In Exercises 19–22, find the slope of the curve at the point i...

Question

In Exercises 19–22, find the slope of the curve at the point indicated.

y = (x − 1) / (x + 1), x = 0

Accepted Answer

Welcome back, everyone. Determine the slope of the tent lines to the curve at the specified point. Y equals x minus 11 divided by X + 11 at X equals 0. Where given4 answer choices A says -1 divided by 11, B 1 divided by 11, C2 divided by E11, and D-2 divided by E11. For this problem, lest recall that the slope of the tangent lines to the curve at the specified point which can be represented by M can be found as limit as x approaches the point of tangency in this case 0 of the value of F of X minus F at the given point. So in this case 0, right, divided by x minus. The point of tangency, so x minus 0. So this is the definition that we want to apply to identify the slope of the tangent line at X equals 0. Let's go ahead and use our functions to solve for the limit. We get limit as X approaches 0 of F of X, which is X minus 11 divided by X + 11. And then we are subtracting F of 0, meaning every x becomes 0, right? So F of 0 is equal to 0 minus 11 divided by 0 + 11, and this gives us -11 divided by 11, which is -1. So we are subtracting -1. And dividing everything by x. So let's simplify. We are going to find the common denominator. We get the limit as x approaches 0 of X minus E11. Notice that 1 minus -1 is simply. Replace with addition, right? We can take X minus 11 divided by X + 11, and instead of saying minus -1, we're just going to add 1, right? So we have rewritten our numerator and we get X minus 11. Plus X + 11 because we're looking for the common denominator which is X + 11. We can clearly see that if we split our new fraction, we get X minus 11 divided by X + 1 + X + 11 divided by X + 11, which gives us that 1, right? And then we are dividing by X. So now let's combine the like terms. We get the limit as X approaches 0 of 2 X. And since we have a double division, we can just combine those denominators, which gives us X multiplied by X + 11. And what we can see is that we can cancel out X as a common factor, and we get limit as X approaches 0 of 2. Divided by X plus 111. Now let's attempt the right substitution, which gives us 2 divided by 0 + 11, which is 2 divided by 11, right? Therefore, we have got our final slope, which is to divided by 11, and this corresponds to the answer to C. Thank you for watching.

In Exercises 19–22, find the slope of the curve at the point indicated.

y = (x − 1) / (x + 1), x = 0

Key Concepts

Derivative

Quotient Rule

Substitution

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