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

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**Calculus I (MAT 145)Dr. Day Wednesday March 20, 2013**• 4.1: Extreme Values • Gateway Quiz #4: Tomorrow MAT 145**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**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**MAXIMUM & MINIMUM VALUES**MAT 145**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**MAXIMUM & MINIMUM VALUES**The graph of the function f(x) = 3x4 – 16x3 + 18x2 -1 ≤ x ≤ 4 is shown here. MAT 145**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**EXTREME VALUE THEOREM**The theorem is illustrated in the figures. Note that an extreme value can be taken on more than once. MAT 145**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**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**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**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**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**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**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**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**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**FERMAT’S THEOREM**Such numbers are given a special name—critical numbers. MAT 145**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**CRITICAL NUMBERS**If f has a local maximum or minimum at x = c, then x = c is a critical number of f. MAT 145**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**CLOSED INTERVAL METHOD**Determine the absolute maximum and absolute minimum values of the function:f(x) = x3 – 3x2 + 1 -½ ≤ x ≤ 4 MAT 145**MAXIMUM & MINIMUM VALUES**• The Hubble Space Telescope was deployed on April 24, 1990, by the space shuttle Discovery. MAT 145**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**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**Assignments**WebAssign • 4.1 (I) due tomorrow night MAT 145