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AP Calculus Section 3.1 Extrema on an Interval

AP Calculus Section 3.1 Extrema on an Interval. Definitions, Theorems, and Examples. Definition. Let f be defined on an interval I containing c. 1.) f(c) is a minimum of f on I if f(c) ≤ f(x) for all x in I. 2.) f(c) is a maximum of f on I if f(c) ≥ f(x) for all x in I. Extrema.

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AP Calculus Section 3.1 Extrema on an Interval

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  1. AP Calculus Section 3.1Extrema on an Interval Definitions, Theorems, and Examples

  2. Definition Let f be defined on an interval I containing c. 1.) f(c) is a minimum of f on I if f(c) ≤ f(x) for all x in I. 2.) f(c) is a maximum of f on I if f(c) ≥ f(x) for all x in I.

  3. Extrema The minimum and maximum of a function on an interval are the extreme values or extrema (singular extremum), of the function on the interval. The minimum and maximum are also called the absolute minimum and absolute maximum on the interval.

  4. A function does not have to have a minimum or a maximum on an interval.

  5. Extrema can occur at interior points or endpoints of an interval. Extrema that occur at the endpoints are called endpoint extrema.

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

  7. Definition 1.) If there is an open interval containing c on which f(c) is a maximum, then f(c) is called a relative maximum of f. f has a relative maximum at (c, f(c)). 2.) If there is an open interval containing c on which f(c) is a minimum, then f(c) is called a relative minimum. f has a relative minimum at (c, f(c)).

  8. Example 1 Find the value of the derivative at each of the relative extrema of A.)

  9. Find the value of the derivative at each of the relative extrema of B.)

  10. Find the value of the derivative at each of the relative extrema of C.)

  11. Definition Let f be defined at c. If f’(c) = 0 or f is not differentiable at c, then c is a critical number of f.

  12. Theorem 3.2 If f has a relative maximum or relative minimum at x = c, then c is a critical number of f.

  13. Guidelines for Finding Extrema on a Closed Interval f is a continuous function on [a, b]. 1.) Find the critical numbers of f in (a, b). 2.) Evaluate f at each critical number. 3.) Evaluate f at each endpoint of [a, b]. 4.) The least of these values is the minimum. The greatest is the maximum.

  14. Example 2 (#23) Find the extrema and critical numbers of on [-1, 2].

  15. Note: On Example 2, the critical numbers x = 0 and x = 1 do not yield a maximum or a minimum. This tells us that the converse of Theorem 3.2 is not true.THE CRITICAL NUMBERS OF A FUNCTION DO NOT HAVE TO PRODUCE RELATIVE EXTREMA.

  16. Example 3 (#25) Find the extrema and critical numbers of on [-1, 1].

  17. Example 4 (#17) Find the extrema and critical numbers of on [0, 2]

  18. Assignment: Section 3.1 p.169-170 #2, 4, 8, 18, 24, 26, 30, 34, 46

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