Consider the function f(x) = x2 – x. Compute the average rate of change of f as x varies from 1 to 1 + h. Suppose now that h is a number that is close to zero, but is slightly larger than 0 or slightly smaller than 0. The average rate of change of f as x varies from 1 to 1 + h is the slope of the line that is passing through the points (1, f(1)) and (1+h, f(1+h)). Imagine that the first point is fixed, but that the second point is a point on the curve that is moving towards the first.
The graph attached shows the function along with two secant lines, one with h=1 and another with h=0.5 and the tangent line. Note how the slopes on the secant lines approach the slope of the tangent line as h approaches zero. Answer the following questions:
What is the average rate of change of f as x varies from 1 to 1 + h?
Find specific values for the average rate of change of f as x varies from 1 to 1 + h for h = 0.1, 0.01, and 0.001. What is your estimate of the slope of the tangent line?
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