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Sec 4.3 – Monotonic Functions and the First Derivative Test

Sec 4.3 – Monotonic Functions and the First Derivative Test. Monotonicity – defines where a function is increasing or decreasing. A function is monotonic if it is increasing or decreasing on an interval. a. c. b. d. Sec 4.3 – Monotonic Functions and the First Derivative Test.

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Sec 4.3 – Monotonic Functions and the First Derivative Test

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  1. Sec 4.3 – Monotonic Functions and the First Derivative Test Monotonicity – defines where a function is increasing or decreasing. A function is monotonic if it is increasing or decreasing on an interval. a c b d

  2. Sec 4.3 – Monotonic Functions and the First Derivative Test The First Derivative Test A function is continuous on an open interval containing critical point(s). If is differentiable on the interval, except possibly at the critical points, then at the critical point(s) can be classified as follows: 1. Local Maximum if changes from positive to negative at m. 2. Local Minimum if changes from negative to positive at n. 3. If there is no sign change, then the critical point is not a local minimum or maximum.    a e m f n g b

  3. Sec 4.3 – Monotonic Functions and the First Derivative Test The First Derivative Test    a e m f n g b Inc. e f Dec. Inc. g

  4. Sec 4.3 – Monotonic Functions and the First Derivative Test Example Problems

  5. Sec 4.4 – Concavity and Curve Sketching Concavity – defines the curvature of a function. A function is concave up on an open interval if is increasing on the interval. A function is concave down on an open interval if is decreasing on the interval. Point of Inflection (poi) – the point on the graph where the concavity changes. poi  a c b

  6. Sec 4.4 – Concavity and Curve Sketching The Second Derivative Test for Concavity The graph of a twice-differentiable function y = f (x) is: Concave up on any interval where , and Concave down on any interval where .    a e f g b Concave down e Concave down f Concave up g

  7. Sec 4.4 – Concavity and Curve Sketching The Second Derivative Test for Local Extrema If (which makes x = c a critical point) and then f has a local maximum at x = c. If (which makes x = c a critical point) and then f has a local minimum at x = c. NOTE: If the second derivative is equal to zero (or undefined) then the Second Derivative Test is inconclusive.   a m n b m Concave down Local max Concave up n Local min

  8. Sec 4.4 – Concavity and Curve Sketching Example Problems

  9. Sec 4.4 – Concavity and Curve Sketching Curve Sketching

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