Use the graph of f to estimate the values of c that satisfy the conclusions of the Mean Value Theorem for the interval [0,8].

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Source: James Stewart's Calculus: Concepts and Contexts, p. 292.

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Solution: Remember that the Mean Value Theorem guarantees that somewhere within a closed, differentiable, continuous interval, the instantaneous rate of change will be equal to the average rate of change over the entire interval. Graphically, the average rate of change is given by the slope of the secant line connecting the beginning and end of the interval (in our example, I have drawn it below in blue). Our job it to approximate (that's all we can do without an actual function!) when the slope of the tangent lines (the instantaneous velocity) are equal to the slope of the secant line (graphically, the lines will be equal as the slopes are the same). I have drawn these tangents in green below. Remember, these are approximations, but my guess is that the values guaranteed by the Mean Value Theorem occur at approximately x = .8, 3.4, 4.4, and 6.2.

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Calculus II (BC): Average Value with a Vengeance       Given the function , find the average area of the triangle formed by the tangent line to f, the x-axis, and the y-axis, for x on [.5, 5].

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Solution:

To find the average area of the triangle, we need a formula which gives the area of a triangle in this instance. Clearly, the x- and y-intercepts of the tangent line will give us the height and base of the triangle, so we need to find them, but first we need the equation of the tangent line. We rewirte f to get f(x)=x ^{- 1}+1. Therefore, f '(x)=-x ^{- 2}. This gives us the slope of the tangent line. A definite point on the tangent line will be (x, f(x)). Utilizing point-slope form and substituting zero in for x and y (one at a time), we can find the y- and x-intercepts, respectively. Here's how to get the y-intercept: Now we know the base and height of the triangle, so (1/2)bh=A, and this is the functions of which we want to the find average value. First, we simplify the area function: All that's left to do now is to find the average value of this much more attractive function: So, the final answer is 12.125. Ugh.

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