to be equal to the limit. As H approaches 0. Of the quantity of 11 plus 2 H. So here I can evaluate the limit by direct substitution. In doing so, this is going to be equal to 11 plus 2 multiplied by 0, which is equal to 11. So that is the slope of our tangent line. And if I compare that to our answer choices, the correct answer choice corresponds to answer choice C. 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) = 3x² - 4x; P(1, -1)

Key Concepts

Tangent Line

Derivative

Point-Slope Form

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