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4 Pages

Test 2 sols

Course: MATH 160, Spring 2012
School: Boise State
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Word Count: 621

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______________ Test Name: #2 Limits, Continuity &amp; Derivatives Find the derivative of the following functions: f ' x =10 x 1 x3 3 4 g ' x = 4 =3 x x 2. g x = 1. f x = x 10 9 4. h x =25 5. f t = ln t h ' x =0 f ' t =1 / t 7. f x =3 x x 10. f x = x ln x x y ' =e x 6. f x =12 x 6 f ' x = =6 x 1 /2 x 9. r s =log 5 s 1 r ' s = log 5 sln 5 8. V x = 2 V ' x = 0 f ' x = 3 ln...

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______________ Test Name: #2 Limits, Continuity & Derivatives Find the derivative of the following functions: f ' x =10 x 1 x3 3 4 g ' x = 4 =3 x x 2. g x = 1. f x = x 10 9 4. h x =25 5. f t = ln t h ' x =0 f ' t =1 / t 7. f x =3 x x 10. f x = x ln x x y ' =e x 6. f x =12 x 6 f ' x = =6 x 1 /2 x 9. r s =log 5 s 1 r ' s = log 5 sln 5 8. V x = 2 V ' x = 0 f ' x = 3 ln 3 3. y =e x 5 x 3 12. P t = ln t 3 2 x 2 x 3 1 5 x 2 2 x 2 x 2 5 x 3 P ' t = =3 y' = 2 2 t t x 2 x 2 5 x 6 x 6 y' = x 2 2 x 2 11. y = f ' x = ln x Find the indicated limit of the functions: For #13 and #14, let 13. Find { 9 x 2 3 x f x = 9 x 2 3 x a x 0 AND x 3 x0 x =3 lim f x x 0 lim f x =3 x 0 Is this function continuous at x = 0? Why, or why not? No, because f(0) does not exist. 14. For what value of a would make this function continuous at x = -3? a=6 15. State the definition of the derivative for a function f(x). f ' x = lim h 0 f x h f x h Use this definition to find the derivative of the function 2 2 f x = x 2 2 f x h = x h = x 2 xh h 2 2 2 f x h f x = x 2 xh h x =2 xh h 2 f x h f x 2 xh h 2 h 2 x h = = =2 x h h h h lim f x h f x = lim 2 x h= 2 x h h 0 h0 16. For the function f x =6 x x 2 , find the equation of the line tangent to the graph of this function a x = 2. f ' x = 6 2 x f 2 =8 f ' 2 = 2 y 8= 2 x 2 For what value(s), if any, is the line tangent to this curve horizontal? f ' x = 6 2 x =0 At x = 3 For each point on the graph, indicate whether or not it: a) has a limit (and, if so, approximately what it is) b) is continuous c) is differentiable 7 6 5 4 3 2 1 -6 17. -4 18.-2 19. 202 21. 4 6 -1 -2 17 18 19 20 21 What the is limit? Approximately 5 Does not exist Approximately 5 Approximately 4.5 Approximately 2.5 Is it continuous? Yes No (limit DNE) Yes No (function not defined) Yes Is it differentiable? No (corner) No (not continuous/limit DNE) Yes No (not continuous) No (vertical tangent) 22. The revenue function R(x) is given by R x =50 x .05 x 2 A) Find R(400) and R(650). R 400 =50 400 .05 400 2 =20000 8000=12000 R 650 =50 650 .05 650 2=32500 21125 =11375 B) Find the marginal revenue at a production level of 400 steam irons. R ' x =50.1 x R ' 400 =50 .1 400 =10 C) Find the marginal revenue at a production level of 650 steam irons. R ' 650 =50 .1 650 =15 D) Using the results above, estimate the revenue for a production level of 401 and 651 steam irons. R 401 R 400 R ' 400 =12000 10=12010 R 651 R 650 R ' 650 =1137515 =11360 23. Consider the function f x = x 3 . A) Find f(10). f 10 =1000 B) Use differentials to estimate the value of 10.13 . f ' x = 3 x 2 f ' 10 = 300 f 10.1 f 10 x f ' 10 =103.1 300 = 1030 For #24 and #25, let lim f x =3 . x 2 24. Evaluate lim x 3 f x 2= 2 3 3 2= 49 x 2 25. If f(2) = 3, is f continuous at x = 2? Explain why or why not. Is f continuous over its domain? lim f x =3= f 2 Yes, f is continuous at x = 2 because . We have no information to know x 2 whether it is continuous at any other point. 26. Let g x = uvw where u, v and w are all functions of x. Find g ' x =u ' vw uv ' w uvw ' 27. h is a polynomial function of degree n. What is the degree of h'? h' has degree n 1. 28. Sketch a function f that satisfies the following conditions: lim lim f x =3 and f x =1 + x 0 x 0The limit exists everywhere else, but f is not continuous at x = -1. g ' x .
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