1 / 14

Warm Up pg 105 QR #1-10all

Warm Up pg 105 QR #1-10all. (can use graphing calc ). Ch 3.1 and 3.2. Derivative of a Function. What is a “derivative”?. Last section we learned that the slope of a curve at any point is given by m = = When this exists, the limit is called the derivative of f. We write f’(x) =

haven
Download Presentation

Warm Up pg 105 QR #1-10all

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. Warm Up pg 105 QR #1-10all (can use graphing calc)

  2. Ch3.1 and 3.2 Derivative of a Function

  3. What is a “derivative”? Last section we learned that the slope of a curve at any point is given by m = = When this exists, the limit is called the derivative of f. We write f’(x) = *The set of points in the domain of f’ for which the limit exists may be smaller than the domain of f. *A function that is differentiable at every point of its domain is a differentiable function.

  4. EX 1A: Applying the definition of the Derivative Find the derivative of f(x) = x³

  5. Is there another way to find the slope of a curve at a specific point? The slope of the secant line here PQ is m = = Therefore the derivative of a function f at the point x=a is the limit:

  6. EX 1B: Applying the alternative definition of the derivative *also show the power rule Differentiate f(x) = using the alternate definition of the derivative )

  7. *also show the power rule You Try! Differentiate f(x) = using • the regular definition b) the alternate definition ) )

  8. What notation should I be familiar with? There are many ways to denote the derivative of a function y = f(x). These are the most common:

  9. Are there places where f’(a) might fail to exist? Yes! A function will NOT have a derivative at a point P(a, f(a)) where the slopes of the secant lines fail to approach a limit as x approaches a.

  10. A Vertical Tangent – where the slopes of secant lines approach either or - from both sides • EX: f(x) = • A Discontinuity – where one or both of the one-sided derivatives is non-existent • EX: f(x) = • A Corner – where the one-sided derivatives differ • EX: f(x) = |x| • A Cusp – where the slopes of the secant lines approach from one side and - from the other • EX: f(x) =

  11. EX 2: Finding where a function is not differentiable a) Find all points in the domain of f(x) = |x - 2|+3 where f is not differentiable. b) Prove with right –hand and left – hand derivatives that the function is not differentiable at point P

  12. You Try! a) Show that the following function does not have a derivative at x = 2 f(x) = b) Show that the function f(x)= has left-hand and right-hand derivatives at x = 0, but there is no derivative there. *First check for continuity, then check derivatives

  13. *Two Important THMS that you need to memorize* • THEOREM 1: Differentiability implies continuity If f has a derivative at x = a, then f is continuous at x = a • THEOREM 2: Intermediate Value Theorem for Derivatives If a and b are any two points in an interval on which f is differentiable, then f’ takes on every value between f’(a) and f’(b)

  14. Ch 3.1/3.2 HW Pg 105 EX #1-7odd, 18-20all, 21, 31 Pg 114 EX: 5-16all (you can use a graphing calc to look at picture and sketch it if you cannot picture it), 35 (foil first before applying derivative definition)

More Related