TO:
FROM:
SUBJECT:
DATE:
AP Calculus AB Students – 2011-12
Mrs. Pompelia
Summer Assignment
May 23, 2011
Welcome to AP Calculus AB! You are expected to complete the attached homework assignment
during the summer. This is because of class time constraints and the amount of material that must
be covered to properly prepare for the AP Calculus AB exam. I will check the assignment, and we will
spend a few days reviewing the material at the beginning of the school year. We will then have a test
on the material. Refer to the examples on the assignment sheet, your pre-calculus notes, or internet
resources for help with any of the problems.
DUE DATE: No later than 1:30 P.M., Tuesday, August 16, 2011, to the senior high school office.
Expect to work a minimum of about 8 hours on this assignment – plan accordingly!
SUMMER HELP SESSIONS: This assignment is challenging. You are being asked to review all of
the algebra, geometry, and trigonometry skills that you have ever learned – and that’s a lot! Please
don’t get too discouraged if you have some trouble; it is to be expected. I will be glad to help you with
questions during the summer, anytime after June 30th, and not the first week of August. You may
either call or text me at 724-787-3644(C) or e-mail me at cindy.pompelia@glsd.k12.pa.us. I can also
meet with you at the school by appointment.
GRAPHING CALCULATORS:, We are planning to purchase new graphing calculators, the TI-Nspire
CX CAS. The calculators will be available for you to use this fall, similar to how you used graphing
calculators this past year. (If you would like to purchase your own, I would recommend ordering
online from either schoolsavers.com or amazon.com for the best price.) These calculators are very
different from the TI-83+/84’s, primarily because they are symbol manipulators. To assist you with the
summer assignment, you will be allowed to keep your TI-83+/84 or Nspire that was loaned to you this
past year. However, you must turn the loaned calculator in with your summer assignment!
NOTEBOOK: You are required to use a loose leaf three-ring binder for taking notes and completing
assignments for this course as there are several handouts, and I collect your assignments. All
homework sections should be labeled, and the original problem should be written out (except for word
problems). I expect high quality work!
TRIGONOMETRY: Trig functions are used extensively in calculus. You must have memorized exact
values of sin(x), cos(x), and tan(x) for x = 0, π/6, π/4, π/3, π/2, π and 3π/2 radians, and be able to
use these to find values in Quadrants II, III, and IV as well as for evaluating sec, csc, and cot
functions. We will work exclusively in radians in calculus. The only trig identity you must memorize is
sin2 θ + cos2 θ = 1.
This assignment is worth 100 points. I will randomly pick seven problems from the worksheet to grade
on correctness and effort; I will also grade your assignment on overall neatness and thoroughness.
Homework is weighted 25% of your total grade in this course.
Please complete all work on a separate sheet of paper, and include the original problem with your
work. All answers, except graphs and tables, should be boxed or circled. I expect high quality work!
I look forward to working with you next year, and remember ~ “Practice makes perfect.”
Good Luck!
1
Complex Fractions
When simplifying complex fractions, there are different ways to simplify, two of which are shown below:
1. work separately with the numerator and denominator, rewriting each with a common
denominator, and then multiplying the numerator by the reciprocal of the denominator; or
2. multiply the entire complex fraction by a fraction equal to 1 which has a numerator and
denominator composed of the common denominator of all the denominators in the complex
fraction.
Example:
7( x  1)  6
7 x  13
7 x  13 x  1 7 x  13
x 1
x 1


=
5
5
x 1
5
5
x 1
x 1
6
6
7 
7 
x 1 
x  1 x  1  7 x  7  6  7 x  13
5
5
x 1
5
5
x 1
x 1
6
x 1 
5
x 1
7 
1)
OR:
2)
2
3x

x ( x  4)
1
5
x4
OR:
2 3 x

x x4
1
5
x4


2( x  4)  3 x 2
x( x  4)
5( x  4)  1
x4
2 3 x

x x  4 x( x  4)
1
x( x  4)
5
x4
Simplify each of the following.
4
25
2
a
x2
1. a
2.
10
5 a
5
x2


3x 2  2 x  8 x  4
3x 2  2 x  8
3x 2  2 x  8
=

x( x  4) 5 x  21
( x)(5 x  21)
5 x 2  21x
2( x  4)  3 x( x)
5( x)( x  4)  1( x)

12
2x  3
3.
15
5
2x  3
4
2 x  8  3 x 2
5 x 2  20 x  x
x

4. x  1
x

x 1
1
x
1
x

3x 2  2 x  8
5 x 2  21x
2x
3x  4
5.
32
x
3x  4
1
Composite Functions
To evaluate a function for a given value, simply plug the value into the function for x.
Recall: f og (x)  f (g(x)) OR f [g(x)] read “f of g of x” means to plug the inside function (in this case


g(x)) in for x in the outside function (in this case, f(x)).
Example: Given f (x)  2x 2  1 and g(x)  x  4 find f(g(x)).
f ( g ( x))  f ( x  4)
 2( x  4) 2  1
 2( x 2  8 x  16)  1
 2 x 2  16 x  32  1
f ( g ( x))  2 x 2  16 x  33
2
Let f (x)  2x  1 and g(x)  2x 2  1 . Find each.
6. f (2)  ____________
7. g(3)  _____________
9. f  g (2)   __________
10. g  f (m  2)   ___________
8. f (t  1)  __________
g ( x  h)  g ( x )
 ______
11.
h
Let f (x)  x 2 , g(x)  2x  5, and h(x)  x 2  1 . Find each.
12. h  f (2)   _______
13. f  g(x  1)   _______
14. g  h(x 3 )   _______
f (x  h)  f (x)
for the given function f.
h
15. f (x)  5  2x
Find
Intercepts and Points of Intersection
To find the x-intercepts, also referred to as the zeros of the function, let y = 0 in
your equation and solve.
To find the y-intercepts, let x = 0 in your equation and solve.
Example: y  x 2  2x  3
x  int . ( Let y  0)
y  int . ( Let x  0)
0  x2  2x  3
0  ( x  3)( x  1)
x  1 or x  3
x  i ntercepts (1, 0) and (3, 0)
y  02  2(0)  3
y  3
y  intercept (0, 3)
Find the x and y intercepts for each.
16.
y  2x  5
17. y  x 2  x  2
18. y  x 16  x 2
19.
y2  x 3  4x
Use substitution or elimination method to solve the system of equations.
Example:
x 2  y 2  16 x  39  0
x2  y 2  9  0
Elimination Method
Substitution Method
2 x  16 x  30  0
Solve one equation for one variable.
2
x  8 x  15  0
( x  3)( x  5)  0
x  3 and x  5
Plug x = 3 and x  5 into one original
2
32  y 2  9  0
52  y 2  9  0
y 2  x 2  16x  39
x 2  (x 2  16x  39)  9  0 Plug what y 2 is equal
to into second equation.
2x 2  16x  30  0
y  0
16  y
x 2  8x  15  0
y0
y  4
(x  3)(x  5)  0
Points of Intersection (5, 4), (5, 4) and (3, 0) x  3 or x  5
2
(1st equation solved for y)
2
3
(The rest is the same as
previous example)
Find the point(s) of intersection of the graphs for the given equations.
20.
xy8
4x  y  7
21.
y  x 2  3x  4
.
y  5 x  11
22.
y  x5
yx
Interval Notation
23. Complete the table with the appropriate notation or graph.
Solution
Interval Notation
Graph
2  x  4
 1, 7 
8
Solve each equation. State your answer in BOTH interval notation and graphically.
24.
2x  1  0
4  2x  3  4
25.
26.
x x
 5
2 3
Domain and Range
Find the domain and range of each function. Write your answer in INTERVAL notation.
27. f (x)  x 2  5
28. f (x)   x  3
29. f (x)  3sin x 30. f (x) 
2
x 1
31. f ( x) 
4
2x  5
Inverses
To find the inverse of a function, simply switch the x and the y and solve for the new “y” value.
Example:
f (x)  3 x  1
Rewrite f(x) as y
y = 3 x 1
Switch x and y
x = 3 y 1
Solve for your new y
x 3  3 y  1 
Cube both sides
x3  y  1
Simplify
3
y  x3  1
1
Solve for y
f (x)  x  1
3
Rewrite in inverse notation
Find the inverse for each function.
33. f (x) 
32. f (x)  2x  1
4
x2
3
Also, recall that to PROVE one function is an inverse of another function, you need to show that:
f (g(x))  g( f (x))  x
Example:
If: f (x) 
x9
and g(x)  4x  9 show f(x) and g(x) are inverses of each other.
4
 x  9
f (g(x))  4 
9
 4 
 x99
x
g( f (x)) 
4x  9   9
4
4x  9  9

4
4x

4
x
f (g(x))  g( f (x))  x therefore they are inverses
of each other.
Prove f and g are inverses of each other.
34. f (x) 
x3
2
g(x)  3 2x
35. f (x)  9  x 2 , x  0
g(x)  9  x
Equation of a line
Slope intercept form: y  mx  b
Vertical line: x = c (slope is undefined)
Point-slope form: y  y1  m(x  x1 )
Horizontal line: y = c (slope is 0)
36. Use slope-intercept form to find the equation of the line having a slope of 3 and a y-intercept of 5.
37. Determine the equation of a line passing through the point (5, -3) with an undefined slope.
38. Determine the equation of a line passing through the point (-4, 2) with a slope of 0.
39. Use point-slope form to find the equation of the line passing through the point (0, 5) with a slope of 2/3.
40. Find the equation of a line passing through the point (2, 8) and parallel to the line y 
5
x 1.
6
41. Find the equation of a line perpendicular to the y- axis passing through the point (4, 7).
42. Find the equation of a line passing through the points (-3, 6) and (1, 2).
43. Find the equation of a line with an x-intercept (2, 0) and a y-intercept (0, 3).
5
Radian and Degree Measure
Note: In calculus, we always use radians, unless noted otherwise!
 radians
180o
Use
to convert from radians
Use
to convert from degrees
 radians
180o
to degrees.
to radians.
5
6
44. Convert to degrees:
a.
45. Convert to radians:
a. 45o
b.
4
5
c. 2.63 radians
b. 17o
c. 237
Unit Circle
You can determine the sine or cosine of a quadrantal angle by using the unit circle. The x-coordinate of the
circle is the cosine and the y-coordinate is the sine of the angle.
2
Example: sin90o  1
cos

2
0
(0,1)
(1,0)
(-1,0)
(0,-1)
-2
46.
a.) sin180o
b.) cos 270o
c.) sin(90o)
d.) sin 
e.) cos 360o
f .) cos( )
47. Without a calculator, determine the exact value of each expression.
a)
sin 0
g) tan
7
4
b) sin
h) tan

2

6
c) sin
3
4
d) cos 
i) tan
2
3
j) sec

3
e) cos
k) csc

3
5
4
f) cos
l) cot
3
4

2
Trigonometric Equations:
Solve each of the equations for 0  x  2 . Isolate the variable, sketch a reference triangle, find all the
solutions within the given domain, 0  x  2 .
1
1
48. sin x  
49. 2cos x  3 50. sin 2 x 
51. 2cos2 x  1  cos x  0 52. 4 cos2 x  3  0
2
2
6
Inverse Trigonometric Functions:
Recall: Inverse Trig Functions can be written in one of ways:


sin 1 x
arcsin x
Inverse trig functions are defined only in the quadrants as indicated below due to their restricted domains.










cos-1x 















sin-1x
cos-1 x
tan-1 x


sin-1 x
tan-1 x
Example:
Express the value of “y” in radians.
1
Draw a reference triangle.
y  arctan
3
3
2
-1


. So, y = –
so that it falls in the interval from
6
6



y
Answer: y = –
2
2
6
This means the reference angle is 30 or
For each of the following, express the value for “y” in radians.
53. y = arcsin
1
1
(or sin 1 )
2
2
54. y = arctan 1
55. y  arcsin
 3
2
 
56. y  arccos 1
Example: Find the value without a calculator.
5

cos  arctan 

6
61
Draw the reference triangle in the correct quadrant first.
5

Find the missing side using Pythagorean Theorem.
Find the ratio of the cosine of the reference triangle. cos 
7
6
6
61
For each of the following find the value without a calculator.


2
12 
57. tan  arccos 
58. sec  sin 1 
3
13 



12 
59. sin  arctan 
5


7
60. sin  sin 1 
8

Circles and Ellipses (No problems to do; just some information for you.)
r 2  (x  h)2  (y  k)2
(x  h)2 (y  k)2

1
a2
b2
Minor Axis
b
CENTER (h, k)
c
FOCUS (h - c, k)
a
Major Axis
FOCUS (h + c, k)
For a circle centered at the origin, the equation is x 2  y2  r 2 , where r is the radius of the circle.
x 2 y2
For an ellipse centered at the origin, the equation is 2  2  1 , where a is the distance from the center to the
a
b
ellipse along the x-axis and b is the distance from the center to the ellipse along the y-axis. If the larger
number is under the y2 term, the ellipse is elongated along the y-axis. For our purposes in calculus, you will not
need to locate the foci.
61. Take a break – you deserve it!
62. Tell someone in your family that you love them and note their reaction.
Vertical Asymptotes
Determine the vertical asymptotes for the function. Set the denominator equal to zero to find the x-value for
which the function is undefined. If the numerator is not equal to zero at that x-value, that is the vertical
asymptote.
1
2x
x2
63. f (x)  2
64. f ( x )  2
65. f (x)  2
x
x 4
x (1  x)
Horizontal Asymptotes
Determine the horizontal asymptotes using the three cases below.
Case I. Degree of the numerator is less than the degree of the denominator. The asymptote is y = 0.
Case II. Degree of the numerator is the same as the degree of the denominator. The asymptote is the ratio of
the leading coefficients.
8
Case III. Degree of the numerator is greater than the degree of the denominator. There is no horizontal
asymptote. The function increases without bound. (If the degree of the numerator is exactly 1 more than the
degree of the denominator, then there exists a slant asymptote, which is determined by long division.)
Determine all Horizontal Asymptotes.
x 2  2x  1
f (x)  3
66.
x x7
67. f (x) 
5x 3  2x 2  8
4x  3x 3  5
68. f (x) 
Logarithms and Exponentials
y  log b x is equivalent to x  b y
log b mn  log b m  log b n
m
log b  log b m  log b n
n
logb m p  p logb m
Product property:
Quotient property:
Power property:
Property of equality: If log b m  logb n , then m = n
log b n
log a n 
Change of base formula:
log b a
logb 1  0, ln1=0, logb a  1, ln e  1
Because logarithms and exponentials are inverse functions of each other:
log b (b x )  x, ln(e x )  x, b logb x  x, eln x  x
69. Solve each exponential or logarithmic equation.
a) 5  125 b) 8
x
2
3
x 1
 16
x
3
4
c) 81  x
1
 2
9
g ) log 4 x  3 h) log 3 ( x  7)  log 3 (2 x  1) i) log x  log( x  3)  1
d) 8
x
e) log 2 32  x
f ) log x
70. Expand each of the following using the properties of logs.
a) log3 5 x 2
b) ln
5x
y2
71. Evaluate the following expressions.
a) e ln 3
b) e (1 ln x )
c) ln 1
d) ln e 7
e) log 3 (1 / 3)
Series
72. Expand and evaluate.
4
n2
a)

n 0 2
3
b)
1
n
n 1
3
9
f) log 1 / 2 8
g) e 3 ln x
4x 5
x2  7
Miscellaneous
73. Simplify. Show the work that leads to your answer.
a)
74.
x4
2
x  3x  4
5 x
b) 2
x  25
f)
1
1

(5a 2 / 3 )( 4a 3 / 2 ) g)
xh x
j)
x xx
3
1
6
h)
2
x2
10
x5
sin 2 x  cos 2 x
k)
cot x
d)
x
x
1
1

i) 3  x 3
x
4 xy 2
e)

1
12 x 3 y 5
j)
(3 x) 2  (3  x) 2
x
l) cot x sec x
Solve for z .
a) 4 x  10 yz  0
75. Expand:
76.
c)
x2  2 x  8
x3  x 2  2 x
b) y 2  3 yz  4 x  8z
( x  y) 3
Solve for x. Show the work that leads to your solution.
x4 1
2
x

3
x

4

14
a)
b)
c) ( x  5) 2  9
0
3
x
e) x 2  2 x  15  0
f) 12 x 2  3x
g)
x 3  7
d) 2 x  1 
5
x2
h) 27 2 x  9 x 3
77. Rationalize the denominator.
(a)
78. Factor:
2
3 2
(b)
1 x
x 1
x 2 ( x  1)  4( x  1)
79. Water is flowing at a rate of 10 m3/s into the cylindrical tank shown below.
a) Find the volume V of the water as a function of the water level h.
b) Find h as a function of the time t during which water has been flowing into the tank.
80. The length of a rectangle is equal to three times the square root of its width. Express the perimeter and area of the
rectangle as a function of its width.
10
.
Graphing
81. Print the remaining pages and attach to your work for the previous problems.
Graph the following functions.
Also, know how to sketch vertical and horizontal translations of theses graphs, such as y = |x+2| - 3.
a)
yx
c)
y
e)
y  sin x
x
b)
y  x2
d)
y x
f)
y  cos x
11
g)
y  tan x
i)
yx
k)
y
h)
1
3
1
x
y  x3
j)
l)
x2  y 2  6
f ( x)   x (greatest integer function)
12
m)
o)
q)
n) y  e x
y  ln x
 2x  3 x  1
y
 x  4 x  1
f ( x) 
x2  4
x2
p)
 4 x

f ( x)   2  3x
 x2

r)
f ( x) 
x0
0 x2
x2
x2
x2  4
Congratulations! You have finished your calculus summer packet. Please use the space below if you would like to
make some comments to your calculus teacher concerning the assignment.
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