Problem 84 - Precalculus with Limits

Question

CONJECTURE Consider the functions \(f(x) = x^3\) and \(g(x) = x^3\). (a) Sketch the graphs of \(f\) and \(f'\) on the same set of coordinate axes. (b) Sketch the graphs of \(g\) and \(g'\) on the same set of coordinate axes. (c) Identify any pattern between the functions \(f\) and \(g\) and their respective derivatives. Use the pattern to make a conjecture about \(h'(x)\) if \(h(x) = x^n\), where \(n\) is an integer and \(n \geq 2\).

Step-by-Step Solution

Answer

When \(h(x) = x^n\), where \(n\) is an integer with \(n \geq 2\), the derivative \(h'(x)\) is \(n \cdot x^{n-1}\).

Step 1: Calculate the derivative of f(x) and sketch f(x)

First, calculate the derivative of \(f(x) = x^3\). Using the power rule for differentiation, the derivative \(f'(x)\) is found to be \(3x^2\). Now, sketch the cubic function \(f(x) = x^3\) on a coordinate plane.

Step 2: Sketch f'(x)

On the same coordinate plane used in the previous step, sketch \(f'(x) = 3x^2\). This is a parabolic function which opens upwards.

Step 3: Repeat process for function g

Following the same steps as before, calculate the derivative of \(g(x) = x^3\). As before, we get \(g'(x) = 3x^2\). Sketch \(g(x) = x^3\) and \(g'(x) = 3x^2\) on the same coordinate axis.

Step 4: Identify pattern

Upon examining the graphs, a pattern indeed emerges. In both cases, the derivative of the cubic function is a quadratic function. The degree of the function (in this case, 3) is reduced by 1 (to 2).

Step 5: Make a conjecture for h'(x)

Based on the pattern witnessed, we can conjecture that when \(h(x) = x^n\) (where \(n\) is an integer and \(n \geq 2\)), the derivative \(h'(x)\) would take the form \(n \cdot x^{n-1}\). This is referred to as the power rule in differentiation and applies for all cases of \(n \geq 2\).

Key Concepts

Derivatives of Functions

The derivative of a function represents the rate at which the function's value changes with respect to changes in its input. For a function like \( f(x) = x^3 \), the derivative \( f'(x) \) can be found using a technique called differentiation. This process provides a new function that explains how \( f \) changes at each point along its graph.

To find the derivative, we often use the **Power Rule**, which is particularly useful for polynomial functions. The Power Rule states that if you have a function \( x^n \), its derivative \( f'(x) \) will be \( n \cdot x^{n-1} \).

For example, applying this rule to \( f(x) = x^3 \) gives us a derivative of \( f'(x) = 3x^2 \). This calculation shows how the slope of the tangent line to the curve changes and offers insight into the behavior of the function.

Polynomial Functions

Polynomial functions are algebraic expressions that consist of variables raised to whole-number exponents and coefficients. These functions can take various forms, such as linear (\( x \)), quadratic (\( x^2 \)), cubic (\( x^3 \)), and so on.

In the exercise, both \( f(x) \) and \( g(x) \) are examples of cubic polynomial functions, specifically \( f(x) = x^3 \) and \( g(x) = x^3 \). These functions form smooth, continuous curves when graphed, showcasing a characteristic S-shaped path for cubic polynomials.

• The leading term, \( x^3 \), determines the overall shape of the curve and its end behavior.
• All other terms and the leading coefficient influence the curve's steepness and where it crosses axes.

Understanding polynomial functions is crucial as they serve as building blocks in calculus for more complex functions.

Graphing Derivatives

When graphing derivatives, it’s important to understand what the derivative graph tells us. For a function \( f(x) = x^3 \), its graph is a curve, and its derivative \( f'(x) = 3x^2 \) is a parabola.

Graphing both functions on the same coordinate axes allows us to visually identify the relationship between a function and its derivative. The key insights include:

• The points where the original graph is horizontal correspond to where the derivative graph crosses zero.
• The derivative graph helps to visualize where the original function is increasing or decreasing.
• Since \( 3x^2 \) is always positive for \( x eq 0 \), the original function is always increasing except at the origin.

Seeing these relationships helps to develop an intuition for the dynamics of polynomial functions and their derivatives. Visualizing these graphs reinforce abstract concepts and help to conceptualize how changes in the function manifest in its derivative.