AP Calculus BC - CD Hylton High School

TO:
FROM:
RE:
AP Calculus BC Students
Mrs. Wilson (formally Brundage), AP Calculus BC Teacher
Summer Assignment
In order to ensure that our AP Calculus classes meet the standards required by the College Board, it is
strongly recommended that all calculus students complete a summer assignment. This must be submitted
by Tuesday Sept. 13 for credit towards 2 extra points on the first quarter grade.
The graded packet assignment should be done after the textbook assignment has been completed. Your
success in calculus will depend on your knowledge of algebra, geometry, trigonometry and functions, as
well as your effort and work habits. This summer assignment is designed to prepare you for a very
successful year.
Our summer assignment helps to compensate for the loss of instructional time due to classes beginning
after Labor Day. The BC Calculus AP exam will be given in early May through out the country,
regardless of when the school year starts. This assignment also helps to meet our objective of giving you
the best opportunity to receive college credit while still in high school. If you need help, please feel free
to contact Mrs. Wilson (Brundage) online during the summer at brundala@pwcs.edu. (this email will
change for the 2011-12 school year.)
READ & STUDY the information in the textbook portion of the assignment, then work out the solutions
to each problem. SHOW ALL WORK and make sure your work is neatly done and complete. You may
work with a friend on the textbook portion. This is a very good opportunity to form study groups for the
year. Just be sure you understand how to do all the problems yourself.
We expect you to share this information with your parents. If they have any questions, they should call
Mrs. Wilson (Brundage) at Hylton before June 17 th . Good luck and have a great summer!
Textbook:
CALCULUS: Graphical, Numerical, Algebraic, Finney, Demana, Waits, Kennedy, 2007.
Textbook Assignment:
Read and understand all of chapter 1 and chapter 2 sections 1, 2, & 3, then complete these problems:
(check the odd answers in the back of the book!)
SHOW ALL WORK COMPLETELY!
p.19 #41-48, 51-53, 59-62 p.27 #22, 23, 31
p.29 #1-4 p.34 #13, 15, 39-42
p.44 #33-44, 54-57 p.52 #5-8, 27-30, 52-55
p.55 #1-4 p.57 #68-70
p.66 #1-4, 37-51,67-70 p.76 #55,56,59,63,quick quiz #1-4
p.84 #1-17
** note: these book work problems will be assigned for homework on the first day of school and will be
due on Tuesday Sept. 13, so you might as well do as many as you can over the summer!
1

AP Calculus BC
Summer Assignment
Show all work in this packet.
1. Find the x-intercept(s): 3x 2 + 2y 2 + 4xy – 12 = 0.
A. ± 6 B. ± 2 C. 4 D. 6 E. None of these
2. Find all the intercepts:
x 1
y = −
x + 3
A. (1 , 0), (0, -1/3) B. (1, 0) C. (-3, 0), (1, 0) D. (-3, 0), (0, -1/3) E. none of these
3. Identify the type(s) of symmetry: x 4 y 2 + 2x 2 y – 1 = 0
A. x-axis B. y-axis C. origin D. both x- and y-axis E. none of these
4. Find the equation of the line that passes through (1, 3) and is perpendicular to the line 2x + 3y + 5 = 0.
A. 3x – 2y + 3 = 0 B. 2x + 3y – 11 = 0 C. 2x + 3y – 9 = 0
D. 3x – 2y – 7 = 0 E. none of these
5. Find the equation of the line through (4, 2) and perpendicular to the line through (9, 7) and (11, 4).
2
3
3
A. y= ( x− ) + B. y=− ( x− ) + C. y ( x )
3
9 7
2
D. y ( x )
=
3
− 4 + 2 E. none of these
2
9 7
=−
2
− 4 + 2
6. Find the domain of f ( x)
A. ⎛ 3 ⎞
−∞ , B. ⎡3 ⎞
⎜ ⎟
,
⎝ 2 ⎠
⎢
∞ ⎟
⎣2
⎠
=
1 .
3−
2x
C. ⎛3 ⎞
⎜ , ∞⎟
⎝2
⎠
D.
⎛ 3 3
⎜−∞, ⎞ ⎟ ⎛ ⎜ , ∞
⎞
⎟
E. none of these
⎝ 2⎠ ⎝2
⎠
2

7. Find
f ( x+ h) − f ( x)
2
if f ( x) = 2x
+ 3
h
A. 4x + 2h B. 2h 2 + 3 C. 2h + 3/h D. 4hx + 2h 2 E. none of these
8. Given f ( x) x 2 3x 4, find f ( x 2) f ( 2)
= − + + − .
A. x 2 – 3x + 4 B. x 2 + x C. x 2 + x – 8 D. x 2 – 3x – 4 E. none of these
f x = 3x − x + 2
9. Is the following function even or odd? ( )
4 2
A. odd B. even C. both D. neither
10. If ( )
2
f x = 1− x and g( x)
( )
1
= , find f g( x ) .
x
A.
1−
x
x
2
B.
1
1−
x
2
C.
1
1− D.
x
1
x
2
+ 1− x E. none of these
⎛ f ⎞
⎜ ⎟
⎝ g ⎠
11. Given f(x) = 6x – 12 and g(x) = x 2 – 4, find ( x)
A.
6
x − 2
B.
( x − )
6 2
x + 2
6
C.
( x+ 2 )( x−2
)
D.
6
x + 2
E. none of these
12. Given f(x) = 2x 2 + 1 and g(x) = x – 2, find f ( g( x )).
A. x 2 – 7 B. 2x 2 + 2x – 1 C. 2x 2 – 1 D. 2x 2 – 8x + 9 E. none of these
13. Match the graph to the right with the correct function.
x + 3
x 1
x −1
=
x + 2x−3
A. f ( x)
= B. f ( x) = x+
3
−
C. f ( x) 2
E. none of these
D. f ( x)
x
=
2
+ 2x−3
x −1
3

14. Find all vertical asymptotes of f ( x)
2x
−1
=
x + 3
A. x = 2 B. x = ½, x = -3 C. x = -3 D. x = ½ E. none of these
15. Using the figure to the right, solve for x:
A.
6
3
B. 6 C. 3 2
3 30º
x
D. 3 E. none of these
16. If
π
2
< θ < π and sin θ=
, find sin2θ .
2 3
A.
4 5
− B. 4 9
3
C. 4 5
9
D. – 4 3
E. none of these
17. Solve the given equation for
≤ θ < π θ + θ = .
2
0 2 : 2sin sin 1
π π 2π 3π
A. 0, , π B. , ,
2
3 3 2
C.
π 5π 3π
, ,
6 6 2
D.
π 3π
,
6 2
E. none of these
18. A rectangle is bounded by the x-axis and the semicircle
rectangle as a function of x.
y
2
= 25 − x . Write the area A of the
A. y = 12 B. y = 3x C. y = 8x D.
y x x
2
= 2 25 − E. none of these
19. Describe how the graph of f ( x) =−2 3− x + 5 can be obtained from the graph of g( x)
= x.
A. reflect through y-axis, shift left 3, reflect through x-axis, vertical stretch by 2, shift up 5.
B. reflect through y-axis, shift right 3, reflect through x-axis, vertical stretch by 2, shift up 5
C. reflect through y-axis, shift right 5, reflect through x-axis, vertical stretch by 2, shift up 3
D. reflect through x-axis, shift left 3, vertical stretch by 2, shift down 5
E. reflect through y-axis, shift left 3, reflect through x-axis, vertical stretch by 2, shift down 5
4

20. Write a formula for the function graphed on the right
A. f ( x)
C. f ( x)
⎧ 5 − x , 0≤ x ≤2
⎪
=
2
⎨
⎪ 5
x− 10, 2 < x≤4
⎪⎩ 2
⎧5 x , 0 ≤ x≤
2
⎪
=
2
⎨
⎪− 5
x + 10, 2 < x ≤ 4
⎪⎩ 2
B. f ( x) =− x + 5
D. f ( x) =− x−
2
E. none of these
21. Give the equation for the graph after the following transformations have been applied to the graph of
y = x 2 : vertical stretch by 2, reflect through the x-axis, shift right 2, shift up 3.
A. y = ( −2x− 2) 2
+ 3 B. y =− 2( x+ 2) 2
+ 3 C. y ( x ) 2
D. y = 2( −x− 2) 2
+ 3 E. y ( x ) 2
=− 2 + 2 − 3
=−2 − 2 + 3
22. f(x) = |6 + 8x| is equivalent to which piecewise function?
A. f ( x)
C. f ( x)
⎧6+ 8 x,
x≥−
= ⎨
⎩ − 8x
− 6, x −
23. Find the value of log8
32 .
3
4
3
4
3
4
3
4
B. f ( x)
D. f ( x)
⎧6+ 8 x,
x≤−
= ⎨
⎩ − 8x
− 6, x >−
⎧6+ 8 x,
x≥−
= ⎨
⎩6 − 8 x,
x

1
− − =
2
A. -1 B. 3 and -2 C. 3 only D. no solutions E. none of these
26. Solve for x: log ( 2
9
x x 3)
27. Solve for x: ( x )
log 15 + 5 − log x=
2
5 5
A. 2 B. ½ C. -1 D. 5 E. none of these
28. Eliminate the parameter: x = t – 3 y = t 2 – 2t – 1
A. y = x 2 + 4x + 2 B. y = x 2 + 4x + 14 C. y = x 2 – 2x + 14
D. y = x 2 – 2x + 2 E. none of these
29. For the equation y = x 2 – 3x + 5, find the y-component of its parametric equation by letting x = t – 2.
A. y = t 2 – 3t + 3 B. y = t 2 – 3t + 15 C. y = t 2 – 7t + 3
D. y = t 2 – 7t + 15 E. none of these
30. This is the grade I will receive in AP Calculus BC
A. A B. B+ C. B D. C+ E. Mrs. Wilson doesn’t want to know
31. If the population of mosquitoes doubles every hour and there are 175 flies initially, approximate how
many mosquitoes will be present after 6.5 hours, assuming that none die.
A. 2,275 B. 11,927 C. 15,839 D. 18,300 E. none of these
ax
32. An exponential curve is in the form of y = Ce . If the curve has a y-intercept of 10 and passes
through the point (5, 5), find the value of a.
A. -0.139 B. -0.693 C. 0.693 D. 0.139 E. none of these
33. The graph of r = 5cos 2θ
has how many loops?
A. none B. 2 C. 4 D. 5 E. none of these
6

34. The graph to the right could be which of the following polar equations?
A. r = 4θ
B. r = 4cos 2θ
C. r = 4 + cosθ
D. r = 1+ 4cosθ
E. none of these
35. f(x) decreases without bound as x approaches what value from the right when f ( x)
+
−
−
+
A. x → 5 B. x → 3 C. x → 5 D. x → 3 E. none of these
=
4
( x−3)( 5−x)
.
36. Evaluate:
x
−
lim
2
4
x→3
A. 1 B. 5 C. -1 D. 5 E. none of these
37. Evaluate:
x
lim
x→4
2
− 5x+
4
x − 4
A. 0 B. -1 C. 3 D. DNE E. none of these
38. Evaluate:
lim
x→3
x − 3
x − 3
A. 0 B. 1 C. 3 D. DNE E. none of these
39. If lim f ( x) = 2 and lim g( x) = 6, then find lim ⎡f ( x) − 2 f ( x) g( x) + g( x)
x→c x→c x→c
2 2 ⎤
⎣ ⎦ .
A. 40 B. -4 C. 16 D. 28 E. none of these
40. Evaluate: lim( 3x
2 + 5)
x→2
A. 41 B. 17 C. 11 D. 0 E. none of these
7

Use the graph to the right to answer #41-42.
41. Which defined function matches the graph?
A. f ( x)
C. f ( x)
⎧x+ 3, x≠3
= ⎨
⎩1, x = 3
⎧3 −x, x≠1
= ⎨
⎩1, x = 1
B. f ( x)
D. f ( x)
⎧x+ 3, x≠1
= ⎨
⎩1, x = 1
⎧3 −x, x≠2
= ⎨
⎩1, x = 2
E. none of these
42. Find lim f ( x)
x→1
.
A. 1 B. 2 C. 3 D. DNE E. none of these
Use the graph to the right to answer #43-44.
43. Which defined function matches the graph?
A. f ( x) =
2
x −1
x −1
B. f ( x)
1
=
x + 1
1
x 1
C. f ( x)
= D. f ( x )
−
E. none of these
=
1
( x + 1) 2
44. Find lim f ( x)
x→−1
.
A. 0 B. 1 C. DNE D. −∞ E. none of these
45. Find
x
lim
x→−1
2
+ 3x+
2
.
2
x + 1
A. 0 B. ∞ C. -1 D. DNE E. none of these
8

46. Find
lim−
x→6
6
3x
−18
− x
A. -1 B. 1 C. 3 D. -3 E. none of these
1
f x = and g x = x − 5
x + 1
47. Let ( ) ( )
2
. Find all values of x for which f ( g( x ))
is discontinuous.
A. -1 B. -1 and ± 5 C. ± 5 D. -2 and 2 E. none of these
48. Determine the value of c so that f(x) is continuous for all real numbers if f ( x)
A. -4 B. -2 C. 2 D. 4 E. none of these
⎧x+ 3, x≤−1
= ⎨
⎩2 x− c, x>−1
⎛ 5 ⎞
49. Find lim⎜2
+
x→0
2 ⎟
⎝ x ⎠ .
A. 7 B. 2 C. ∞ D. - ∞ E. none of these
50. Find
1
lim
x
+
x→0
A. ∞ B. 0 C. - ∞ D. DNE, since the right and left hand limits are not equal E. none of these
51. Find
⎛
lim 2 −
x→1
⎜
⎝
5
( x − 2) 2
⎞
⎟
⎠
A. - ∞ B. ∞ C. -3 D. 2 E. none of these
52. Describe the slope at the indicated point P on the graph to the right.
A. no slope B. positive C. negative
P
D. zero E. none of these
9

53. At which point is the slope of the tangent line closest to zero on the graph shown below?
A. A B. B C. C D. D E. E
4 3
54. Given f ( x) = 3x − 6x + 3x− 2, find '( )
f x .
A
B
C
D
E
f ' x = 12x − 18x + 3x−
2
2 2
4 3
3 2
A. f '( x) = 3x − 6x
+ 3 B. f '( x) = 3x − 6x + 3x
C. ( )
f ' x = 12x − 18x
+ 3 E. none of these
D. ( )
3 2
55. Find the point(s) on the graph of the function ( )
3
f x = x − 2 where the slope is three.
A. (1, 3) & (-1, 3) B. (1, -1) & (-1, -3) C. ( 3 2 , 0) D. (1, 3) E. none of these
56. Where is the function f ( x) =
2
x
5
−2x−15
discontinuous?
A. x = -5 and x = -3 B. x = -5 and x = 3 C. x = -3 and x = 5
D. x = 3 and x = 5 E. x = 2 and x = 15
57. Find
lim
x→∞
2
3x
+ 2x+
1
x + 1
.
A. 3 B. ∞ C. 3 D. 0 E.
3
2
3/2
58. If f ( x) = x , then find '4 ( )
f .
A. -6 B. -3 C. 3 D. 6 E. 8
f x = 2x
+ 1, then find
59. If ( )
2
lim
x→0
( ) − ( 0)
f x f
x
2
.
A. 0 B. 1 C. 2 D. 4 E. DNE
10

−1
60. Evaluate tan ( 1)
.
A. 1 B. -1 C.
π
2
D.
π
4
E. 0
61. Solve the equation: sin x= cos x for 0≤ x < 2π
.
A.
π 3π 5π 7π
, , ,
4 4 4 4
π π
B. ,
3 6
C.
π 5π
,
4 4
D.
π 3π
0, , π , E. none of these
2 2
63)
64) Use synthetic division to divide 5x 4 – 13x 2 – 6x + 9 by x – 3
65) Factor completely:
A) 10(x + 8)(x – 7) 2 (x – 1) B) (x + 8)(x – 7) 2 (10x – 10)
C) (x + 8)(x – 7) 2 (5x – 5) D) 2(x + 7)(x – 8) 2 (5x – 5)
11

66) Simplify by writing as a single quotient with positive exponents.
67) Express as a single logarithm:
68) Evaluate the infinite geometric sum:
69) Find the equation of the tangent line to the graph of f at the given point.
70) The graph of f(x) is given on the right.
lim f x
Evaluate: ( )
x→3
A. 3 B. 5 C. 4
D. DNE E. none of these
12