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Solved Problems (lecture 3-11-2013) 1. Answer true or false. If y2 = 7x, dy/dx = 7. A) True B) False 2. If x2 + y2 = 1, find dy/dx. A) dy x  dx y B) dy y  dx x C) dy y  dx x D) dy  2x  2 y dx E) dy x  dx y

3. If cos(xy) = 36, find dy/dx. A) dy x  dx y B) dy   xdy  ydx  sin  xy  dx C) dy y  dx x D) dy y  dx x E) dy x  dx y

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4. If x2 - y2 = 82, find dy/dx. A) dy x  dx y B) dy y  dx x C) dy y  dx x D) dy  2x  2 y dx E) dy x  dx y

5. Find A) B) C) D) E)

6.

Find A) B) C) D) E)

y2 dy if  38 . dx x dy 38  dx 2 y dy y  dx x dy 2 y  dx x dy 2 yxdy  y 2dx  dx x2 dy 2 yxdy  y 2dx  dx 38x 2

dy if y sin x = 17. dx dy  y cot x dx dy  – y cot x dx dy  – y csc x dx dy  – y tan x dx dy  y tan x dx

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7. Answer true or false. If y = x3 - 2x, dy/dx = 2x. A) True B) False 8. Find dy/dx if y3 = 7x. A) dy 7  dx 3 y B) dy 7  2 dx y C) dy 7  dx 3 y 2 D) dy 3 y 2  dx 7 E) dy  21x 2 dx

9. Find dy/dx if xy = 96. A) dy y  dx x B) dy x  dx y C) dy x  dx y D) dy  x dy  y dx dx E) dy y  dx x

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10. Find dy/dx if 4xy = 81. A) dy y  dx x B) dy x  dx y C) dy x  dx y D) dy  x dy  y dx dx E) dy y  dx x 11. Find dy/dx if 8x2y2 = 49. A) dy y  dx x B) dy y  dx x C) dy x  dx y D) dy  16xy 2dx  16x 2 y dy dx E) dy x  dx y 12. Find dy/dx if y8 = 4x. A) dy 4  dx 8 y 7 B) dy 4  7 dx 8y C) dy 4  7 dx y D) dy 4  8 dx 8 y E) dy 4  8 dx y

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13. Find the value c such that the conclusion of Rolle's Theorem are satisfied for f(x) = 3x2  1 on [-4, 4]. A) 0 B) -1 C) 1 D) 0.5 E) -0.5

14. Answer true or false. Rolle's Theorem is used to find the slope of a function. A) True B) False 15. Answer true or false. The Mean-Value Theorem can be used on f(x) = |x  3| on [-5, 5]. A) True B) False 16. If f (x )  4 x on [0, 81], find the value c that satisfies the Mean-Value Theorem. (Round to three decimal places.) A) 6.750 B) 12.757 C) 4.241 D) 9.546 E) 0.472 17. Find the value for which f(x) = x2 + 2 on [4, 6] satisfies the Mean-Value Theorem. A) 4.5 B) 5 C) 5.5 D) 4.66 E) 5.33 18. Find the value for which f(x) = x3  2 on [3, 5] satisfies the Mean-Value Theorem. A) 3.215 B) 2.517 C) 7.000 D) 4.041 E) 4.950

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19. Answer true or false. A graphing utility can be used to show that Rolle's Theorem can be applied to show that f(x) = (x - 2)2 has a point where f (x) = 0. A) True B) False 20. Find the value of c in the interval [0, 1] that satisfies the Mean Value Theorem. f(x) = x7 21. Find the value of c in the interval [-1, 1] that satisfies the Mean Value Theorem. f(x) = x2  8x + 7

a. Verify that f(x) = x3  x satisfies the hypothesis of Rolle's Theorem on the interval [-1, 1] and find all values of C in (-1, 1) such that f (C) = 0 b. Verify that f(x) = x3  5x + 2 satisfies the hypothesis of the Mean-Value Theorem over the interval [-2, 3] and find all values of C that satisfy the conclusion of the theorem.

Answer Key 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

B E C A A B B C E E B A A B B B B D B

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20. f (1)  f (0) 1  0 =1  1 0 1 f '(x) = 7x6 7x6 = 1 1 x6  7 5

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7 6 1 7 6 of which only is in (0, 1) x 6  7 7 7 21. f(1) = 12 - 8(1) + 7 = 0 f(-1) = (-1)2 - 8(-1) + 7 = 16 0  16 16  = -8 1   1 2 f '(x) = 2x - 8 2x - 8 = -8 2x = -0 0 x    -0 2

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