Find G if .

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Begin by cross-multiplying; the equation should be true if you plug in x = 2. You can verify your answer by plugging it into the function; the resulting limit is, indeed, 1.

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Calculus II (BC): Infinite limits

(b)    How many times is the Mean Value Theorem satisfied on the interval [3,3]?

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(a)    According to the graph, f(1) is approximately 1.5, since the graph appears to contain the point (1,1.5). Your answer may vary a little bit, since this is only an approximation. To approximate f '(1), you have to draw an approximate tangent line at x = 1, like the red one below. The derivative at x = 1 will be the slope of this tangent line. To find the slope, choose two points which appear to be on the line. It appears that (0,1) and (1.5,3) are on the line, so calculate the change in y divided by the change in x: (31)/(1.50) = 4/1.5 = 2.667. Again, answers may vary, but they should be relatively close to 2.667.

(b)    The Mean Value Theorem states that the slope of the tangent line somewhere on the interval [3,3] will equal the slope of the secant line from x = 3 to x = 3.

The secant line is drawn in blue, and the four tangent lines which have the identical slopes as (are tangent to) the secant line are drawn in purple. Four such tangent lines exist, so the answer is 4.

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