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Miss Battaglia AP Calculus AB/BC

3.1 Extrema on an Interval Objective: Understand the definitions and find extrema and relative extrema of a function on an interval. Miss Battaglia AP Calculus AB/BC. Extrema of a Function. Min & max are the largest and smallest value that the function takes at a point

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Miss Battaglia AP Calculus AB/BC

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  1. 3.1 Extrema on an IntervalObjective: Understand the definitions and find extrema and relative extrema of a function on an interval. Miss BattagliaAP Calculus AB/BC

  2. Extrema of a Function • Min & max are the largest and smallest value that the function takes at a point Let f be defined as an interval I containing c. • f(c) is the min of f on I if f(c)<f(x) for all x in I • f(c) is the max of f on I if f(c)>f(x) for all x in I

  3. Not a max max max min min Not a min f is continuous [-1,2] is closed f is continuous (-1,2) is open g is not continuous [-1,2] is closed

  4. 3.1 The Extreme Value Theorem If f is continuous on a closed interval [a,b] then f has both a minimum and a maximum on the interval.

  5. Relative Extrema • Think of a relative max as occurring on a “hill” on the graph and a relative min as occurring on a “valley” of a graph. If there is an open interval containing c on which f(c) is a max, then f(c) is called a relative max of f, or you can say f has a relative max at (c,f(c)) If there is an open interval containing c on which f(c) is a min, then f(c) is called a relative min of f, or you can say f has a relative min at (c,f(c)) AKA local max and local min

  6. Ways Relative Extrema Can Look f’(c) = o or undefined

  7. The Value of the Derivative at Relative Extrema Find the value of the derivative at the relative max (3,2).

  8. The Value of the Derivative at Relative Extrema Find the value of the derivative at the relative min (0,0).

  9. The Value of the Derivative at Relative Extrema Find the value of the derivative at the relative max (π/2,1) and relative min (3π/2,-1)

  10. Definition of a Critical Number • Let f be define at c. If f’(c)=0 or if f is not differentiable at c, then c is a critical number of f. c has to be in the domain of f, but does not have to be in the domain of f’.

  11. Thm 3.2 Relative Extrema Occur Only at a Critical Number • If f has a relative min or a relative max at x=c, then c is a critical number of f. • Is the converse true? Think about y=x3.. Is 0 a critical value? Is it a relative min or max?

  12. Guidelines for Finding Extrema on a Closed Interval To find the extrema of a continuous function f on a closed interval [a,b], use the following steps: • Find the critical numbers of f in (a,b) • Evaluate f at each critical number in (a,b) • Evaluate f at each endpoint of [a,b] • The least of these values is the minimum. The greatest is the maximum.

  13. Finding Extrema on a Closed Interval Find the extrema of f(x)=3x4-4x3 on the interval [-1,2]

  14. Finding Extrema on a Closed Interval Find the extrema of f(x)=2x-3x2/3 on the interval [-1,3]

  15. Finding Extrema on a Closed Interval Find the extrema of f(x)=2sinx – cos2x on the interval [0,2π]

  16. Classwork/Homework • Read 3.1 Page 169 #11-27 odd, 39

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