MAT 232-Calc-1-A4 - Calculus I Calculus: Early...

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Calculus I Calculus: Early Transcendental Functions , 4th ed., by Ron Larson, Robert Hostetler, and Bruce H. Edwards (Boston: Houghton Mifflin, 2007; ISBN-10: 0-618-60624-6). Written Assignment 4 The written assignment draws on even-numbered exercises from the textbook. Answer all assigned exercises, and show all work. Section Exercises 4.1 24, 28 4.4 12, 14, 16 4.5 22, 30, 64, 68, 74 4.6 16, 26, 32, 34 4.7 2(a-c), 8, 14, 20, 30 Section 4.1 Exercise 24 ƒ(x) = x 2 +2x – 4, [-1, 1] Differentiate with respect to x ƒ’(x) = 2x + 2 ƒ’(x) = 0 when 2x + 2 = 0 x = -1 Critical number x = -1 left end point of the interval ƒ(-1) = (-1) 2 +2(-1) – 4 = 1 – 2 – 4 = -5 ƒ(1) = (1) 2 +2(1) – 4 = 1 + 2 – 4 = -1 Absolute maximum (1, -1) Absolute minimum is (-1, -5) Exercise 28 g(x) = 3 √x [-1, 1] Differentiate with respect to x g’(x) = x 1/3 g’(x) = 1/3 x -2/3 g’(x) = 1/ [3 x 2/3 ]
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g’(x) ≠ 0 for all x ≠ 0 x = 0 is the critical number Calculating the value of ƒ(x) at the critical number and end points (-1,1) g(-1) = -1 g(0) = 0 g(1) = 1 Absolute maximum (-1, 1) Absolute minimum is (1,1) Section 4.4 Exercise 12 ƒ(x) = 2x 3 - 3x 2 – 12x + 5 ƒ’(x) = 6x 2 -6x – 12 first derivative ƒ’’(x) = 12x - 6 second derivative ƒ’’(x) = 0 when 12x – 6 = 0, x=1/2 ƒ’’(x) < 0 when x<1/2 and ƒ’’(x) > 0 when x>1/2 ƒ’’(x) changes at x = ½ therefore the point of inflection is [1/2, ƒ(1/2)] = [1/2, -3/2] graph of ƒ(x) concave downward for ( - ∞, ½) and concave upward for (1/2, ∞) Exercise 14 ƒ(x) = 2x 4 - 8x + 3 ƒ’(x) = 8x 3 - 8 ƒ’’(x) = 24x 2 ƒ’’(x) = 0 when x = 0 ƒ’’(x) = 24x 2 > 0 for all x< 0 and x > 0 graph of ƒ(x) concave upward for ( - ∞, ∞) and there is no point of inflection Exercise 16 ƒ(x) = x 3 (x – 2) ƒ’(x) = 3x 2 (x – 2) + (x – 2) ƒ’(x) = (x – 2) + (3x 2 + x – 2) ƒ” (x) = (x – 2) + (6x + x – 2) ƒ” (x) = (x – 2) + 2(3x + x) ƒ” (x) = 2(x – 2) (x + 1/3) ƒ” (x) = 0 when x = 2, 1/3 ƒ’’(x) = 2 > 0 for all x< 0 and x > 0 graph of ƒ(x) concave upward for ( - ∞, ∞) and there is no point of inflection
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Section 4.5 Exercise 22 lim 3x 3 +2 / 9x 3 2x 2 +7 x > ∞ lim [(3x 3 +2) / x 3 ] / [9x 3 2x 2 +7 / x 3 ] x > ∞
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This note was uploaded on 09/29/2010 for the course MAT MAT 231 taught by Professor Hannah during the Spring '10 term at Thomas Edison State.

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MAT 232-Calc-1-A4 - Calculus I Calculus: Early...

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