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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) =

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Warm Up pg 105 QR #1-10all

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  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)

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