Use definition (1) (p. 133) to find the slope of the line tang...

Question

Use definition (1) (p. 133) to find the slope of the line tangent to the graph of f at P.

f(x) = 2/√x; P(4,1)

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 6 divided by the square root of X, and P is located at 36.1. And we're given the definition that m tangent is equal to the limit as x approaches A of the quantity of FX minus F of A in quantity, divided by the quantity of X minus A. We give 4 possible choices as their answers. For choice A, we have -1 divided by 36. For choice B, we have minus 1 divided by 72. For choice C, we have 1 divided by 24, and for choice D, we have 12. Now, in order to use our definition that we're given the problem, we need to identify what A is, as well as F of A. So, A is going to be our X coordinate of our point P. So that means that A is going to be equal to 36. So F of A Is going to be equal to F of 36. Which is going to be equal to 1, since 1 is the Y coordinate of our point P. So now we can actually apply the definition that we're given and the problem to find the slope of the tangent line. We'll have M tangent. It's equal to the limit, as X approaches 36, since A is equal to 36 of the quantity of F of X, which we're given in the problem, so it's gonna be 6 divided by the square root of X. Minus F of A, which in this case is 1, and that whole quantity is divided by the quantity of X minus A, which is 36. So now, at this point, we could try to apply direct substitution to evaluate the limit. And if we were to do that, we would get 6 divided by the square root of 36. -1, and that whole quantity is divided by the quantity of 36 minus 36. And if we were to evaluate this expression, we'd get 0 in the numerator, that is divided by 0 in the denominator. So we get 0 divided by 0, which is an indeterminate form. So, what we're going to try to do now is to eliminate the X minus 36 in the denominator. So, we're gonna start off by looking at the expression that we just had, which is the limit, as X approaches 36. Of the quantity of 6 divided by the square root of X minus 1 in quantity divided by the quantity of X minus 36, and we're going to want to eliminate the fraction out of the numerator. So we're going to do that by multiplying both the numerator and denominator by the square root of X. So doing so, this is going to be equal to the limit. As X approaches 36. Of the quantity of 6 minus square root of X. In quantity, divide by square root of X, multiplying the quantity of X minus 36. So now what we want to do is to get rid of the square root sign in the numerator. And so that means we're going to multiply both the numerator and denominator by 6 plus square root of X. And so, doing so, We will get that this is equal to the limit, as X approaches 36. Of the quantity of 36 minus X. In 20 divided by. The quantity of the square root of Xs. Multiplying the quantity of 6 plus square root of X. In quantity, multiplying the quantity of X minus. 36 in quantity. So we can factor out a -1 out of the numerator, so this is going to be equal to. The limit As X approaches 36. Of minus the quantity of X minus 36 in quantity, divided by the quantity of the square root of X, multiplying the quantity of 6 plus square root of X. In quantity, multiplying the quantity of X minus 36. So now we see that the factors of X minus 36 will cancel out, and at this point, we can apply direct substitution to evaluate the limit. This is going to be equal to -1 divided by. The square root of 36, multiplying the quantity of 6, plus the square root of 36. In quantity And when we evaluate this expression, this is going to be equal to. -1, divided by. 72. So, this is the answer to the question. And if we look at our answer choices, and compare our answer that we calculated to the choices, the correct answer choice corresponds to answer choice B. So I hope that this explanation has been useful, and I'll see you in the next video.

Use definition (1) (p. 133) to find the slope of the line tangent to the graph of f at P.
f(x) = 2/√x; P(4,1)

Key Concepts

Tangent Line

Derivative

Limit Definition of Derivative

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