Problem 58 - 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. $$ y=2-x-x^{3}, \quad-2 \leq x \leq 2, \quad x_{0}=\frac{3}{2} $$

Step-by-Step Solution

Answer

The function is one-to-one since it is strictly decreasing. The inverse is found numerically. Tangent lines at symmetric points are determined using slopes.

Step 1: Plot the Function and its Derivative

First, we plot the function \( y = f(x) = 2 - x - x^3 \) over the interval \(-2 \leq x \leq 2\). Next, we find its derivative, \( f'(x) = -1 - 3x^2 \) and plot this as well. By observing the plot of the derivative, determine where \( f'(x) \) does not change sign over the interval, indicating that \( f(x) \) is one-to-one. Since \( f'(x) < 0 \) everywhere in the given interval, \( f \) is strictly decreasing and hence one-to-one.

Step 2: Solve for the Inverse Function

Solve \( y = 2 - x - x^3 \) for \( x \) in terms of \( y \). This is a cubic equation in \( x \) that needs solving. The inverse function \( g(y) \) does not have a simple analytical expression but can be expressed using numerical or symbolic methods as necessary.

Step 3: Find the Tangent Line to f at x_0

To find the tangent line to the function \( f\) at \( x_0 = \frac{3}{2} \), we use the point \( \left(x_0, f(x_0)\right) = \left(\frac{3}{2}, f(\frac{3}{2})\right) = \left(\frac{3}{2}, -\frac{11}{8}\right) \). The slope at this point is \( f'(\frac{3}{2}) = -1 - 3\left(\frac{3}{2}\right)^2 = -\frac{35}{4} \). Thus, the equation of the tangent line is \( y + \frac{11}{8} = -\frac{35}{4}(x - \frac{3}{2}) \). Simplifying gives the tangent line equation.

Step 4: Find the Tangent Line to g at the Symmetric Point

The point symmetric to \( \left(\frac{3}{2}, -\frac{11}{8}\right) \) across \( y = x \) is \( \left(-\frac{11}{8}, \frac{3}{2}\right) \). Since the slope of \( f \)'s tangent is \( f'(\frac{3}{2}) \), the slope of \( g \)'s tangent at this symmetric point is the reciprocal, \( \frac{1}{f'(\frac{3}{2})} = -\frac{4}{35} \). Therefore, the tangent line equation at \( g \) is obtained using this slope.

Step 5: Plotting and Discussing Symmetries

Plot the functions \( f \), \( g \), the identity function \( y = x \), and the two tangent lines. Additionally, draw the line segment joining the points \( \left(\frac{3}{2}, -\frac{11}{8}\right) \) and \( \left(-\frac{11}{8}, \frac{3}{2}\right) \). These plots allow us to visually verify the symmetry about the line \( y = x \), as the graph of \( g \) is a reflection of \( f \) across this line.

Key Concepts

Derivatives

Derivatives are a fundamental concept in calculus, representing the rate at which a function's value changes as its input changes. Suppose you have a function \( f(x) = 2 - x - x^3 \). To determine where this function is increasing or decreasing, you find its derivative by applying differentiation rules. The derivative is calculated as \( f'(x) = -1 - 3x^2 \).
This derivative tells us how \( f(x) \) behaves. If \( f'(x) < 0 \) across an interval, the function is decreasing - which is the case here for all \( x \) in \(-2 \leq x \leq 2\). This constant negative derivative confirms that our function is one-to-one on this interval. A single unchanged sign indicates consistent monotonic behavior, making it possible to define an inverse function over the given domain.
Remember:

• A positive derivative implies a function is increasing.
• A negative derivative implies a function is decreasing.
• A zero derivative might indicate a constant function or a local extremum point.

Understanding derivatives is essential as they provide insights into the shape and behavior of graphs.

Tangent Lines

Tangent lines are straight lines that touch a curve at a single point without crossing it immediately. They represent the instantaneous rate of change or the slope of the curve at that specific point. For a function \( f \), the tangent line at \( x_0 \) has the slope \( f'(x_0) \).
In the exercise, the point of interest is \( x_0 = \frac{3}{2} \). The function \( f(x) = 2 - x - x^3 \) has the point \( \left( \frac{3}{2}, f\left(\frac{3}{2}\right) \right) = \left( \frac{3}{2}, -\frac{11}{8} \right) \). The derivative at this point gives the slope of \( -\frac{35}{4} \). The equation of the tangent line, using the point-slope formula \( y - y_1 = m(x-x_1) \), becomes:\[y + \frac{11}{8} = -\frac{35}{4}(x - \frac{3}{2})\]
This linear equation provides a simple way to approximate \( f \) near \( x_0 \). Approximations are valuable in solving complex problems, where analyzing linear behavior simplifies understanding regional behavior of non-linear functions.

Symmetry

Symmetry in functions, especially regarding inverse functions, is intriguing. Reflective symmetry about the line \( y = x \) is most common when discussing inverses. When viewing the graph of a function and its inverse, if you can flip the curve over the \( y = x \) line and they look identical, you have confirmed the symmetry.
In this exercise, the function \( f(x) \) and its inverse \( g(y) \) exhibit such symmetry. The symmetric reflection is marked by the line \( y = x \). If point \( (a, b) \) is on f, then \( (b, a) \) is on \( g \). For instance, \( \left(\frac{3}{2}, -\frac{11}{8}\right) \) on \( f \) reflects to \( \left(-\frac{11}{8}, \frac{3}{2}\right) \) on \( g \).
This symmetry provides a clear visual demonstration that the two functions are inverse, supporting conceptual understanding of inverse relationships.
To identify symmetry, check for:

• Reflective symmetry across 45-degree line or \( y = x \).
• Matching points that swap x and y coordinates.

Symmetrical interpretations simplify predictions about functions and their interactions.

One-to-One Functions

One-to-one functions, or injective functions, are those where each output is mapped uniquely by exactly one input. This ensures that no two different inputs produce the same output, crucial for defining inverse functions.
For \( f(x) = 2 - x - x^3 \), we established that its derivative \( f'(x) < 0 \) over \(-2 \leq x \leq 2\), showing that it is strictly decreasing and, therefore, one-to-one within this interval. Only such functions can have meaningful inverses that are also functions rather than multivalued.
Indications of a one-to-one function include:

• Consistent increase or decrease of the function's values.
• Strict monotonicity indicated by derivatives not changing sign.
• Passing the "Horizontal Line Test" where any horizontal line crosses the graph at most once.

Understanding whether a function is one-to-one is vital when solving equation inversions or understanding underlying data mappings in real-world contexts.