Problem 57 - Thomas Calculus

Question

In Exercises \(53-60,\) you will explore some functions and their inverses together with their derivatives and linear approximating functions 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. b. Solve the equation \(y=f(x)\) for \(x\) as a function of \(y,\) and name the resulting inverse function \(g\) . c. Find the equation for the tangent line to \(f\) at the specified point \(\left(x_{0}, f\left(x_{0}\right)\right) .\) d. Find the equation for the tangent line to \(g\) at the point \(\left(f\left(x_{0}\right), x_{0}\right)\) located symmetrically across the \(45^{\circ}\) line \(y=x\) (which is the graph of the identity function). Use Theorem 1 to find the slope of this tangent line. e. Plot the functions \(f\) and \(g\) , the identity, the two tangent lines, and the line segment joining the points \(\left(x_{0}, f\left(x_{0}\right)\right)\) and \(\left(f\left(x_{0}\right), x_{0}\right) .\) Discuss the symmetries you see across the main diagonal. $$ y=x^{3}-3 x^{2}-1, \quad 2 \leq x \leq 5, \quad x_{0}=\frac{27}{10} $$

Step-by-Step Solution

Answer

Plot functions and derivatives; confirm one-to-one; solve inverse; find tangent lines; observe symmetries in plots.

Step 1: Plot the Function and Derivative

Begin by graphing the function \(y = f(x) = x^3 - 3x^2 - 1\) as well as its derivative \(f'(x) = 3x^2 - 6x\) over the interval \([2, 5]\). The graph shows that the function is strictly increasing over this interval because its derivative is positive throughout. By the definition of a one-to-one function, if a function is strictly increasing or decreasing, it must be one-to-one over that interval.

Step 2: Solve for the Inverse Function

Solve the equation \(y = x^3 - 3x^2 - 1\) for \(x\) in terms of \(y\). This requires expressing \(x\) as a function of \(y\), which can be complex analytically. Using numerical methods or software, you obtain \(x\) as the function \(g(y)\), the inverse of \(f\). Assuming \(g\) is correctly computed, it reverses the roles of \(x\) and \(y\).

Step 3: Find the Tangent Line to f

The tangent line to the function \(f\) at \(x_0 = \frac{27}{10}\) is found using the derivative. Calculate \(f'\left(\frac{27}{10}\right)\) to find the slope, \(m_f\), of the tangent. Compute \(f\left(\frac{27}{10}\right)\) to obtain the point through which the tangent passes. The equation of the tangent line is \(y - f(x_0) = m_f(x - x_0)\).

Step 4: Find the Tangent Line to g

The tangent line to the inverse function \(g\) at the point \(\left(f(x_0), x_0\right)\) uses Theorem 1, which states the slope of the tangent to \(g\) is the reciprocal of that to \(f\). Hence, the slope of the tangent line to \(g\) at this point is \(1/m_f\). Its equation is \(y - x_0 = (1/m_f)(x - f(x_0))\).

Step 5: Plot All Functions and Lines

Finally, plot the original function \(f\), its inverse \(g\), the line \(y=x\), both tangent lines from Steps 3 and 4, and the line segment joining the points \(\left(x_0, f(x_0)\right)\) and \(\left(f(x_0), x_0\right)\). Observe how both tangent lines are symmetrical around the line \(y = x\), illustrating the reflection property of inverse functions.

Key Concepts

Derivatives

Derivatives are a fundamental concept in calculus, representing the rate at which a function changes at any given point. In our exercise, the derivative of the function \( y = x^3 - 3x^2 - 1 \) is \( f'(x) = 3x^2 - 6x \). This derivative helps us understand the behavior of the function over a particular interval.

By plotting the function and its derivative over the interval \([2, 5]\), we can see that the derivative is positive, indicating that the function is strictly increasing. This information is critical. Why? Because if the function is always increasing, it means that there are no repeated \( y \)-values, and thus every \( x \)-value maps to a single, unique \( y \)-value. This is a key property of a one-to-one function.

• The steepness of the graph represents how rapidly or slowly \( y \) changes with respect to \( x \).
• A higher derivative implies steeper slopes, while a lower derivative implies gentler slopes.
• Understanding derivatives helps in identifying points of interest, such as local maxima, minima, and inflection points.

Tangent Lines

Tangent lines provide essential information about how a function behaves around a specific point. For the function \( f(x) \), the tangent line at \( x_0 \) offers a linear approximation of the function at that point.

The slope of the tangent line to \( f(x) \) at \( x_0 = \frac{27}{10} \) is given by evaluating the derivative \( f'(x) \) at this point. The line's equation is determined using the point-slope form: \( y - f(x_0) = m_f(x - x_0) \), where \( m_f \) is the slope \( f'(x_0) \).

• Tangent lines make complex functions simpler to analyze locally.
• They are used to approximate the function values close to the point of tangency, providing a linear snapshot of the curve.
• In physics, tangent lines can help understand rates of speed or acceleration for changing quantities.

One-to-One Functions

A one-to-one function ensures that each input corresponds to one unique output. This is crucial for a function to have an inverse. In the context of our exercise, determining that a function is one-to-one was done by examining its derivative.

When the derivative is positive or negative over an interval, it indicates that the function is strictly monotonic (always increasing or decreasing). Thus, in the interval \([2, 5]\), since the derivative \( f'(x) \) is positive, we confirm the function is one-to-one.

• Monotonic functions simplify determining whether a function is one-to-one.
• If a function is continuously increasing or decreasing, it cannot "loop back" on itself.
• One-to-one functions are necessary for defining an inverse since an inverse \( g(y) \) will map each \( y \)-value back to a unique \( x \)-value.

Function Inverse Relationship

The relationship between a function and its inverse revolves around swapping the roles of \( x \) and \( y \). Inverse functions essentially "undo" what the original function does. If \( f(x) \) takes \( x \) and converts it to \( y \), then \( g(y) \) will take \( y \) and return it to \( x \).

For our function \( y = x^3 - 3x^2 - 1 \), finding the inverse involves solving it for \( x \) as a function of \( y \). This may not always be straightforward, often requiring numerical tools or computational algebra systems (CAS). Once obtained, the inverse function \( g(y) \) can be analyzed similarly to \( f(x) \).

• The graph of the inverse function is symmetrical to the original function across the line \( y = x \).
• Reflecting graphical properties between \( f \) and inverse \( g \) can visualize their relationship.
• The slope of the tangent line for the inverse is the reciprocal of the slope of the tangent line for the original function.