Math 220 March 7 I. If 1200 cm2 of material is available to make a ...

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$Math 220 October 30 I. Find f 1. f (x) = x √ x2 + sin(x) + cos(x), f(0 ...$

$Math Times - Mathematics - University of Illinois at Urbana-Champaign$

D = (x − 4) 2 + (y − 0) 2 D(x) = (x − 4) 2 + ( √ x) 2 D(x) = √ x 2 − 8x + 16 + x D(x) = √ x 2 − 7x + 16 D ′ 2x − 7 (x) = 2 √ x2 − 7x + 16 2x − 7 0 = 2 √ x2 − 7x + 16 0 = 2x − 7 x = 7 2 D ′ (x) < 0 when x < 7/2 D ′ (x) > 0 when x > 7/2 Thus there is a minimum at x = 7/2 The point (7/2, 7/2) is closest. III. At which points on the curve y = 1 + x + 60x 3 − x 5 does the tangent line have the largest slope? Answer: slope = y ′ = 180x 2 − 5x 4 4
m ′′ (x) = 360 − 60x 2 m = 180x 2 − 5x 4 m ′ (x) = 360x − 20x 3 m ′ (x) = 20x(18 − x 2 ) m ′′ (0) > 0 m ′′ ( √ 18) < 0 m ′′ ( √ 18) < 0 0 = 20x(18 − x 2 ) x = 0, ± √ 18 There is a maximum at x = ± √ 18 So the tangent line will have the largest slope at the points ( √ 18, 1+ √ 18+60(18) 3/2 −(18) 5/2 ) and (− √ 18, 1+ √ 18+60(18) 3/2 −(18) 5/2 ) IV. A farmer whats to fence in area of 400 square feet in a rectangular field and then divide it 3 sections with a two fence parallel to one of the sides of the rectangle. How can he do this so as to minimize the cost of the fence? Answer: 400 = xy Cost = 4x + 2y 5
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Page 7 and 8: T (x) = 1 5 T ′ (x) = 1 5 + 1 (x)
Page 9 and 10: 4. 5. Answer: Answer: Answer: lim x
Page 11 and 12: 9. 10. 11. Answer: Answer: 2x − s
Page 13 and 14: 14. Answer: ln(y) = ln( lim x→0 +
Page 15: x 2 ln(y) = ln( lim e + x x→∞ 1

Math 220 March 7 I. If 1200 cm2 of material is available to make a ...

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