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)=(x²−1)sin^−1 x on [−1,1]

Accepted Answer

Hello. In this video, we are going to be approximating a value of X that corresponds to a horizontal tangent line. So the problem tells us to consider the function H of X, which is equal to 2 X + 1, multiplied by sin inverse of X on the interval from -1 to 1. We want to approximate the value of X that corresponds to a point where H of X has a horizontal tangent line by graphing HX using a graphing utility. So, how can we approximate a value of X for a horizontal tangent line? Well, since we are looking for a value that is located on a horizontal tangent line, this indicates that we are going to have to take the derivative, since we are looking for a value of a tangent line in general. So, we have this function H X equal to 2 X plus 1 multiplied by sine inverse of X. Now, if we want to approximate the value of X at a horizontal tangent line, we must take the derivative. Notice that H of X is given to us in the form of a product of two functions of X. What this means is that we are going to have to find the derivative by using the product rule. So, we will go ahead and label the first function 2 X + 1 as our F of X for the product rule, and we will labelsine inverse of X as our second function G of X for the product rule. Now, the product rule states that we will multiply F of X with the derivative of G of X, then add G of X multiplied by the derivative of F of X. Now, we have no problem taking the derivative of 2 X plus 1, but what about the derivative of sine inverse of X? Well, let's go ahead and review what the derivative of sine inverse of X is. The derivative of sine inverse of X is defined as positive 1 divided by the square root of 1 minus X squad. So the standard derivative of sine inverse is given to us as this definition, and we will be using this definition in the product rule. So let's go ahead and take the derivative. Of our H of X function. So, we will take the original F of X, the first function, which is 2 X + 1, then multiply this by the derivative of sin inverse of X, which is going to be positive 1 divided by the square root of 1 minus X2. Then we will add sine inverse of X, our second function, and multiply this by the derivative of the first function. And by the power rule, the derivative of 2 X + 1 will just be 2. Now let's go ahead and simplify. Multiplying the first two functions together will simplify to be 2 x + 1 divided by the square root of 1 minus X2, and sine inverse of X multiplied by 2 will give us 2 multiplied by sine inverse of X. This is going to be the simplest form of the H X derivative. Now, we want to approximate the value of X at the horizontal tangent line, and a horizontal tangent line means that we are looking for a tangent line that has no rise or run, no change in slope, and at this point, the derivative H of X will equal to zero. So this is where we're going to need to use our graphing utility. Let's go ahead and take our HX function and plug it into a graphing utility. So what I have here Is the graph of the H X function we just solved for. Now, where is H X equal to 0 on this graph? Well, since this is the graph of H X, this graph will equal to 0 at the X intercept located at the point negative 0.25276.0. What this means is that the approximate value of X rounded to two decimal places for the original H of X function is going to be negative 0.25. This is going to be the approximate value of X for H X located at the horizontal tangent line. And with that being said, the overall solution is going to be a. So I hope this video helps you in understanding how to approach this problem, and I will 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)=(x²−1)sin^−1 x on [−1,1]

Key Concepts

Derivative and Critical Points

Horizontal Tangent Lines

Graphing Functions

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