1 / 11

4.2 Mean value theorem

4.2 Mean value theorem. Rolle’s theorem. Exploration: Sketch a rectangular coordinate plane on a piece of paper. Label the points (1, 3) and (5, 3). Draw the graph of a differentiable function that starts at (1, 3) and ends at (5, 3). WHAT DO YOU NOTICE ABOUT YOUR GRAPH?. Rolle’s theorem.

kesia
Download Presentation

4.2 Mean value theorem

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 4.2Mean value theorem

  2. Rolle’s theorem • Exploration: • Sketch a rectangular coordinate plane on a piece of paper. • Label the points (1, 3) and (5, 3). • Draw the graph of a differentiable function that starts at (1, 3) and ends at (5, 3). • WHAT DO YOU NOTICE ABOUT YOUR GRAPH?

  3. Rolle’s theorem • If • f is continuous on [a, b], • f is differentiable on (a, b) AND • f(a) = f(b) • Then there is at least one number c in (a, b) such that f’(c) = 0. In other words, if the stipulations above hold true, there is at least one point where the derivative equals 0 (there exists at least one critical point in (a, b) ALSO f has at least an absolute max or min on (a, b)).

  4. Rolle’s theorem • Example: Show that f(x) = x4 – 2x2satisfies the stipulations to Rolle’s Theorem on [-2, 2] and find where Rolle’s Theorem holds true. • f is continuous on [-2, 2] • f is differentiable on (-2, 2) • f(-2) = f(2) = 8

  5. Mean Value theorem • Consider the graph below: Slope of tangent at c? f’(c) Slope of secant from a to b? a c b

  6. Mean value theorem • If • f is continuous at every point on a closed interval [a, b] • f is differentiable at every point on its interior (a, b) • Then there is at least one point c in (a, b) at which In other words, if the stipulations above hold true, there is at least one point where the instantaneous slope equals the average slope.

  7. Mean value theorem • Example: a.) Show that the function f(x) = x2 + 2x – 1 on [0, 1] satisfies the Mean Value Theorem. b.) Find each value of c that satisfies the MVT. a.) f is continuous and differentiable on [0, 1] b.) So,

  8. Mean value theorem • Example: Given f(x) = 5 – (4/x), find all values of c in the open interval (1, 4) that satisfies the Mean Value Theorem. So,

  9. Increasing/decreasing functions Let f be a function defined on an interval I and let x1 and x2 be any two points in I. • fincreases on I if x1 < x2 results in f(x1) < f(x2) • fdecreases on I if x1 > x2 results in f(x1) > f(x2) More importantly relating to derivatives: • Let f be continuous on [a, b] and differentiable on (a, b): • If f’ > 0at each point of (a, b), then fincreases on [a, b] • If f’ < 0 at each point of (a, b), then fdecreases on [a, b] • If f’ = 0 at each point of (a, b), then f is constant on [a, b] (called monotonic)

  10. Increasing/decreasing functions • Example: Find the open intervals on which • is increasing or decreasing. 1.) Find critical points

  11. Antiderivatives • Find a function F whose derivative is f(x) = 3x2 • F(x) = x3 • A function F is an antiderivative of f on an interval I if F’(x) = f(x) for all x in I. To find an antiderivative of a power function, increase the exponent by 1 and divide the original coefficient by the new exponent. What is the antiderivative of 4x5? (2/3)x6 Look at example 8 on pg. 201!!!

More Related