Download Section IV: Logarithm Review

AP Calculus AB Summer Assignment
Name:_______________
Welcome to AP Calculus!! You have finally reached the pinnacle of high school mathematics.
It will be a challenging time, but one that will pay off if you keep your nose to the grindstone.
While you are enjoying your summer, I would like to ask you to take time to complete the
attached assignments. They are designed to help you make the transition into this challenging
course as smooth as possible. One thing is for sure – the more you do over the summer, the easier
it will be when school starts and the more comfortable you will feel with the pace of the class.
If you need to, you may use reference materials to assist you and refresh your memory (old
notes, textbooks, online resources, etc.). While the graphing calculators will be used in class for
some things, there are no calculators allowed on this packet. You should be able to do everything
without a calculator.
Your summer assignment consists of problems that cover the following 4 areas:
I. Algebra
II. Logarithms
III. Trigonometry
IV. Graphing
It is very important that you complete the summer work and have a firm understanding of
the prerequisite skills, as we will be building on these concepts throughout the course. This
assignment is DUE ON THE FIRST DAY OF SCHOOL. All work must be neatly shown for each
problem. Use graph paper for the problems that require you to graph.
*There will be a test over the summer packet material on the second ‘normal’ day of school.
IMPORTANT NOTE: NO RETAKES!!!! This means if you fail this test your grade in AP
Calculus is in serious jeopardy! It matters A LOT!
Give us your best work while giving yourself the opportunity to get off to a great start!
Name:
Show all work – no credit will be awarded for answers missing appropriate work.
No calculators!
Section II: Algebra Review
Identify the following statements as true or false.
x y x y
1
1 1
1.
  _____
2.
  _____
2
2 2
pq p q
4. 3 
a 3a

_____
b b
5. 3 
a  b 3a  b

_____
c
c
3.
2k
k

_____
2x  h x  h
6. a 2  b 2  a  b _____
Identify the following statements as true or false over the set of real numbers. Give a
counter example for any false statement.
7. x 3  1  x 3 _____
8. x 3  x  x 3 _____
9. x 2  0 _____
10. x 2  x _____
11. 2 x  x _____
13.  x  0 _____
14.
16. Solve xy'  y  1  y ' for y ' .
1
 x _____
x
12.
x  0 _____
15. x  x _____
17.Solve ln y  kt for y .
16._____________
17._____________
18. Factor: y 3  27
19. Factor: x 2 ( x  1)  4( x  1)
18._____________
19._____________
Simplify each expression.
x 
20.
2 3
x
x7
1
21. x  3 x  x 6 ___________________
_________________
5  x  h   5x 3
22.
_________________
h
3  x  h   3x 2
23.
_________________
h
x2  1
24. x __________________
x 1
x3
1 4
 2
25. x x _______________________
1
3
x
3
2
Simplify by rationalizing the numerator.
Example:
x4 2

x
26.
x4 2
x42
x44
x




x
x42 x x42 x x42
x9 3
_____________________
x

 
27.

1
x4 2
xh  x
________________________
h
Solve each equation or inequality for x over the set of real numbers.
28. 2 x 4  3x 3  2 x 2  0 _____________________
29.
2x  7
2x

_________________________
x 1
x4
3x  5
 0 _____________________
( x  1)( x 4  7)
31.
x 2  9  x  1 _________________________
30.
32. 2 x  3  14 ___________________________
33. x 2  2 x  8  0 _________________________
Section III: Trigonometry Review
34. You must have the first quadrant of the unit circle (all 6 functions and arc-functions)
memorized to the point where you can answer any question within 10 seconds. This part of
the test will be a slide show with 10 seconds per slide. I suggest creating flash cards and
using them MANY times over several weeks (this is harder to master than you think).
examples
 3
 
 
a . tan    ____________ b. arcsin 
  ____________ c. csc    ____________
6
4
 2 
Complete each of the following using trigonometric identities and formulas.
35. sin 2 x  cos2 x  _________
36. 1  cot 2 x  ____________
Solve each trigonometric equation for 0  x  2 .
3
38. sin x 
____________
2
40. cos
x
2

____________
2
2
37. 1  tan 2 x  ____________
39. tan 2 x  1 ___________
41. 2sin 2 x  sin x  1  0 ____________
Section IV: Logarithm Review
Solve each exponential or logarithmic equation.
3
43. 8x 1  16x __________
44. 814  x _________
45. 8 3  x __________
46. log 2 32  x __________
47. log x
48. log 4 x  3 __________
49. log 3 ( x  7)  log 3 (2 x  1) ____________
42. 5x  125 __________
2
1
 2 __________
9
Expand each of the following using the laws of logs.
50. log 3 5 x 2 ________________________
51. ln
5x
_______________________
y2
Section V: Graphing Review
This section will also be timed to 10 seconds per problem. You must be able to go both
ways (draw a rough graph given the function and write the function when shown a rough
graph).
Sketch the following functions. Draw and label your own axes.
52. f ( x )  x
53. f ( x )  x 2
54. f ( x )  x 3
55. f ( x)  x
56. f ( x)  x
57. f ( x ) 
1
x
1
x2
59. f ( x ) 
1
1  x2
58. f ( x ) 
60. f ( x )  x
61. f ( x )  e x
62. f ( x )  ln x
63. f ( x )  1  x 2
64. f ( x )  sin x
65. f ( x )  cos x
66. f ( x )  csc x
67. f ( x )  sec x
68. f ( x )  tan x
69. f ( x )  cot x