Consider the functions f(x) = x2 and g(x) = x3 . (a) Graph f and f′ on the same set of axes. (b) Graph g and g′ 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′(x) if h(x) = xn , where n is an integer and n ≥ 2. (d) Find f′(x) if f(x) = x4 . Compare the result with the conjecture in part (c). Is this a proof of your conjecture? Explain.
Rate of Change
The relation between two quantities which displays how much greater one quantity is than another is called ratio.
Slope
The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:
Consider the functions f(x) = x2 and g(x) = x3 . (a) Graph f and f′ on the same set of axes. (b) Graph g and g′ 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′(x) if h(x) = xn , where n is an integer and n ≥ 2. (d) Find f′(x) if f(x) = x4 . Compare the result with the conjecture in part (c). Is this a proof of your conjecture? Explain.
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