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Warm Up Feb. 28 th

Warm Up Feb. 28 th. For each of the following… find the intervals where the graph is increasing/decreasing, find all extrema 1. f(x) = -x 4 + 4x 2. g(x)= x 5 – 15x 3 + 10 Write the equation of the tangent line to the graph of f (x) = -3x 2 + 4x – 1 at x = 2. Homework Questions….

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Warm Up Feb. 28 th

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  1. Warm Up Feb. 28th • For each of the following… • find the intervals where the graph is increasing/decreasing, • find all extrema • 1. f(x) = -x4 + 4x 2. g(x)= x5 – 15x3 + 10 • Write the equation of the tangent line to the graph of f(x) = -3x2 + 4x – 1 at x = 2

  2. Homework Questions… • Y = 6x + 1 • Y = 3x – 5 • Y = -2x + 8 • X = -1 • X = 0 and x = ½ • X = 2 • I: (0, ∞) D: (-∞, 0) Min @ x = 0 • I: (-∞, 2) U (3, ∞) D: (2, 3) Max @ x = 2, Min @ x = 3 • I: (-2, 0) U ( 2, ∞) D: (-∞, -2) U (0, 2) Max @ x = 0, Min @ x = -2 and x = 2

  3. Second Derivatives & Concavity Using derivatives and knowledge about concavity to help accurately graph a polynomial

  4. More Derivatives • f '(x) represents the first derivative • Slope of the tangent line, instantaneous velocity or rate of change • f '' (x) represents the second derivative • Classify extrema or acceleration • Find the first and second derivative of the following. 1) f(x) = x9 + 2x5 – 5x3 + 9 2) g(x) = 8x3 – 4x2 + 3x + 16

  5. Concavity • Concave up Concave down The second derivative of a function can tell us whether a function is concave upward or concave downward. If a) f ''(x) > 0 for all x in an interval I, the graph is concave up on I. b) f ''(x) < 0 for all x in an interval I, the graph is concave down on I.

  6. Point ofinflection Point of Inflection: the point where the graph changes from concave up to concave down or vice versa

  7. Approximate each of the following: • the point(s) of inflection of f(x) • Interval(s) where f(x) is concave up • Interval(s) where f(x) is concave down

  8. Helps us get a rough idea of the graph of the function Extrema: f '(x) = 0 Increasing: f '(x) > 0 Decreasing: f '(x) < 0 Point of inflection: f''(x) = 0 Concave up: f''(x) > 0 Concave Down: f''(x) < 0 Allows us to graph more accurately!

  9. Examples: • g(x) = 1/3x3 – x2 – 3x + 2 • Describe the end behavior of the graph. • What is/are the point(s) of inflection? • Where is the graph concave up?

  10. Examples (cont.): • h(x) = 0.25x4 – x3 + 1 • Describe the end behavior of the graph. • Where is the graph increasing? • What are the extrema? • What is/are the point(s) of inflection? • Where is the graph concave up?

  11. Given the graph of f(x) below, what do you know about f’ and f” at each indicated point? D E F C H A G B

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