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UAE_Math_Grade_12_Vol_1_SE_718383_ch3
Discuss the possibility that this is a representative graph: 13:38 ′ ′
30. f(1) = 0, lim f(x) = 2,f (x) < 0 for x < 1, f (x) > 0 for x > 1,
that is, is it possible that there are any important points not f (1) = 0. x→∞
′
shown in this window?
′
31. f(−1) = f(2) = 0, f (x) < 0 for x < −1 and 0 < x < 2
4. Suppose that the function in exercise 3 has three distinct and x > 2, f (x) > 0 for −1 < x < 0, f (−1) does not
′
′
critical numbers. Explain why the graph is not a represen- exist, f (2) = 0.
′
tative graph. Discuss how you would change the graphing
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window to show the rest of the graph. 32. f(0) = 0, f(3) =−1, f (x) < 0 for x > 3, f (x) > 0 for x <
′
′
0 and 0 < x < 1 and 1 < x < 3, f (0) = 0, f (1) does not
′
exist and f (3) = 0.
............................................................
In exercises 1–10, find (by hand) the intervals where the func-
tion is increasing and decreasing. Use this information to deter-
mine all local extrema and sketch a graph. In exercises 33–38, find (by hand) all asymptotes and extrema,
and sketch a graph.
3
3
2
1. y = x − 3x + 2 2. y = x + 2x + 1 2
x
x
2
3. y = x − 8x + 1 4. y = x − 3x − 9x + 1 33. y = x − 1 34. y = x − 1
4
3
2
2
2
5. y = (x + 1) 2∕3 6. y = (x − 1) 1∕3 35. y = 2 x 2 36. y = x 4
2
7. y = sin x + cos x 8. y = sin x x − 4x + 3 38. y = 1 − x
2
x + 2
x
2
2
9. y = e x −1 10. y = ln(x − 1) 37. y = √ x + 1 (x + 1) 2
2
............................................................ ............................................................
In exercises 11–20, find (by hand) all critical numbers and use In exercises 39–42, estimate critical numbers and sketch graphs
the First Derivative Test to classify each as the location of a local showing both global and local behavior.
maximum, local minimum or neither. 2
39. y = x − 30 40. y = x − 8
2
5
4
11. y = x + 4x − 2 12. y = x − 5x + 1 x − 1 x − 1
3
4
4
13. y = xe −2x 14. y = x e 41. y = x + 60 42. y = x − 60
2 −x
2
x − 1
2
x + 1
( ) ............................................................
15. y = tan (x ) 16. y = sin −1 1 − 1
−1
2
x 2
17. y = x 18. y = x 43. Give a graphical example showing that the following state-
1 + x 3 1 + x 4 ment is false. If f(0) = 4 and f is a decreasing function, then
√
19. y = x + 3x 2 20. y = x 4∕3 + 4x 1∕3 the equation f(x) = 0 has exactly one solution.
3
............................................................
44. Assume that f is an increasing function with inverse func-
−1
tion f . Show that f −1 is also an increasing function.
In exercises 21–26, approximate the x-coordinates of all ex-
trema and sketch graphs showing global and local behavior of 45. State the domain for sin −1 x and determine where it is in-
the function. creasing and decreasing.
4
2
21. y = x − 15x − 2x + 40x − 2 ( )
3
46. State the domain for sin −1 2 tan −1 x and determine where

22. y = x − 16x − 0.1x + 0.5x − 1 it is increasing and decreasing.
3
2
4
3
5
23. y = x − 200x + 605x − 2 47. If f and g are both increasing functions, is it true that f(g(x))
4
24. y = x − 0.5x − 0.02x + 0.02x + 1 is also increasing? Either prove that it is true or give an ex-
2
3
ample that proves it false.
2
25. y = (x + x + 0.45)e −2x
48. If f and g are both increasing functions with f(5) = 0,
26. y = x ln 8x 2 find the maximum and minimum of the following values:
5
............................................................ g(1), g(4), g(f(1)), g(f(4)).
2
In exercises 27–32, sketch a graph of a function with the given { x + 2x sin(1∕x) if x ≠ 0
properties. 49. For f(x) =
′
0 if x = 0 show that f (0) > 0,
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27. f(0) = 1, f(2) = 5, f (x) < 0 for x < 0 and x > 2, but that f is not increasing in any interval around 0. Ex-
′
Copyright © McGraw-Hill Education 28. f(−1) = 1,f(2) = 5,f (x) < 0 for x < −1 and x > 2,f (x) > 0 50. For f(x) = x , show that f is increasing in any interval
f (x) > 0 for 0 < x < 2.
plain why this does not contradict Theorem 4.1.
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′
3
′
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for −1 < x < 2,f (−1) = 0,f (2) does not exist.
′
around 0, but f (0) = 0. Explain why this does not contradict
′
Theorem 4.1.
′
29. f(3) = 0,f (x) < 0 for x < 0 and x > 3,f (x) > 0 for 0 < x <
′
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3,f (3) = 0,f(0) and f (0) do not exist.
51. Prove Theorem 4.2 (the First Derivative Test).
267