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Lesson 4-TR Chapter 4 Test Review

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Relationship between f(x), f’(x), f’’(x) f(x) x x x f’(x) f’’(x) f’(x) shows slope of f(x) f(x) increasing from (-∞, c 1 ) f(x) decreasing from (c 1, c 2 ) f(x) increasing from (c 1, ∞) f’’(x) shows concavity of f(x) [and the slope of f’(x)] f(x) concave down from (-∞, 0) f(x) concave up from (0, ∞) c1c1 c2c2 f’(c 1 ) = 0 = f’(c 2 ) Red dashed lines show critical numbers (possible locations of extrema) f’’(c 1 ) 0 Concave up f’’(x) > 0 min Concave down f’’(x) < 0 max f’’(0) = 0 inflection point!

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Optimization Problem #2, 4-7, pg 336 Find two numbers whose difference is 100 and whose product is a minimum. (P): P = xy (S): D = 100 = x - y 100 + y = x (P): P = (100 + y)y = 100y + y² P’ = 100 + 2y 0 = 100 + 2y 2y = -100 y = -50 P’’ = 2 Concave up for all y min y = -50 x = 50

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Optimization Problem #6, 4-7, pg 336 A farmer wants to fence an area of 1.5 million sq ft in a rectangular field and then divide it in half with a fence parallel to one of the sides of the rectangle. How can he do this as to minimize the cost of the fence? (P): C = 3x + 4y (S): A = 1.5 = x ∙ 2y 1.5 / 2y = x (P): C = 3(1.5/2y) + 4y = 2.25y -1 + 4y C’ = -2.25y -2 + 4 0 = -2.25y -2 + 4 4 = 2.25y -2 y = .5625 =.75 C’’ = 4.5y -3 + 0 Concave up for all y>0 min y = 0.75 million ft x = 1 million ft y x

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L’Hospital’s Rule x² Lim --------- = x→∞ e x e x Lim ------ x→∞ 2x 1 - e x Lim ----------- x→0 sec x cos x - 1 Lim --------------- x→0 x + 1 2x Lim --------- = x→∞ e x 2 Lim ------ = 0 x→∞ e x ∞ --- = ∞ --- = ∞ e x Lim ------ = x→∞ 2 ∞ --- = ∞ --- = DNE 2 1 - 1 = --------- = 0 (L’Hospital DNA) 1/1 1 - 1 = --------- = 0 (L’Hospital DNA) 0 + 1

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Intervals - Table Notation Intervals(-∞,0)00,2)2(2,3)3(3,4)4(4,∞) f(x)+0--16--27-0+ f’(x) - slope-0--16-0+64+ f’’(x) - concavity+0-0+36+96+ Notes:IP y=0 IPmin y=0 f(x) = x 4 – 4x 3 = x 3 (x – 4) f’(x) = 4x 3 – 12x 2 = 4x 2 (x – 3) f’’(x) = 12x 2 – 24x = 12x (x – 2)

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Chapter 4 Test Sections 4-1 through 4-7 Quiz 1 covered 4-1 to 4-4 Quiz 2 covered 4-5 to 4-7 Part 1 of test should be finished in class Part 2 (graphing problem and bonus) is take- home due on Tuesday before 1 st period class Sections 4-8 and 4-9 not covered Section 4-10 will be picked up with Ch 5

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Section 4.1 Using First and Second Derivatives. Let’s see what we remember about derivatives of a function and its graph –If f’ > 0 on an interval than.

Section 4.1 Using First and Second Derivatives. Let’s see what we remember about derivatives of a function and its graph –If f’ > 0 on an interval than.

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