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Section 13.3 – 13.4 Higher Derivatives, Concavity, and Curve Sketching

Section 13.3 – 13.4 Higher Derivatives, Concavity, and Curve Sketching. Definition. f ’( x ) is called the (first) derivative of f ( x ). The derivative of f ’( x ) is called the second derivative of f ( x ), denoted by

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Section 13.3 – 13.4 Higher Derivatives, Concavity, and Curve Sketching

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  1. Section 13.3 – 13.4Higher Derivatives, Concavity, and Curve Sketching

  2. Definition • f ’(x) is called the (first) derivative of f (x). • The derivative of f ’(x) is called the second derivative of f (x), denoted by • The derivative of f ’’(x) is called the third derivative of f (x), denoted by • The fourth derivative of f (x) is denoted by • Notation

  3. Definition • A function f (x) is concave up on an interval (a,b) if the graph of f (x) lies above its tangent line at each point on (a,b). • A function f (x) is concave down on an interval (a,b) if the graph of f (x) lies below its tangent line at each point on (a,b). • A point where a graph changes its concavity is called an inflection point. • If the slopes of the tangent lines are increasing, then the function is concave up. • If the slopes of the tangent lines are decreasing, then the function is concave down. Facts

  4. Test For Concavity • If f ’’(x) > 0 on an interval (a,b), then f (x) is concave up on (a,b). • If f ’’(x) < 0 on an interval (a,b), then f (x) is concave down on (a,b). • If f ’’(x) changes signs at c, then f(x) has an inflection point at c. Example: Find the intervals where the function is concave up or concave down, and find all inflection points

  5. How to find inflection points and interval of concavity: • Find all numbers c such that f ’’(c) = 0 or undefined • Put all values found in step 1 on the number line and use test values to determine the sign of the second derivative for each interval. • Determine the interval of concavity based on the sign of the second derivative.

  6. Examples Find the intervals of concavity and all inflection points.

  7. Second Derivative test Let c be a critical number of a function f (x) . • If f ’’(c) > 0, then f (x) has a local minimum at c. • If f ’’(c) < 0, then f (x) has a local maximum at c. • If f ’’(c) = 0 or dne, then this test fails  must use the first derivative test! Example: Find all local extrema

  8. Curve Sketching • Domain of the function • Vertical and horizontal asymptotes • Intervals of increase/decrease • Local extrema • Intervals of concavity • Inflection points • Plot additional points • Sketch the graph

  9. Examples • a) Find the intervals where the function is increasing/decreasing, and find all local extrema. • b) Find the intervals where the function is concave up or concave down, and find all inflection points

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