Chapter 1: Limits and Their Properties

Transcription

1 1. Decide whether the following problem can be solved using precalculus, or whether calculus is required. If the problem can be solved using precalculus, solve it. If the problem seems to require calculus, use a graphical or numerical approach to estimate the solution. Find the distance traveled in 11 seconds by an object moving with a velocity of vt () = 6 + 3cost. A) precalculus, D) precalculus, B) calculus, E) precalculus, C) calculus, Decide whether the following problem can be solved using precalculus, or whether calculus is required. If the problem can be solved using precalculus, solve it. If the problem seems to require calculus, use a graphical or numerical approach to estimate the solution. Find the distance traveled in 1 seconds by an object traveling at a constant velocity of 1 feet per second. A) precalculus, 88 D) calculus, 88 B) precalculus, 144 E) calculus, 164 C) calculus, 144 Copyright Houghton Mifflin Company. All rights reserved. 17

2 3. Decide whether the following problem can be solved using precalculus, or whether calculus is required. If the problem can be solved using precalculus, solve it. If the problem seems to require calculus, use a graphical or numerical approach to estimate the solution. A cyclist is riding on a path whose elevation is modeled by the function f ( x) = 0.09( 18 x x ) where x and f ( x ) are measured in miles. Find the rate of change of elevation when x = 4.5. A) precalculus, 0.1 D) precalculus, 0.09 B) calculus, 0.09 E) calculus, 0.81 C) precalculus, Decide whether the following problem can be solved using precalculus, or whether calculus is required. If the problem can be solved using precalculus, solve it. If the problem seems to require calculus, use a graphical or numerical approach to estimate the solution. A cyclist is riding on a path whose elevation is modeled by the function f ( x) = 0.16x where x and f ( x ) are measured in miles. Find the rate of change of elevation when x = 4.0. A) calculus, 1.8 D) precalculus, 0.16 B) precalculus, 1.8 E) precalculus, 0.41 C) calculus, Copyright Houghton Mifflin Company. All rights reserved.

3 5. Decide whether the following problem can be solved using precalculus, or whether calculus is required. If the problem can be solved using precalculus, solve it. If the problem seems to require calculus, use a graphical or numerical approach to estimate the solution. Find the area of the shaded region. A) precalculus, D) calculus, 15 B) calculus, E) precalculus, 18 C) precalculus, Decide whether the following problem can be solved using precalculus, or whether calculus is required. If the problem can be solved using precalculus, solve it. If the problem seems to require calculus, use a graphical or numerical approach to estimate the solution. Find the area of the shaded region bounded by the triangle with vertices (0,0), (3,4), (7,0). A) precalculus, 14.0 D) calculus, 8.0 B) precalculus, 8.0 E) precalculus, 4.0 C) calculus, 4.0 Copyright Houghton Mifflin Company. All rights reserved. 19

4 7. Consider the function f ( x ) = x and the point P(100,10) on the graph of f. Consider the secant lines passing through P(100,10) and Q(x,f(x)) for x values of 97, 99, and 101. Find the slope of each secant line to four decimal places. (Think about how you could use your results to estimate the slope of the tangent line of f at P(100,10), and how to improve your approximation of the slope.) A) , , D) 0.05, 0.051, B) 0.05, 0.051, E) , , C) , , Use the rectangles in each graph to approximate the area of the region bounded by the following. y = 9/x, y = 0, x = 1, and x = 9. A) , D) , B) , E) , C) , Use the rectangles in the following graph to approximate the area of the region bounded by y = sin x, y = 0, x = 0, and x = π. A) 1.41 B) 3.79 C) D) E) Copyright Houghton Mifflin Company. All rights reserved.

5 10. Use the rectangles in the following graph to approximate the area of the region bounded π π by y = cos x, y = 0, x =, and x =. A) B).6055 C) D) E) Complete the table and use the result to estimate the it. x 5 x 5 x 0 x + 75 x f(x) A) B) C) D) E) Complete the table and use the result to estimate the it. x x 7 x + 7 x f(x) A) B) C) D) E) Complete the table and use the result to estimate the it. 1 1 x x 5 x 5 x f(x) A) B) C) D) E) Copyright Houghton Mifflin Company. All rights reserved. 1

6 14. Complete the table and use the result to estimate the it. sin x x 0 x x f(x) A) 1 B) 0.5 C) 1 D) 0.5 E) Complete the table and use the result to estimate the it. cos ( 4x) 1 x 0 4x x f(x) A) 1 B) 0 C) 1 D) E) 16. Determine the following it. (Hint: Use the graph of the function.) 5 x ( x) A) 7 B) 5 C) D) 3 E) Does not exist Copyright Houghton Mifflin Company. All rights reserved.

7 17. Let 3 x, x 1 f( x) =. 0 x = 1 Determine the following it. (Hint: Use the graph of the function.) f ( x) x 1 A) B) 0 C) 3 D) 4 E) Does not exist 18. Determine the following it. (Hint: Use the graph of the function.) x ( x + ) 3 A) Does not exist B) C) 3 D) 7 E) 0 Copyright Houghton Mifflin Company. All rights reserved. 3

8 19. Let x + 4, x 1 f( x) =. 1, x = 1 Determine the following it. (Hint: Use the graph of the function.) f ( x) x 1 A) 5 B) 1 C) 4 D) 16 E) Does not exist. 0. Determine the following it. (Hint: Use the graph of the function.) 1 x 3 x 3 A) 0 B) Does not exist C) 3 D) 3 E) 6 1. Let f( x) = 3 x 5 and (a) f ( x) x 4 (b) gx ( ) gx ( ) x =. Find the its: (c) x x 4 g( f( x)) A) 1, 4, 3 D) 1,, 5 B) 4,, 16 E) 7, 4, 89 C) 8,, 3 4 Copyright Houghton Mifflin Company. All rights reserved.

9 . Let f( x) = x + 1 and gx ( ) = 3x. Find the its: (a) f ( x) x 5 (b) li m gx ( ) (c) g( f( x)) x x 3 A), 3, 6 B) 5,, 7 C) 6, 6, 30 D) 5,, 1 E) 6,, 8 3. Let = x and gx ( ) x+ 3 f ( x) 5 + =. Find the its: (a) f ( x) x (b) li m gx ( ) (c) g( f( x)) x 5 x 5 A) 9, 8, 33 B), 5, 5 C) 6,5, 30 D) 4, 3, 8 E) 6, 5, Let f( x) 5 x + 3 x and (a) f ( x) x A) = gx x x 5 ( ) = x 4. Find the its: (b) li m gx ( ) (c) g( f( x)) 4,, 134 D) 3 3 4,, 134 B) 3 3 8,, 134 E) None of the above C) 3 3 8,, Find the it: sin x 3π x 4 A) 1/ B) 1/ C) 1/ 4 D) Does not exist E) 1/ 4 6. Find the it: π x cos x 5 6 1/ 3 A) B) 4 1/ 3 C) 1/ 3 D) 0 E) 1/ Find the it: 5x tan x π 6 1/ A) 3 B) 1/ 1/ 3 C) 6 D) Does not exist E) 1/ 6 Copyright Houghton Mifflin Company. All rights reserved. 5

10 8. Suppose that f( x) = 7 and gx ( ) = 4. Find the following it: x c x c ( ) f( x) g x x c A) 11 B) 3 C) 0 D) 8 E) Suppose that f ( x ) = 8 and gx ( ) = 7. Find the following it: x c x c f ( x) + g( x) x c [ ] A) 56 B) 15 C) 0 D) 1 E) Suppose that f ( x ) = 9 and gx ( ) = 14. Find the following it: x c x c f ( x) g( x) x c [ ] A) 0 B) 5 C) 3 D) 16 E) Suppose that f ( x ) = 7 and gx ( ) = 13. Find the following it: x c x c 9g( x) x c [ ] A) 117 B) 63 C) 9 D) 13 E) 7 3. Suppose that f ( x ) = 14 and gx ( ) = 15. Find the following it: x c x c f ( xgx ) ( ) x c [ ] A) 14 B) 9 C) 1 D) 10 E) Suppose that f ( x ) = 6 and gx ( ) = 9. Find the following it: x c x c f ( x) x c gx ( ) A) 3 B) 54 C) 3 D) Does not exist. E) Find the following it (if it exists). Write a simpler function that agrees with the given function at all but one point. 3 x + 51 x 8 x + 8 A) 19, x 8 x+ 64 D) Limit does not exist. B) 64, x + 8 x+ 64 E) 19, x 8 x+ 64 C) 64, x 8 x 64 6 Copyright Houghton Mifflin Company. All rights reserved.

11 35. Find the following it (if it exists). Write a simpler function that agrees with the given function at all but one point. 1 x 179 x+ 154 x 14 x 14 A) Does not exist. D) 157, 1 x + 11 B) 179, 1 x + 11 E) 157, 1 x 11 C) 179, 1 x Find the it (if it exists): x 8 x 8 64 x A) 3 B) 1 16 C) Find the it (if it exists): D) 8 E) 1 3 x 0 A) ( x+ x) ( x+ x) 6 ( x x 6) x 3 x x x B) 3 x x 6x C) 0 D) x 1 E) x x Determine the it (if it exists): ( x) sin x 1 cos x 0 8 x A) 0 B) 1 C) Does not exist. D) E) Determine the it (if it exists): ( x) 1 cos x 0 x A) 0 B) 4 C) 8 D) Does not exist E) Determine the it (if it exists): 5 sin x x 0 4 x A) B) 1 C) 0 D) Does not exist. E) Copyright Houghton Mifflin Company. All rights reserved. 7

12 41. Use the graph as shown to determine the following its, and discuss the continuity of the function at x = 4. (i) f ( x) + x 4 (ii) f ( x) x 4 (iii) f ( x) x 4 A) 4, 4, 4, Not continuous D) 3, 3, 3, Continuous B) 3, 3, 3, Not continuous E),,, Not continuous C),,, Continuous 4. Use the graph as shown to determine the following its, and discuss the continuity of the function at x = 3. (i) f ( x) + x 3 (ii) f ( x) x 3 (iii) f ( x) x 3 A) 1, 1, 1, Not continuous D),,, Continuous B) 1, 1, 1, Continuous E) 3, 3, 3, Continuous C),,, Not continuous 8 Copyright Houghton Mifflin Company. All rights reserved.

13 43. Use the graph to determine the following its, and discuss the continuity of the function at x = 4. (i) f ( x) + x 4 (ii) f ( x) x 4 (iii) f ( x) x 4 A), 0, Does not exist, Not continuous D) 4, 0, Does not exist, Not continuous B) 0,, Does not exist, Not continuous E),, Does not exist, Not continuous C) 0,, 0, Continuous 44. Find the it (if it exists). Note that f ( x) x 8 + ( x ) 4 7 A) 39 B) 35 C) Does not exist D) 39 E) 35 = x represents the greatest integer function. 45. Find the x-values (if any) at which the function f( x) = 3 x 7 x 11 is not continuous. Which of the discontinuitites are removable? A) Continuous everywhere D) 7 x =. Not removable. 6 B) x = 11. Removable E) both B and C C) 7 x =. Removable x Find the x-values (if any) at which the function f( x) = is not continuous. x + 64 Which of the discontinuitites are removable? A) 8 and -8. Not removable. D) Discontinuous everywhere B) Continuous everywhere E) None of the above C) 8 and -8. Removable. Copyright Houghton Mifflin Company. All rights reserved. 9

14 47. x 5 Find the x-values (if any) at which the function f( x) = x 6 x+ 5 is not continuous. Which of the discontinuitites are removable? A) No points of discontinuity. B) x = 5 (Not removable), x = 1 (Removable) C) x = 5 (Removable), x = 1 (Not removable) D) No points of continuity. E) x = 5 (Not removable), x = 1 (Not removable) 48. Find constants a and b such that the function 9, x 5 f ( x) = ax+ b, 5< x< 1 9, x 1 is continuous on the entire real line. A) a = 3, b = 0 D) a = 3, b = 6 B) a = 3, b = 6 E) a = 3, b = 6 C) a = 3, b = Find the constant a such that the function sin x 4, x < 0 f( x) = x a+ 15 x, x 0 is continuous on the entire real line. A) 4 B) 4 C) 15 D) 15 E) x Determine whether f( x) = approaches or as x approaches 9 from the x 81 left and from the right by completing the tables below. x f(x) x f(x) A) f( x) = D) f( x) = B) C) x 9 x 9 + x 9 f( x) = E) Both A and C f( x) = F) Both B and D + x 9 30 Copyright Houghton Mifflin Company. All rights reserved.

15 51. x 5 Find the vertical asymptotes (if any) of the function f( x) =. x 15 x+ 50 A) x = 10 B) x = 10 C) x = 5 D) x = 5 E) x = x + 8 x+ 1 Find the vertical asymptotes (if any) of the function f( x) =. 3 x + x 6 x+ 4 A) x = 4 B) x = 1 C) x = 4 D) x = 1 E) A and B F) C and D 53. Find the vertical asymptotes (if any) of the function f ( x) = tan( 15 x). A) k + 1 D) No vertical asymptotes x = π ( k = 0, ± 1, ±,...) 30 B) k + 1 E) k x = π ( k = 0, ± 1, ±,...) x = π ( k = 0, ± 1, ±,...) C) k x = π ( k = 0, ± 1, ±,...) Find the it: x 8 + x 6 x + 6 A) B) C) 0 D) 1 E) Find the it: ( x 144 )( x 1) x 1 + x A) 4 B) 56. Find the it: 1x 1 4 C) 4 D) 1 4 E) x x 0 x A) 0 B) C) 1 D) 1 E) Copyright Houghton Mifflin Company. All rights reserved. 31

16 Answer Key 1. C. B 3. E 4. D 5. D 6. A 7. C 8. A 9. D 10. E 11. A Section: C Section: D Section: A Section: B Section: D Section: A Section: D Section: A Section: B Section: E. C 3 Copyright Houghton Mifflin Company. All rights reserved.

17 3. A 4. D 5. B 6. C 7. A 8. E 9. D 30. C 31. A 3. D 33. C 34. A 35. E 36. C 37. D 38. C 39. A 40. C 41. E Section: A Section: B Section: D Section: A Section: 1.4 Copyright Houghton Mifflin Company. All rights reserved. 33

18 46. B Section: C Section: E Section: B Section: E Section: B Section: E Section: A Section: B Section: D Section: E Section: Copyright Houghton Mifflin Company. All rights reserved.