In Exercises 67–72, you will explore some functions and their ...

Question

In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:

e. Plot the functions f and g, the identity, the two tangent lines, and the line segment joining the points (x_0, f(x_0)) and (f(x_0), x_0). Discuss the symmetries you see across the main diagonal y=x.

72. y= 2-x-x³, -2 ≤ x ≤ 2, x_0 = 3/2

Accepted Answer

Welcome back everyone. Find the slope of the tangent line to the inverse function g of x equals f inverse of x at the point x f x0. F x is equal to 3 minus 2 x minus x cubed, and x0 is equal to 1. A -5, B5, C 15, and D 1/5. So for this problem we can recall that the slope m. Of the tangent line to the inverse function g of x we can call it m subscript inverse. It is equal to 1 divided by. The derivative of the original function f at the point x knot. So what we are going to do in this problem is identify the derivative of the original function f of x. F x is equal to the derivative of 3 minus 2x minus x cubed. The derivative of 3 is 0. The derivative of x is 1, so we're going to get -2 minus, according to the power rule, the derivative of x cubed is 3 x 2. And now we can find F at x0, which is F at 1 because x is a 1. And we're going to get the -2 minus 3 multiplied by. 1 squared and this is equal to -5, right? So the slope. Of the tangent line to the inverse function is equal to 1 divided by this value, right, which is 1 divided by -5, and that would be equal to -15. Therefore, the correct answer corresponds to the answer of choice C. Thank you for watching.

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