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Name ______________________________________________ Date __________________________________Thurgood Marshall Academy Public Charter High School – Calculus
Exam Review #2
OBJECTIVES ABOUT COMPOSITION OF FUNCTIONS:
10.1 Find values of composite functions using equations and graphs.
10.2 Write a function as a composite of simpler functions.
PRACTICE EXERCISES ABOUT COMPOSITION OF FUNCTIONS:
1. Suppose f ( x )=3 x−5 and g ( x )=x2−2 x. Find g (f (2 ) ).
2. Suppose m (x )=−2x+5 and n ( x )=3+2 x. Find m (n (a+3 ) ). Express the answer in simplest form.
3. Suppose f ( x )=3x and g ( x )=cos x. Express k ( x )=3cosx as a composite of f and g.
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4. Express k ( x )=√sin(x−5) as a composite of three simpler functions.
5. Suppose a ( x )=|x−70|, b ( x )=2x, and c ( x )=x2. Find a (c (b (3 ) ) ).
6. Suppose f ( x )=x+5 and g ( x )=3√ x. Find f (g ( x ) ).
7. Suppose f ( x )=|x|, g ( x )=x2−5, and h ( x )=cos x . Express k ( x )=(cos|x|)2−5 as a composite of f, g, and h.
8. The graph of f is at the left. The graph of g is at the right. Find g (f (2 ) ).mrmerlintma.wordpress.com
9. Suppose p ( x )=2x−5 and q ( x )=x+7. Find q ( p (a ) ). Express the answer in simplest form.
10. Suppose f ( x )=√x+5, g ( x )=sin x, and h ( x )=x2. Find g ( f (h (x ) ) ).
11. Express k ( x )=|x|+4 as a composite of two simpler functions.
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12. The graph of f is at the left. The graph of g is at the right. Find f (g (g (4)) ).
OBJECTIVES ABOUT HOLES:
6.1 Graph a function with a hole in it.
6.2 Given the graph of a function with a hole in it, write its equation.
6.3 Given the equation of a function with a hole in it, give the coordinates of the hole.
PRACTICE EXERCISES ABOUT HOLES:
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13. Draw the graph of f ( x )=(2x+1)(x+2)(x+2)
.
14. Write the equation of the function.
15. Draw the graph of f ( x )=(x−1)(|x+2|)(x−1)
.
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16. Find the coordinates of the hole in f ( x )= x2−5 x+6x2−x−2
.
17. Write the equation of the function.
18. Draw the graph of f ( x )= x2−2 x−3x+1
.
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19. Write the equation of the function.
20. Draw the graph of f ( x )= x (|x|−3)x
.
21. Find the coordinates of the hole in f ( x )= xx2+11 x
.
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22. Write the equation of the function.
OBJECTIVES ABOUT LIMITS AND CONTINUITY:
7.1 Construct a table to determine the probable behavior of a given function as x approaches a given number, and describe that behavior.
7.2. State whether limx→c
f (x ) exists (and what it equals if it does exist) given information about f (c ) and the behavior of
f (x) as x approaches c from the positive side and negative side.
8.1 Given a graph of f (x) and a value of c, determine limx→c
f (x ), or explain why it does not exist.
9.1 Categorize function behavior at a specific number as a nonremovable discontinuity, a removable discontinuity, or continuous.
11.1 Find limit as x→c of a function at a point of continuity.
11.2 Find limits of sums, products, and quotients.
12.1 Find limit as x→c of a function at a hole, recognizing that there is a hole analytically.
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PRACTICE EXERCISES ABOUT LIMITS AND CONTINUITY:
23. The graph of f is shown. Describe the behavior of f at x=3. Choices: continuity, discontinuity (non-removable), discontinuity (removable).
24. Suppose f ( x )= x2−4x−2
. Describe the behavior of f at x=2. Choices: continuity, discontinuity (non-removable),
discontinuity (removable).
25. The graph of f is shown. Find limx→−3
f (x).
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26. Suppose f has the following properties.
As x approaches 2 from the negative side, f (x) approaches 3. As x approaches 2 from the positive side, f (x) approaches 5. f (2 )=8.
What is limx→2
f (x )?
27. Suppose limx→2
f (x )=7 and limx→2
g (x)=8. Find limx→2
[ f ( x )−2g (x)].
28. Find limx→π
(cos x ).
29. Find limx→2
(5x−2 ).
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30. Find limx→5 ( 1
x−5 ).
31. Find limx→0 ( sin x5 x ).
32. A table of values for f is shown. Describe the probable behavior of f at x=3. Choices: continuity, discontinuity (non-removable), discontinuity (removable).
x 2.9 2.99 2.999 3 3.001 3.01 3.1
f (x) 7.9 7.99 7.999 8 8.001 8.01 8.1
33. Suppose f has the following properties.
As x approaches 5 from the negative side, f (x) approaches 2. As x approaches 5 from the positive side, f (x) approaches 2. f (2 ) does not exist.
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Describe the behavior of f at x=5. Choices: continuity, discontinuity (non-removable), discontinuity (removable).
34. A table of values for f is shown. Find the probable value of limx→3
f (x ).
x 2.9 2.99 2.999 3 3.001 3.01 3.1
f (x) 8.9 8.99 8.999 10 9.001 9.01 9.1
35. Complete the table of values for f in any reasonable way based on the graph.
x 1.9 1.99 1.999 2 2.001 2.01 2.1
f (x)
36. Find limx→27
.
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37. Find limx→2
3 xx−4 .
38. Find limx→5
x−5|x−5|.
39. Find limx→5
x−5(x−5)(x−2)
.
40. Find limx→∞
(−x2+4).
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41. The graph of f is shown. Describe the behavior of f at x=3. Choices: continuity, discontinuity (non-removable), discontinuity (removable).
42. Suppose f ( x )= 1
x−3
. Describe the behavior of f at x=0
. Choices:
continuity, discontinuity (non-removable), discontinuity (removable).
43. The graph of f is shown. Find limx→0
f (x ).
44. Suppose f has the following properties.
As x approaches 6 from the negative side, f (x) approaches 4.mrmerlintma.wordpress.com
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As x approaches 6 from the positive side, f (x) approaches 4. f (6 ) does not exist.
What is limx→6
f (x )?
45. Suppose limx→10
f (x )=5 and limx→ 10
g (x)=4. Find limx→10
[ f ( x )g( x)].
46. Find limx→ π
2
¿¿.
47. Find limx→3
|x+2|x+2
.
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48. Find limx→2 ( 1
x2−4 x+4 ).
49. Find limx→0 ( sin xx +cos x+2x).
50. A table of values for f is shown. Describe the probable behavior of f at x=3. Choices: continuity, discontinuity (non-removable), discontinuity (removable).
x 2.9 2.99 2.999 3 3.001 3.01 3.1
f (x) 10 1000 100000 undefined 100000 1000 10
51. Suppose f has the following properties.
As x approaches 4 from the negative side, f (x) approaches 6.
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As x approaches 4 from the positive side, f (x) approaches 6. f (4 )=6.
Describe the behavior of f at x=4. Choices: continuity, discontinuity (non-removable), discontinuity (removable).
52. A table of values for f is shown. Find the probable value of limx→3
f (x ).
x 2.9 2.99 2.999 3 3.001 3.01 3.1
f (x) 5.1 5.01 5.001 10 9.999 9.99 9.9
53. Complete the table of values for f in any reasonable way based on the graph.
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f (x)
54. Find limx→ 05.
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55. Find limx→3
(−x3−2 ).
56. Find limx→0
|x|x
.
57. Find limx→−2
x2+3 x+2x2−x−6
.
58. Find limx→−∞
2x.
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x 1.9 1.99 1.999 2 2.001 2.01 2.1
f (x)
x 2.9 2.99 2.999 2 3.001 3.01 3.1
f (x)
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