Evaluate the summations and products in ( 1 1 + 1 ) ( 2 2 + 1 ) ( 3 3 + 1 ) ⋯ ( k k + 1 ) ; k = 3 .

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Chapter 5.1, Problem 35ES

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Evaluate the summations and products in (11+1)(22+1)(33+1)⋯(kk+1); k=3.

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Consider the function f(x) = x²-1. (a) Find the instantaneous rate of change of f(x) at x=1 using the definition of the derivative. Show all your steps clearly. (b) Sketch the graph of f(x) around x = 1. Draw the secant line passing through the points on the graph where x 1 and x-> 1+h (for a small positive value of h, illustrate conceptually). Then, draw the tangent line to the graph at x=1. Explain how the slope of the tangent line relates to the value you found in part (a). (c) In a few sentences, explain what the instantaneous rate of change of f(x) at x = 1 represents in the context of the graph of f(x). How does the rate of change of this function vary at different points?