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:
a. Plot the function y=f(x) together with its derivative over the given interval. Explain why you know that f is one-to-one over the interval.
72. y= 2-x-x³, -2 ≤ x ≤ 2, x_0 = 3/2

Verified step by step guidance

1. First, identify the function given: \(y = f(x) = 2 - x - x^{3}\), defined on the interval \([-2, 2]\) with the point \(x_0 = \frac{3}{2}\).
2. Calculate the derivative of the function \(f(x)\) with respect to \(x\). Use the power rule for derivatives: \(f'(x) = \frac{d}{dx}(2) - \frac{d}{dx}(x) - \frac{d}{dx}(x^{3})\) which simplifies to \(f'(x) = 0 - 1 - 3x^{2} = -1 - 3x^{2}\).
3. Plot both the original function \(f(x)\) and its derivative \(f'(x)\) over the interval \([-2, 2]\). This visual will help you analyze the behavior of the function and its slope at different points.
4. To determine if \(f\) is one-to-one on the interval, examine the sign of the derivative \(f'(x)\) over \([-2, 2]\). If \(f'(x)\) does not change sign (always positive or always negative), then \(f\) is strictly monotonic and therefore one-to-one on that interval.
5. Finally, use the derivative at the point \(x_0 = \frac{3}{2}\) to find the slope of the tangent line at that point. Then, write the equation of the tangent line using the point-slope form: \(y - f(x_0) = f'(x_0)(x - x_0)\).

Key Concepts

One-to-One Functions

A function is one-to-one if it never takes the same value twice, meaning each output corresponds to exactly one input. This property is essential for a function to have an inverse. Over a given interval, checking if the function is strictly increasing or decreasing helps confirm it is one-to-one.

Derivative and Monotonicity

The derivative of a function indicates its rate of change. If the derivative is positive over an interval, the function is increasing; if negative, it is decreasing. This monotonic behavior ensures the function is one-to-one on that interval, which is crucial for the existence of an inverse.

Tangent Line Approximation

The tangent line approximation uses the derivative at a specific point to approximate the function near that point. It is given by the equation y = f(x₀) + f'(x₀)(x - x₀), providing a linear estimate of the function’s behavior close to x₀, useful for understanding local changes.