MATH 101 Week 1 Derivatives Question

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< AA chegg.com + X Chegg eReader Chegg eReader: Applied Mathematics for the Manageri... < Q 608 CHAPTER 9 THE DERIVATIVE 9.1 Concept Questions 1-2 4. What is an indeterminate form? Illustrate with an example. 5. Explain in your own words the meaning of lim f(x) = L and lim f(x) = M. X-3 1. Explain what is meant by the statement lim f(x) = 3. 2. a. If lim f(x) = 5, what can you say about f(3)? Explain. b. If f(2) = 6, what can you say about lim f(x)? Explain. 3. Evaluate the following, and state the property of limits that you use at each step. 2x2 + x + 5 a. lim Vx(2x2 + 1) b. lim x + 1 x x-2 -50 3/2 1-4 9.1 Exercises In Exercises 1-8, use the graph of the given function fto determine lim f(x) at the indicated value of a, if it exists. xa In Exercises 9-16, complete the table by computing f(x) at the given values of x. Use these results to estimate the indicated limit (if it exists). 9. f(x) = x² + 1; lim f(x) 1. y = f(x) y 2. y = f(x) 3 2. . 1.9 1.99 1.999 2.001 2.01 2.1 2 1 f(x) . -3 -2 -1 1 2 3 -3-2 1 2 3 -1 10. f(x) = 2x² - 1; lim f(x) a= -2 a=1 0.9 0.99 0.999 1.001 1.01 1.1 3. AY 4. f(x) y = f(x) 4 3 1.x 3 2 11. f(x) lim f(x) y = f(x) X 1-0 1 . -0.1 -0.01 -0.001 0.001 0.01 0.1 -2 -1 1 X 2 3 a = 1 f(x) -3 -2 -1 1 2 3 a = 3 x 12. f(x) 11 "; lim f(x) 5. 6. x-1 y=f(x) 3+ 4 x 0.9 0.99 0.999 1.001 1.01 1.1 2 3- f(x) y = f(x) 1+ 2 1 1 -3-2-1 1 2 3 13. f(x) = (x lim f(x) 1)" . a= -2 24 -3 -2 -1 12 a = -2 . 0.9 0.99 0.999 1.001 1.01 1.1 7. AY 8. Y 3 y = f(x) 3 2 y = f(x) 14. f(x) 1 lim f(x) x - 212 2 1 X 1.9 1.99 1.999 2.001 2.01 2.1 -4-3-2-1 12 -3-2-1 +4 3 1 2 -2 a = -2 a = 0 x2 + x - 2 15. f(x) lim f(x) X-1 . 0.9 0.99 0.999 1.001 1.01 1.1 f(x) Unless otherwise noted, all content on this page is Cengage Learning Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapters) Editorial review has deemed that any suppressed content does not materially affect the overall leaming experience. Cengage Leaming reserves the right to remove additional content at any time if subsequent rights restrictions require it. 2:24 PM Mon Sep 28 100% < AA chegg.com + X Chegg eReader Chegg eReader: Applied Mathematics for the Manageri... . < Q 9.1 LIMITS 609 X-1 16. f(x) lim f(x) 45. lim Vg(x) 46. lim V 5f(x) + 3g(x) x-1-1 x- X- . 0.9 0.99 0.999 1.001 1.01 1.1 $(x) 2f(x) - g(x) 47. lim f(x)g(x) g(x) - f(x) 48. lim -a f(x) + Vg(x) In Exercises 17-22, sketch the graph of the function f, and eval- uate lim f(x), if it exists, for the given value of a. In Exercises 49-62, find the indicated limit, if it exists. r? - 1 x2 - 4 49. lim 50. lim X-X-1 X-2 x + 2 x-a 17. f(x) x-1 1-1 if x < 0 if x > 0
(a = 0)
3x
x – x
51. lim
.
2.x2
52. lim
X
18. f(x)
Sx-1 if x < 3 - 2x + 8 if x > 3
(a = 3)
x2 – 25
53. lim
—5x + 5
b + 1
54. lim
–3b + 3
X
19. f(x)
if x < 1 if x = 1 if x > 1
0
-X + 2
(a = 1)
X
55. lim
2-3-1
x + 2
56. lim
2+28 – 2
– x – 6
57. lim
*–2x + x – 2
58. lim
20. f(x)
– 8
2.
– 2x + 4 if x 1
Slxl if x = 0
(a = 0)
1 if x = 0
X-4
60. lim
3-4 VX-2
21. f(x)
Vr-1
59. lim
1-X-1
Vx+1
Hint: Multiply by
Vx+1
=
Hint: See Exercise 59.
x – 1|
22. f(x)
{
if x # 1
if x = 1
(a = 1)
X-1
61. lim
*- x + x2 – 2x
4 – x2
62. lim
-~-2 2×2 + x3
In Exercises 23-40, find the indicated limit.
23. lim 3
24. lim -3
X-2
In Exercises 63-68, use the graph of the function f to determine
lim f(x) and lim f(x), if they exist.
X2
25. lim x
26. lim – 3x
–2
63.
64.
AY
f(x) = 2×2 – 10
f(x) = x3 – x
27. lim (1 – 2x)
28. lim (412 – 2t + 1)
29. lim (2x – 3×2 + x + 2)
10
30. lim (4×5 – 20×2 + 2x + 1)
-1
1 2 3
10
3
31. lim (2s2 – 1)(2s + 4) 32. lim (x2 + 1)(x2 – 4)
3-0
x2
65.
66.
2x + 1
33. lim
*2 x + 2
x2
x3 + 1
34. lim
– 2×3 + 2
36. lim V 5x + 2
1-2
35. lim Var + 2
f(x) =
1
37. lim V2x4 + x2
38. lim
x-12
121² + 4
x? + 1
-3 -2 -1
-2
1
2
メ→-3
3
67.
AY
V x2 + 8
39. lim
*–1 2x + 4
XV x2 + 7
40. lim
**3 2x – 2x + 3
f(x) = 2 – 4
3
In Exercises 41-48, find the indicated limit given that
lim f(x) = 3 and lim g(x) = 4.
A
X
-1
41. lim (f(x) – g(x)]
42. lim 2f(x)
43. lim (2f(x) – 3g(x)]
44. lim [f(x)g(x)]
Unless otherwise noted, all content on this page is Cengage Learning
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or Chapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
2:25 PM Mon Sep 28
100%
< AA chegg.com + X Chegg eReader Chegg eReader: Applied Mathematics for the Manageri... < Q 610 CHAPTER 9 THE DERIVATIVE 68. y 81. Toxic WASTE A city's main well was recently found to be contaminated with trichloroethylene, a cancer-causing chem- ical, as a result of an abandoned chemical dump leaching chemicals into the water. A proposal submitted to city coun- cil members indicates that the cost, measured in millions of dollars, of removing x% of the toxic pollutant is given by 1 V-X SV- if x < 0 f(x) = x if x > 0
-2 – 1
1
2
3
x + 1
C(x)
0.5x
100 – x
(0

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