Consider the functions and (a) Graph and on the same set of axes. (b) Graph and on the

Chapter 2, Problem 100

(choose chapter or problem)

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.

Text Transcription:

f(x) = x^2

g(x) = x^3

f^prime

g^prime

h^prime (x)

h(x) = x^n

n geq 2

f^prime (x)

f(x) = x^4