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Calculus I (MAT 145) Dr. Day Wednesday March 20, 2013

Calculus I (MAT 145) Dr. Day Wednesday March 20, 2013. 4.1: Extreme Values Gateway Quiz # 4 : Tomorrow. MAXIMUM & MINIMUM VALUES. A function f has an absolute maximum (or global maximum) at x = c if f ( c ) ≥ f ( x ) for all x in D , where D is the domain of f .

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Calculus I (MAT 145) Dr. Day Wednesday March 20, 2013

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  1. Calculus I (MAT 145)Dr. Day Wednesday March 20, 2013 • 4.1: Extreme Values • Gateway Quiz #4: Tomorrow MAT 145

  2. MAXIMUM & MINIMUM VALUES • A function f has an absolute maximum (or global maximum) at x = cif f(c) ≥ f(x) for all x in D, where D is the domain of f. • The number f(c) is called the maximum value of f on D. MAT 145

  3. MAXIMUM & MINIMUM VALUES • Similarly, f has an absolute minimum at x = cif f(c) ≤ f(x) for all x in D and the number f(c) is called the minimum value of f on D. • The maximum and minimum values of fare called the extreme values of f. MAT 145

  4. MAXIMUM & MINIMUM VALUES MAT 145

  5. MAXIMUM & MINIMUM VALUES A function f has a local maximum (or relative maximum) at x = cif f(c) ≥ f(x) when x is near c. • This means that f(c) ≥ f(x) for all x in some open interval containing c. Similarly, f has a local minimum at x = cif f(c) ≤ f(x) when x is near c. MAT 145

  6. MAXIMUM & MINIMUM VALUES The graph of the function f(x) = 3x4 – 16x3 + 18x2 -1 ≤ x ≤ 4 is shown here. MAT 145

  7. EXTREME VALUE THEOREM • If f is continuous on a closed interval [a, b], then f attains an absolute maximum value f(c) and an absolute minimum value f(d) at some numbers x = cand x = din [a, b]. MAT 145

  8. EXTREME VALUE THEOREM The theorem is illustrated in the figures. Note that an extreme value can be taken on more than once. MAT 145

  9. EXTREME VALUE THEOREM The figures show that a function need not possess extreme values if either hypothesis (continuity or closed interval) is omitted from the theorem. MAT 145

  10. EXTREME VALUE THEOREM • The function f whose graph is shown is defined on the closed interval [0, 2] but has no maximum value. Notice that the range of f is [0, 3). The function takes on values arbitrarily close to y = 3, but never actually attains the value 3. MAT 145

  11. EXTREME VALUE THEOREM This does not contradict the theorem because f is not continuous. • Nonetheless, a discontinuous function could have maximum and minimum values. MAT 145

  12. EXTREME VALUE THEOREM The function g shown here is continuous on the open interval (0, 2) but has neither a maximum nor a minimum value. • The range of g is (1, ∞). • The function takes on arbitrarily large values. • This does not contradict the theorem because the interval (0, 2) is not closed. MAT 145

  13. FERMAT’S THEOREM • If f has a local maximum or minimum at x = c, and if f’(c) exists, then f’(c) = 0. MAT 145

  14. FERMAT’S THEOREM The following examples caution us against reading too much into the theorem. • We can’t expect to locate extreme values simply by setting f’(x) = 0 and solving for x. MAT 145

  15. FERMAT’S THEOREM If f(x) = x3, then f ’(x) = 3x2, so f’(0) = 0. • However, f has no maximum or minimum at x = 0, as you can see from the graph. • Alternatively, observe that x3 > 0 for x > 0 but x3 < 0 for x < 0. MAT 145

  16. FERMAT’S THEOREM The function f(x) = |x| has its (local and absolute) minimum value at x = 0. • However, that value can’t be found by setting f ’(x) = 0. • This is becausef’(0) does not exist. MAT 145

  17. FERMAT’S THEOREM The theorem does suggest that we should at least start looking for extreme values of fat the numbers x = cwhere either: • f’(c) = 0 • f’(c) does not exist MAT 145

  18. FERMAT’S THEOREM Such numbers are given a special name—critical numbers. MAT 145

  19. CRITICAL NUMBERS A critical numberof a function f is a number x = cin the domain of f such that either: f ’(c) = 0 or f’(c) does not exist. MAT 145

  20. CRITICAL NUMBERS If f has a local maximum or minimum at x = c, then x = c is a critical number of f. MAT 145

  21. CLOSED INTERVAL METHOD To determine the absolute maximum and minimum values of a continuous function f on a closed interval [a, b]: • Find the values of f at the critical numbers of fin (a, b). • Find the values of f at the endpoints of the interval. • The largest value from 1 and 2 is the absolute maximum value. The smallest is the absolute minimum value. MAT 145

  22. CLOSED INTERVAL METHOD Determine the absolute maximum and absolute minimum values of the function:f(x) = x3 – 3x2 + 1 -½ ≤ x ≤ 4 MAT 145

  23. MAXIMUM & MINIMUM VALUES • The Hubble Space Telescope was deployed on April 24, 1990, by the space shuttle Discovery. MAT 145

  24. MAXIMUM & MINIMUM VALUES A model for the velocity of the shuttle during this mission—from liftoff at t = 0 until the solid rocket boosters were jettisoned at t = 126 s—is given by: v(t) = 0.001302t3 – 0.09029t2 + 23.61t – 3.083 (in feet per second) MAT 145

  25. MAXIMUM & MINIMUM VALUES Using this model, estimate the absolute maximum and minimum values of the acceleration of the shuttle between liftoff and the jettisoning of the boosters. v(t) = 0.001302t3 – 0.09029t2 + 23.61t – 3.083 MAT 145

  26. Assignments WebAssign • 4.1 (I) due tomorrow night MAT 145

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