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Math141_Section 4.3

# Math141_Section 4.3 - f ‘(x is decreasing implies f is...

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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 - - = ) ( ) ( ) ( 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 - - = x x x f

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3. Concavity f’’(x) > 0 implies f ‘(x) is increasing implies f is concave up f ‘’(x) < 0 implies
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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 ) (...
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