Lecture_Section 4_3

Lecture_Section 4_3 - f ‘ (x) is decreasing implies f is...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
Section 4.3 Derivatives and Shapes of Curves 1. The Mean Value Theorem If f is differentiable on the interval [a, b], then there exists a number c between a and b such that a b a f b f c f - - = 29 ( 29 ( 29 ( Sketch: Example: Given 2 ) ( x x f = , find the c guaranteed by the Mean Value Theorem on the interval [0, 4]. Sketch a graph . 2. Increasing and Decreasing a) f’(x)>0 implies f is increasing b) f’(x)<0 implies f is decreasing easiest example : 2 ) ( x x f = The First Derivatives Test Suppose c is a critical number of a continuous function, f, If f’ switches from positive to negative on either side of c then f has a local max at x =c. If f’ switches from negative to positive on either side of c then f has a local min. at x = c. Example: #8 p.287 1 4 ) ( 4 - - = ξ φ
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
3. Concavity f’’(x) > 0 implies f ‘(x) is increasing implies f is concave up f ‘’(x) < 0 implies
Background image of page 2
Background image of page 3
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: f ‘ (x) is decreasing implies f is concave down If concavity switches at a point, it is called and inflection point. Easiest example: 3 ) ( x x f = 2 nd Derivative Test Suppose f ‘’ is continuous near c. a) if f ‘(c) = 0 and f ‘’(c) > 0 then f has a local min. at x = c b) if f ‘(c) = 0 and f ‘’(c) < 0 then f has a local max. at x = c Example: Go back to the function in the above example 1 4 ) ( 4--= x x x f And discuss concavity and inflection points. More Examples: Find all vertical and horizontal asymptotes. Find the interval of increase or decrease. Find the local maximum and minimum values. Fin the intervals of concavity and the inflection points. Use the information to sketch a graph. 1. #30 2 2 ) 2 ( ) (-= x x x f 2. #40 x e x x f-= 2 ) (...
View Full Document

This note was uploaded on 04/12/2008 for the course MA 141 taught by Professor Wears during the Spring '07 term at N.C. State.

Page1 / 3

Lecture_Section 4_3 - f ‘ (x) is decreasing implies f is...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online