Question

Conjecture Consider the functions \(f(x)=x^{2}\) and \(g(x)=x^{3}\). (a) Graph \(f\) and \(f^{\prime}\) on the same set of axes. (b) Graph \(g\) and \(g^{\prime}\) on the same set of axes. (c) Identify a pattern between \(f\) and \(g\) and their respective derivatives. Use the pattern to make a conjecture about \(h^{\prime}(x)\) if \(h(x)=x^{n},\) where \(n\) is an integer and \(n \geq 2\) (d) Find \(f^{\prime}(x)\) if \(f(x)=x^{4}\). Compare the result with the conjecture in part (c). Is this a proof of your conjecture? Explain.

Short answer

The conjecture that the derivative of \(h(x)=x^{n}\) is \(h^{\prime}(x)=nx^{n-1}\) is confirmed for the functions \(x^{2}, x^{3}, x^{4}\), though this does not constitute a proof, which would necessitate showing it to be true for all \(n\).

Step-by-step solution

Graphing \(f(x)=x^{2}\) and \(f^{\prime}(x)\)

First, graph the function \(f(x)=x^{2}\). The derivative of \(f(x)=x^{2}\) is \(f^{\prime}(x)=2x\). Graph this derivative function on the same axes as the original function.

Graphing \(g(x)=x^{3}\) and \(g^{\prime}(x)\)

Next, graph the function \(g(x)=x^{3}\). The derivative of \(g(x)=x^{3}\) is \(g^{\prime}(x) = 3x^{2}\). Graph this derivative function on the same axes as the original function.

Identifying pattern and formulating the conjecture

In the graphs of \(f(x)\) and \(g(x)\) and their derivatives, observe the pattern: When \(f(x)=x^{n}\), it appears that \(n\) becomes a coefficient in the derivative, and the exponent decreases by 1, yielding \(f^{\prime}(x)=nx^{n-1}\). Based on this observation, form the conjecture.

Verifying the conjecture

To verify the conjecture, take the function \(h(x)=x^{4}\). Compute the derivative to be \(h^{\prime}(x)=4x^{3}\). This confirms the conjecture: When \(h(x)=x^{n}\), then \(h^{\prime}(x)=nx^{n-1}\). However, this not a proof of the conjecture. A proof would require showing this to be true for all \(n\), not just some specific examples.

Key concepts

Graphing Functions

Understanding the visual representation of functions is a fundamental skill in calculus. When graphing functions like f(x) = x^2 and g(x) = x^3, we create a visual layout that helps to reveal the relationship between the input values (x) and the output values (f(x) or g(x)). The curve of f(x) = x^2 is a parabola opening upwards, showing that as x increases or decreases away from zero, f(x) grows at an increasing rate. On the other hand, g(x) = x^3 introduces an inflection point, where the direction of the curve changes.

Graphing these functions on the same axes as their derivatives allows students to see how the rate of change of the function, which is the derivative, behaves relative to the function itself. In general, graphing is a crucial tool for understanding the behavior of functions and their derivatives, offering an intuitive approach to calculus problems.

Identifying Patterns in Derivatives

A pattern in derivatives refers to a consistent rule or behavior that occurs when taking the derivative of different functions. In the exercise, we identify a clear pattern when differentiating power functions like f(x) = x^n. The derivative of f(x) = x^2 is 2x, and the derivative of g(x) = x^3 is 3x^2. This pattern shows that the exponent n becomes a coefficient in the derivative, and the new exponent is n - 1.

This observation can be extended to hypothesize that for any power function h(x) = x^n, the derivative will follow suit as h'(x) = nx^{n-1}. Identifying such patterns is valuable as they help simplify complex derivative calculations and are indicative of underlying principles in calculus.

Power Rule for Derivatives

The Power Rule is one of the most fundamental and widely used rules for finding derivatives in calculus. It states that if f(x) = x^n, where n is a real number, then the derivative of f with respect to x is f'(x) = nx^{n-1}. This rule quickly and efficiently calculates the slope of the tangent line to the function at any point x.

Applying the Power Rule to the exercise, for the function f(x) = x^4, the derivative f'(x) is found to be 4x^3. The consistent decrease of the exponent by 1 and the multiplication by the original exponent provide a quick verification of patterns noted in previous exercise steps, reinforcing the practical utility of the Power Rule in derivative calculations.

Conjecture Verification

Conjecture verification in mathematics involves taking a hypothesis, based on observed patterns or specific cases, and testing its validity. It is important to recognize that verifying a conjecture through specific examples, as done with h(x) = x^4 matching the hypothesized pattern h'(x) = nx^{n-1}, does not constitute a mathematical proof. True verification requires proof that the conjecture holds for all possible cases within its domain.

In this exercise, we used specific functions to bolster our confidence in the conjecture that the derivative of x^n is nx^{n-1}, but we need to perform a formal proof to confirm this conjecture universally. Nonetheless, the exercise of conjecture verification is a crucial step in the scientific method and mathematical theory development, serving as a precursor to rigorous proof.