Hypothesis: Conclusion: The Mean Value Theorem Sec 3.2: #2,4,16,20,23,38-48 evens, 51,52 We begin with a common-sense geometrical fact: …somewhere between two zeros of a continuous function f, the function must change direction For a differentiable function, the derivative is 0 at the point where f changes direction. Thus, we expect there to be a point c where the tangent is horizontal. These ideas are stated in Rolle's Theorem: Rolle’s Theorem Hypothesis: Conclusion: (1) Let f be differentiable on an OPEN interval (a,b); then there is at least (2) and f is continuous on a CLOSED interval [a,b]; one point c in (a,b) (3) If f(a) = f(b) (think of zeros shifted up or down) for which f ’(c) = 0

Hypothesis: Conclusion: Rolle’s Theorem Hypothesis: Conclusion: (1) Let f be differentiable on an OPEN interval (a,b); then there is at least (2) and f is continuous on a CLOSED interval [a,b]; one point c in (a,b) (3) If f(a) = f(b) for which f ’(c) = 0 Notice that both conditions (1) and (2) on f are necessary. Without either one, the statement is false! For a discontinuous function, the conclusion of Rolle’s Theorem may not hold. For a continuous, but NOT differentiable function, again the conclusion may not hold.

The Mean Value Theorem is a generalization of Rolle's Theorem: We now let f(a) and f(b) have values that are not equal to each other and look at the secant line through (a, f(a)) and (b, f(b)). We expect that somewhere between a and b there is a point c where the tangent is parallel to this secant. Hypothesis: (1) Let f be differentiable on an OPEN interval (a,b); (2) and f is continuous on a CLOSED interval [a,b]; Conclusion: Then, there is at least one point c in (a, b) such the slope of the secant line = the slope of the tangent line. We can write the conclusion as the formula …

How to apply the Mean Value Theorem To apply the Mean Value Theorem to the function on the interval (a = -1, b = 3) First calculate the slope of the secant line: Then, take the derivative …. Equate the derivative to the slope of the secant line: Solve this equation for x=c, the mean value

Secant Line has slope = -3. Check that you have the correct mean value x=1 by looking at the graph of on the interval (a = -1, b = 3) Secant Line has slope = -3. The tangent line is parallel to the secant line because it has the same slope. The tangent line is connected to f(x) at the “mean value” c = 1

Sec 3-2 #37: Apply the Mean Value Theorem to the function on the interval (a = -1, b = 2) First, calculate the slope of the secant line: Next, take the derivative …. Finally, equate the derivative to the slope of the secant line: Solve this equation for the mean value, x

To check that you have the correct mean value, x=c, by doing the following: Write the equation of the secant line that goes through (-1,f(-1) ) & (2, f(2) ) Write the equation of the tangent line to connected at the mean value, x=c. Graph the function, the secant line and the tangent line on a domain that includes the interval, -1 < x < 2

Go back and compare Rolle’s Theorem and the Mean Value Theorem. What is similar about the hypothesis in these two theorems? What is different about the hypothesis? What is similar about the conclusion in these two theorems? What is different about the conclusion? Examples: Explain why Rolle’s Theorem does not apply to the function even though there exists a and b such that f(a)=f(b). #1. #2.

Examples: Determine whether the Mean Value Theorem can be applied to f on the closed interval [a, b]. If the Mean Value Theorem can be appplied, find all values of c in the open interval (a, b) such that . If the Mean Value Theorem can not be applied, explain why not. #39. #45.