file · web viewname _date _thurgood marshall academy public charter high school...

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.

mrmerlintma.wordpress.com

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.


x

y

x

y

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:


x

y

x

y

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)

.


x

y

x

y

x

y

16. Find the coordinates of the hole in f ( x )= x2−5 x+6x2−x−2

.


18. Draw the graph of f ( x )= x2−2 x−3x+1

.


x

y

x

y


20. Draw the graph of f ( x )= x (|x|−3)x

.

21. Find the coordinates of the hole in f ( x )= xx2+11 x

.


x

y

x

y


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.


x

y

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).


x

y

x

y

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 ).


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


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.


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

.


x

y

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).


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 ).


As x approaches 6 from the negative side, f (x) approaches 4.mrmerlintma.wordpress.com

x

y

x

y

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

.


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


As x approaches 4 from the negative side, f (x) approaches 6.


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.

x 2.9 2.99 2.999 3 3.001 3.01 3.1

f (x)

54. Find limx→ 05.


x

y

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.



x

y

x

y

x

y

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)


file · web viewname _____date _____thurgood marshall academy public charter high school...

Documents

file · web viewname _date _thurgood marshall academy public charter high school...