Problem 39 - Thomas Calculus

Question

\begin{equation} \begin{array}{l}{\text { a. Show that } f(x)=x^{3} \text { and } g(x)=\sqrt[1]{x} \text { are inverses of one }} \\ {\text { another. }} \\\ {\text { b. Graph } f \text { and } g \text { over an } x \text { -interval large enough to show the }} \\ {\text { graphs intersecting at }(1,1) \text { and }(-1,-1) \text { . Be sure the picture }} \\ {\text { shows the required symmetry about the line } y=x \text { . }}\\\\{\text { c. Find the slopes of the tangent lines to the graphs of } f \text { and } g \text { at }} \\\ {(1,1) \text { and }(-1,-1) \text { (four tangent lines in all). }} \\ {\text { d. What lines are tangent to the curves at the origin? }}\end{array} \end{equation}

Step-by-Step Solution

Answer

f(x) and g(x) are inverses; tangents at (1,1) and (-1,-1) have slopes 3 (f) and 1 (g); at origin, tangent to f is horizontal, to g is y=x.

Step 1: Check if f(x) and g(x) are inverses

To prove that two functions \( f(x) \) and \( g(x) \) are inverses, we need to show that \( f(g(x)) = x \) and \( g(f(x)) = x \). Start by evaluating \( f(g(x)) \). Given \( g(x) = x \), then \( f(g(x)) = f(x) = x^3 \). Now evaluate \( g(f(x)) \), which results in \( g(x^3) = x \). Both satisfy the conditions, so \( f(x) \) and \( g(x) \) are indeed inverses.

Step 2: Graph f(x) and g(x)

Graph \( f(x) = x^3 \) and \( g(x) = x \) over the interval, for example \(-2 \leq x \leq 2\). The graph of \( f(x) \) is a cubic curve and \( g(x) \) is a line with a 45-degree angle to the axes, intersecting the cubic curve at points \((1,1)\) and \((-1,-1)\). The graphs should be symmetrical about the line \( y = x \).

Step 3: Find tangent slopes at (1,1) and (-1,-1)

The slope of the tangent line to a curve at a point is given by the derivative of the function at that point. For \( f(x) = x^3 \), the derivative is \( f'(x) = 3x^2 \). At \( x = 1 \), the slope is \( 3 \), and at \( x = -1 \), it is also \( 3 \). The slope of the tangent line for \( g(x) = x \) is \( g'(x) = 1 \). At \( (1, 1) \) and \( (-1, -1) \), the slopes are both \( 1 \).

Step 4: Find tangent lines at the origin

At the origin \((0, 0)\), the slope of the tangent line for \( f(x) = x^3 \) using \( f'(0) = 0 \) is horizontal. For \( g(x) = x \), since its derivative is always \( 1 \), the tangent at the origin is a line of slope \( 1 \), which is \( y = x \).

Key Concepts

Cubic Functions

Cubic functions are polynomials of degree three and have the form \( f(x) = ax^3 + bx^2 + cx + d \), where \( a eq 0 \). These functions are known for their distinct 'S' shaped curve and can cross the x-axis up to three times. The classic example of a cubic function is \( f(x) = x^3 \), which is simple yet exhibits the key characteristics of cubic behavior.

• They always have at least one real root (since they are polynomials of odd degree).
• They can have one or two turning points, requiring calculus to locate them.
• The graph of a cubic function might have points of inflection, where the curve changes concavity.

Understanding cubic functions is important when evaluating transformations, growth patterns, or relationships in algebraically defined systems.

Derivatives

Derivatives represent the rate of change of a function with respect to a variable. They are essential for finding slopes of tangent lines to curves at specific points.
For a cubic function like \( f(x) = x^3 \), the derivative \( f'(x) = 3x^2 \) gives us the slope of the tangent line at any x-value.

• The derivative is a fundamental tool in calculus, used to compute instantaneous rates of change.
• To find the slope of the tangent line at a particular point, plug the x-value into the derivative expression.
• For example, at \( x = 1 \), \( f'(1) = 3(1)^2 = 3 \), indicating the slope of the tangent line is 3.

Derivatives not only help determine slopes, but they also assist in finding maxima, minima, and points of inflection of functions.

Function Graphs

Graphing functions provides a visual representation of mathematical relationships. For the functions \( f(x) = x^3 \) and \( g(x) = x \), graphing helps illuminate their properties and behaviors.

• The cubic function \( f(x) = x^3 \) displays a steep curve increasing exponentially as x moves away from zero in either direction.
• The line \( g(x) = x \) represents a simple linear relationship with an angle of 45 degrees to both axes.
• These functions intersect at points where their values and derivatives may tell us more about their relationships, specifically at \((1, 1)\) and \((-1, -1)\).

Graphing is indispensable for verifying intersections, checking function behavior, and illustrating symmetries, such as those about the line \( y = x \).

Tangent Lines

Tangent lines are straight lines that touch a curve at only one point and have the same slope as the curve did at that point. For any function graph, the tangent line provides a linear approximation to the curve near the point of tangency.

• To find the tangent line at a point, calculate the derivative and evaluate it at that point to find the slope.
• The equation of the tangent line can be written in point-slope form: \( y - y_1 = m(x - x_1) \), where \( m \) is the slope, and \((x_1, y_1)\) is the point of tangency.
• At the origin, the tangent for \( f(x) = x^3 \) is horizontal with a slope of 0, while for \( g(x) = x \), it is \( y = x \), showing how derivatives differ in effect between linear and cubic functions.

Understanding tangent lines is key for approximating values and solving optimization problems.