Math 103 Day 13: The Mean Value Theorem and Tuesday How Derivatives OctoberShape 26, 2010 a Graph1 / 12

Transcription

1 Math 103 Day 13: The Mean Value Theorem and How Derivatives Shape a Graph Ryan Blair University of Pennsylvania Tuesday October 26, 2010 Math 103 Day 13: The Mean Value Theorem and Tuesday How Derivatives OctoberShape 26, 2010 a Graph1 / 12

2 Outline 1 The Mean Value Theorem 2 How Derivatives Shape a Graph Math 103 Day 13: The Mean Value Theorem and Tuesday How Derivatives OctoberShape 26, 2010 a Graph2 / 12

3 The Mean Value Theorem Theorem (Rolle s Theorem) Let f be a function that satisfies the following three hypothesis: 1 f is continuous on the closed interval [a,b]. 2 f is differentiable on the open interval (a,b). 3 f (a) = f (b). Then there is a number c in (a,b) such that f (c) = 0 Math 103 Day 13: The Mean Value Theorem and Tuesday How Derivatives OctoberShape 26, 2010 a Graph3 / 12

4 The Mean Value Theorem Theorem (Rolle s Theorem) Let f be a function that satisfies the following three hypothesis: 1 f is continuous on the closed interval [a,b]. 2 f is differentiable on the open interval (a,b). 3 f (a) = f (b). Then there is a number c in (a,b) such that f (c) = 0 Example: f (x) = 1 x 2 on [ 1,1]. Math 103 Day 13: The Mean Value Theorem and Tuesday How Derivatives OctoberShape 26, 2010 a Graph3 / 12

5 The Mean Value Theorem Example Verify that the function f (x) = 5 12x + 3x 2 satisfies the hypothesis of Rolle s Theorem on [1,3]. Then find all c in [1,3] such that f (c) = 0. Math 103 Day 13: The Mean Value Theorem and Tuesday How Derivatives OctoberShape 26, 2010 a Graph4 / 12

6 The Mean Value Theorem Theorem (Mean Value Theorem) Let f be a function that satisfies the following hypothesis: 1 f is continuous on the closed interval [a,b]. 2 f is differentiable on the open interval (a,b). Then there is a number c in (a,b) such that f (c) = f (b) f(a) b a Math 103 Day 13: The Mean Value Theorem and Tuesday How Derivatives OctoberShape 26, 2010 a Graph5 / 12

7 The Mean Value Theorem Theorem (Mean Value Theorem) Let f be a function that satisfies the following hypothesis: 1 f is continuous on the closed interval [a,b]. 2 f is differentiable on the open interval (a,b). Then there is a number c in (a,b) such that Example: f (x) = 1 x 2 on [ 1,1]. f (c) = f (b) f(a) b a Math 103 Day 13: The Mean Value Theorem and Tuesday How Derivatives OctoberShape 26, 2010 a Graph5 / 12

8 The Mean Value Theorem ExampleVerify that f (x) = 3x 2 + 2x + 5 satisfies the hypothesis of the Mean Value Theorem on [ 1,1]. Then find all numbers c satisfying the conclusion of the Mean Value Theorem. Math 103 Day 13: The Mean Value Theorem and Tuesday How Derivatives OctoberShape 26, 2010 a Graph6 / 12

9 The Mean Value Theorem Exercise Show that f (x) = x c has at most one real root in [ 2,2]. Math 103 Day 13: The Mean Value Theorem and Tuesday How Derivatives OctoberShape 26, 2010 a Graph7 / 12

10 How Derivatives Shape a Graph How Derivatives Shape a Graph Increasing/Decreasing Test 1 If f (x) > 0 on an interval, then f is increasing on that interval. 2 If f (x) < 0 on an interval, then f is decreasing on that interval. Math 103 Day 13: The Mean Value Theorem and Tuesday How Derivatives OctoberShape 26, 2010 a Graph8 / 12

11 How Derivatives Shape a Graph First Derivative Test Suppose that c is a critical number of a continuous function f. 1 If f changes from positive to negative at c, then f has a local maximum at c. 2 If f changes from negative to positive at c, then f has a local minimum at c. 3 If f does not change sign at c, then f has no local maximum or minimum at c. Math 103 Day 13: The Mean Value Theorem and Tuesday How Derivatives OctoberShape 26, 2010 a Graph9 / 12

12 How Derivatives Shape a Graph Definition If a graph of f lies above all of its tangents on an interval I, then is is called concave up on I. If a graph of f lies below all of its tangents on an interval I, then is is called concave down on I. Concavity test 1 If f (x) > 0 for all x in I, then the graph of f is concave up on I. 2 If f (x) < 0 for all x in I, then the graph of f is concave down on I. Math 103 Day 13: The Mean Value Theorem andtuesday How Derivatives October 26, Shape 2010a Graph10 / 12

13 How Derivatives Shape a Graph Definition A point P on a curve y = f (x) is called and inflection point if f is continuous there and the curve changes from concave down to concave up or from concave up to concave down at P. Math 103 Day 13: The Mean Value Theorem andtuesday How Derivatives October 26, Shape 2010a Graph11 / 12

14 How Derivatives Shape a Graph The Second Derivative Test Suppose f is continuous near c. 1 If f (c) = 0 and f (c) > 0, then f has a local minimum at c. 2 If f (c) = 0 and f (c) < 0, then f has a local maximum at c. Math 103 Day 13: The Mean Value Theorem andtuesday How Derivatives October 26, Shape 2010a Graph12 / 12