62–65. {Use of Tech} Graphing f and f'
c. Verify th...

Question

62–65. {Use of Tech} Graphing f and f'

c. Verify that the zeros of f' correspond to points at which f has a horizontal tangent line.

f(x) = (x−1) sin^−1 x on [−1,1]

Accepted Answer

Hello. In this video, we are going to be solving the following problem. We are told to consider the function G of X equal to 2 X + 1 multiplied by cosine inverse of X on the interval from -1 to 1. We want to approximate the value of X that corresponds to a point where G of X has a horizontal tangent line by graphene G of X using a graphing utility. So, what exactly are we supposed to do here? Well, we want to approximate the value of X on the interval, but specifically, we want to approximate the value of X where it corresponds to a horizontal tangent line. Since we are looking for a tangent line, what we need to do first is we need to come up with the equation for the derivative of the given G of X function. So again, G of X is equal to the quantity 2 X plus 1 multiplied by cosine inverse of X. So how do we take the derivative of this function? Well, notice that we have a quantity of X multiplied by cosine inverse of X. What we have here is a function in the form of the product of two functions. So in order to use or to take the derivative, we will have to use the product rule. So, let's go ahead and state the 1st and 2nd function. The first function, which we will call F of X, will be 2 X + 1, and the second function G of X will be cosine inverse of X, and the product rule states that we will take the original first and multiply it by the derivative of the second, then add the original second multiplied by the derivative of the 1st. Now, we have no problem taking the derivative of 2 X plus 1, but what about the derivative of cosine inverse of X? Well, before we jump into the product rule, let's just quickly review cosine inverse of X's derivative. If we want to take the derivative of cosine inverse of X, the derivative will equal to the negative 1 divided by the square root of 1 minus X2. This is going to be the derivative of cosine inverse of X. So, let's go ahead and use the product rule to get G of X. So we will take the original first function, which is 2 X + 1, and multiply this by the derivative of the second function, which is negative 1 divided by the square root of 1 minus X2. Then we will add the original second function, which is cosine inverse of X, and multiply this by the derivative of the first function, which by the power rule will just be 2. Now let's go ahead and simplify the derivative. Multiplying the first two functions together will leave us with negative 2 X + 1 divided by the square root of 1 minus X2. And cosine inverse of X multiplied by 2 will leave us with positive 2 multiplied by cosine inverse of X. This is going to be the simplest form of the derivative. And now we want to go ahead and approximate the derivative where the horizontal tangent line exists. Now, the horizontal tangent line means that we are looking for a slope. That has no rise or run, no increase or decrease. So this is going to be a part of the graph where G of X is equal to 0. Let's go ahead and plug in our derivative into a graphing utility. So, what we have here is the graph of the derivative of G X. Where is the graph equal to zero for the derivative? Well, that point is going to be located on the X axis, and the point with the horizontal tangent line located on the X axis is going to be located at 0.46402. Whoops, everything is just breaking. 464,020. And if we take the approximate value to two decimal places of the X value, X will approximately equal to 0.46. And this is going to be the solution to this problem. And the overall solution is going to be a. So I hope this video helps you in understanding how to approach this problem, and I'll go ahead and see you all in the next video.

62–65. {Use of Tech} Graphing f and f'
c. Verify that the zeros of f' correspond to points at which f has a horizontal tangent line.
f(x) = (x−1) sin^−1 x on [−1,1]

Key Concepts

Derivative and Critical Points

Horizontal Tangent Lines

Graphing Functions

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