1 / 5

Concavity

2.4 Geometrical Application of Calculus. Concavity. The second derivative gives us information about the curves shape. f’’(x) > 0 - curve is concave upward. f’’(x) < 0 - curve is concave downward. 2.4 Geometrical Application of Calculus. Inflection.

loc
Download Presentation

Concavity

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. 2.4 Geometrical Application of Calculus Concavity The second derivative gives us information about the curves shape. f’’(x) > 0 - curve is concave upward. f’’(x) < 0 - curve is concave downward.

  2. 2.4 Geometrical Application of Calculus Inflection The second derivative gives us information about the curves shape. f’’(x) = 0 - point of horizontal inflection.

  3. 2.4 Geometrical Application of Calculus Exploring Stationary Points. Yes, points of inflection as concavity does change. f’’(x) - / 0 / + + / 0 / - No points of inflection as concavity does not change. f’’(x) + / 0 / + - / 0 / -

  4. 2.4 Geometrical Application of Calculus Exploring stationary points. 1. Does the curve y = x4 have a point of inflection? Stationary point when 12x2 = 0 f(x) = x4 f’(x) = 4x3 i.e. x = 0 f(0) = 0 Stationary point @ (0, 0) f’’(x) = 12x2 Test Concavity x -1 0 1 f’’(x) +12 0 +12 Concavity does not change - Not Inflection point.

  5. 2.4 Geometrical Application of Calculus Exploring stationary points. 2. Does the curve y = x3 have a point of inflection? Stationary point when 6x = 0 f(x) = x3 f’(x) = 3x2 i.e. x = 0 f(0) = 0 Stationary point @ (0, 0) f’’(x) = 6x Test Concavity x -1 0 1 f’’(x) -6 0 +6 Concavity does change - Inflection point.

More Related