Welcome to AP Calculus AB!
Erie High School
Anna Panakhyo
[email protected]
Introduction to AP Calculus:
AP Calculus is a rigorous course in which we will be covering all topics included on the AP Calculus AB
Course Exam (more information at https://apstudent.collegeboard.org/apcourse/ap-calculus-ab). The
major textbook is the 10th edition of Larson Calculus of a Single Variable AP Edition. My goal is to
prepare you to do well on the AP exam, but also to become better problem solvers, critical thinkers
and further develop your study skills. All of these are crucial for the transition into college.
All students are expected to take the AP exam at the end of the year. The cost was $93.00 for the
2016 exam. The AP exams are graded on a scale from 1 to 5. A score of 3 (qualified) may or may not
earn you credit at colleges you are interested in. You will need to check with your particular college of
interest to see what score you will need to be given credit. Some colleges and programs may require
you to re-take Calculus 1 at their school. The exposure here will be helpful, but you never know what
topic your college/school will want you to know.
We will be using graphing calculators in this course and you can use graphing calculators on part of
the AP Exam. The calculator I use and recommend is a TI-84. You are expected to bring your
calculator everyday to class whether we use it that day or not. You are expected to know how to use
certain functions of the graphing calculator including finding the intersection of two functions, finding
the maximum and minimum of a function, using SOLVER to solve equations, fitting a window of a
graph, changing graphing settings, and adjusting a table. If you need help using or learning these
functions I will be available Thursday, May 19 after school for a graphing calculator training. I highly
suggest anyone who does not know how to use these functions of the graphing calculator come to the
calculator training as there is very little time in class to teach these graphing calculator functions.
Additionally, I suggest you get an AP Calculus review book to supplement the practice in class and
from our textbook. On Schoology (schoology.svvsd.org), I will have a list of books that I suggest. This is
not a requirement, but HIGHLY recommended.
Schoology (schoology.svvsd.org) will be used for the class webiste. On Schoology, I will post calendars,
homework assignments, study aides, online resources and important information. Class notes will be
posted on the website and students will be expected to print notes before class. The website is
schoology.svvsd.org.
Summer Homework:
In order to get through all of the material by May, you have been assigned a summer assignment. In
this assignment you will be reviewing concepts from previous math courses, practicing graphing using
your calculator, and introducing yourself to the course material.
This summer assignment will count for a major portion of your semester homework grade; therefore,
this is not an optional assignment. Your summer assignment will be due the first day of class and we
will then take a quiz over the material. Students who do not, or cannot complete this assignment
should definitely ask themselves if they should be taking AP Calculus.
I suggested you complete the assignment in parts over the summer to avoid cramming the entire
assignment until August. All of these problems were chosen to help students review and prepare over
the summer, as well as familiarize you with the AP exam. Students are encouraged to work together,
1
but any final work must be that of the individual student. You are encouraged to review and use your
notes from previous math courses to help with the assignment.
Legibility and understandability are important. If your work is illegible, I cannot find your answer, or
there is not support (work) for your answer, you will receive not credit for the problem. You can use
your calculator to complete the summer homework, but please keep in mind that more than half of the
AP exam is non-calculator. Therefore, do not rely too much on your calculator.
Schoology:
You will be added to a Schoology (schoology.svvsd.org) course for the summer. Most likely this will be
a different Schoology course than we will use during the school year. I will use this Schoology site to
share important information and resources throughout the summer. There will also be a discussion
board where you can post questions for your peers and the teacher throughout the summer. It is an
expectation that you check Schoology on a regular basis over the summer.
Calculus Readiness Test:
Please use the Fresno State University Calculus Readiness Test
(http://www.fresnostate.edu/csm/math/students/placement-exams/calc-readiness.html) to prepare
for the upcoming year of calculus. There are two practice test available, and you will get immediate
feedback. The problems on this readiness test are similar to the summer homework and first day quiz.
Calc Camp:
Calc Camp will be August 1-3 in T109 from 1-3 pm. On August 1, we will review sections A-H and
calculator tips (section P). On August 2, we will review sections I-O and limits. On August 3, we will
review limits and derivatives. It is highly suggested that you attend these sessions. These are not
required, but highly suggested. Please note, I will not be teaching the concepts in this summer
homework, but reviewing the concepts. We will also begin discussions on limits and derivatives.
On August 4, the session will be for Calculus BC students on parametric equations and other concepts
needed for Calculus BC.
Other Online Resources:
www.purplemath.com
www.kahnacademy.com
http://tutorial.math.lamar.edu/Classes/CalcI/Cal
cI.aspx
http://www.mastermathmentor.com/calc/RURea
dy.ashx
https://www.wolframalpha.com/
If you have any questions, you can contact me over the summer at [email protected]. I will
be out of town part of the summer, but generally check this e-mail every 2-3 days. Please send me a
brief e-mail introducing yourself and so I have a way of contacting you over the summer.
I am looking forward to an exciting year in AP Calculus AB next year and getting to know all of you!
Have a good summer and see you in the fall,
Anna Panakhyo
2
AP Calculus Summer Homework
Due: First Day of Class
-
There will be a quiz over this material the first day of class.
You are encouraged to work with other students, but each individual student must have their own work.
It is suggested that you work on this assignment in parts over the summer. It is NOT suggested to wait until
August to complete this assignment.
You may complete all your work on this paper. Show ALL work! Box all answers.
A graphing calculator is required for some questions.
Round answers to the nearest thousandth.
A. Solve Linear Equations
- Can you solve an equation requiring distribution, combining like terms, and clearing the fraction?
- Can you solve for a variable in a literal equation?
1. Solve for x.
!
"
+
$
3. Solve for r.
𝐴 = 2𝜋𝑟 $
4𝑥 + 9 = 0
%
4. Solve for d.
3𝑦 $ 𝑑 + 2𝑦𝑑 = 5𝑑 + 2𝑥
2. Solve for y.
"
+
+
,
+-$
=
%
$+
B. Equation of a Line
- Can you write the equation of a line in point-slope, standard, and slope-intercept form?
- Can you determine the slope of a line given two points, a function, or a graph?
- Can you determine the intercepts of a line?
5. For the points (3, -4) and (-5, 2), write the equation of a line passing through the two points in point-slope and
slope-intercept form.
6. Write the equation of a line parallel to 2𝑥 − 3𝑦 = 4 and passing through (8, 1) in point-slope form.
!
7. Write the equation of a line perpendicular to 𝑦 = − 𝑥 + 2 and passing through (-2, 3) in point-slope form.
%
3
8. What are the x- and y- intercepts of a line passing through (4, 5) and (3, 8)?
9. Given 𝑓 𝑥 = 𝑥 $ − 2, find the slope of a line passing 𝑓(2) and 𝑓(7).
C. System of Equations and Inequalities
- Can you identify the solution of a system of equations or a system of inequalities from a graph?
- Can you solve a system of equations by graphing, substitution, and elimination?
10. Solve for x and y.
9𝑥 + 𝑦 = −62
4𝑥 + 5𝑦 = −23
11. Solve for x and y.
4𝑦 − 3𝑥 = 19
2𝑦 = 5𝑥 + 1
12. Shade the solution to the system of equations on a
3𝑥 − 𝑦 − 7 < 0
coordinate plane.
𝑥 + 5𝑦 + 10 ≥ 0
4
D. Piecewise Functions
- Can you graph a system of equation?
- Can you evaluate a system of equations?
13. Graph 𝑦 =
4𝑖𝑓𝑥 < −4
𝑥 𝑖𝑓 − 4 ≤ 𝑥 ≤ 4
4𝑖𝑓𝑥 > 4
14. Graph 𝑦 =
−5𝑥 + 2𝑖𝑓𝑥 < 2
2𝑥 − 2𝑖𝑓𝑥 ≥ 2
15. Graph 𝑦 =
−𝑥 $ + 5𝑖𝑓𝑥 ≠ 2
−2𝑖𝑓𝑥 = 2
5
𝑥 $ − 3𝑥 + 7,𝑥 < −4
16. Evaluate ℎ 𝑥 = 2𝑥 − 7,𝑥 = 7
8,𝑥 ≠ 7𝑎𝑛𝑑𝑥 ≥ −4
a) ℎ 7
b) ℎ 0
c) ℎ −4
d) ℎ −5
E. Domain and Range
- Can you determine the domain and range given the function or graph?
17. 𝑓 𝑥 = 4 − 𝑥
Domain:
Range:
18. 𝑓 𝑥 = 7
Domain:
Range:
19. 𝑓 𝑥 =
23. 𝑓 𝑥 = 𝑥 − 4
Domain:
Range:
JKL%
$K-!
Domain:
Range:
20. 𝑓 𝑥 =
!
K M LNKLO
Domain:
Range:
21. 𝑓 𝑥 =
NK
K K M L"O
Domain:
Range:
27. Find the domain and range for the following
piecewise function.
Domain:
Range:
22. 𝑓 𝑥 = 2 K − 3
Domain:
Range:
24. 𝑓 𝑥 = sin 2𝑥
Domain:
Range:
25. 𝑓 𝑥 = ln(𝑥 + 1)
Domain:
Range:
26. 𝑓 𝑥 =
$
K-"
Domain:
Range:
28. Find the domain and range of for the following
piecewise function.
Domain:
Range:
6
F. Function Behavior
- Can you identify when a function is increasing and decreasing?
- Can you find the zeroes of a function?
- Can you identify points of discontinuity and name the types of discontinuity?
29.
-
Sketch a function with the following characteristics
A vertical asymptote at 𝑥 = 3
A jump at 𝑥 = −1
Passes through 0, 2
Decreasing on the interval (−∞, −1)
30. Sketch a function with the following characteristics
- A horizontal asymptote at 𝑦 = −4
- A removable discontinuity at 𝑥 = 2
31.
-
Sketch a function with the following characteristics
Domain is [-5, 6]
Range is [-8, 2]
The function is increasing on (-4, 0) only
f(0) = 2
7
G. Factoring
- Can you factor a quadratic?
- Can you factor a polynomial completely?
32. Factor completely: 4𝑥 $ − 28𝑥 − 32
35. Factor completely: 64𝑥 $ − 121
33. Factor completely: 𝑥 , − 16𝑥 "
36. Factor completely: 𝑥 " − 1
34. Factor completely: 4𝑥 % − 8𝑥 $ − 25𝑥 + 50
H. Quadratics
- Can you solve a quadratic equation be factoring, square roots (if possible), completing the square, and the
quadratic formula?
- Can you graph quadratic functions?
- Can you determine the domain and range of quadratic functions?
37. Solve by factoring.
5𝑥 $ + 14𝑥 + 14 = 2 − 2𝑥
For #41-43: Solve using square roots (if possible), completing the square, or quadratic formula.
38. Solve 3𝑥 $ + 18𝑥 − 7 = 0
39. Solve 121𝑥 $ = 49
40. Solve 𝑥 $ + 24𝑥 = −8
8
41. Consider the parent function: 𝑦 = 𝑥 $ . Graph the parent function
along with any necessary asymptotes in BLACK. Identify the
domain and range of the parent function. Then on the same
coordinate plane, graph the following translations and identify the
domain and range of each translation.
In BLACK: 𝑦 = 𝑥 $
Domain:
Range:
$
In RED: 𝑦 = 𝑥 − 4
Domain:
Range:
$
- In BLUE: 𝑦 = 𝑥 − 3
Domain:
Range:
$
- In GREEN: 𝑦 = −𝑥 + 5 Domain:
Range:
$
- In ORANGE: 𝑦 = 2𝑥
Domain:
Range:
I.
Rational Expressions and Equations
- Can you simplify a rational expression?
- Can you add/subtract/multiply/divide rational expressions?
- Can you graph rational functions?
- Can you solve rational equations?
42. Simplify the following completely. State any restrictions on the domain.
K M L"
K M -JK-,
43. Simplify the following completely. State any restrictions on the domain.
%K U L%K
K M L"KLJ
44. Simplify the following completely. State any restrictions on the domain.
V V
–
W Y
M
WY
45. Simplify the following completely. State any restrictions on the domain.
"K+
V
UZ
46. Simplify the following completely. State any restrictions on the domain.
JL
V
MZ
U
Z[M
47. Simplify the following completely. Leave answer in factored form. State any restrictions on the domain.
$
K-!
−
%
K
9
48. Solve the following. Check for extraneous solutions.
"
K-$
+
%
=
KL$
!
K M L"
49. Solve the following. Check for extraneous solutions.
2𝑥 + 1 =
J
K-$
!
50. Consider the parent function: 𝑦 = . Graph the parent
K
function along with any necessary asymptotes in BLACK.
Identify the domain and range of the parent function. Then
on the same coordinate plane, graph the following
translations and identify the domain and range of each
translation.
In BLACK: 𝑦 =
!
!
K
In RED: 𝑦 = − 4
K
In BLUE: 𝑦 =
!
KL%
In GREEN: 𝑦 = −
In ORANGE: 𝑦 =
!
K-J
$
K
Domain:
Range:
Domain:
Range:
Domain:
Range:
Domain:
Range:
Domain:
Range:
J. Exponents
- Can you simplify exponents?
-
\
Can you write exponents in the form 𝑥 ] in radical form?
For #51-52: Simplify the exponent expressions so there are no negative or rational exponents.
51. 3𝑥 $ 𝑦 L"
%
52. 3𝑥 2𝑦 $ 4𝑥 L% (5𝑦 ^ )
For #53-54: Re-write the exponent expressions in radical form.
V
53. 4𝑥 M 54. 5𝑦
L
U
M
For #55-56: Re-write the expression with rational exponents.
55.
$
K-%
56.
U
!
K
10
K. Exponential and Logarithmic Functions
- Can you graph exponential functions?
- Can you identify the domain and range of exponential and logarithmic functions?
- Can you simplify logarithms?
- Can you solve logarithmic and exponential equations (we will mainly work in base e)?
57. Use properties of logarithms to condense: 2 ln(𝑥 − 3) + ln 𝑥 + 2 − 6 ln 𝑥
58. Express y in terms of x.
ln 𝑦 = 𝑥 + 2
59. Solve for x.
ln 𝑒 % = 𝑥
60. Solve for x.
ln 6 + ln 𝑥 = 3
61. Consider the parent function: 𝑦 = 𝑒 K . Graph the parent
function along with any necessary asymptotes in BLACK.
Identify the domain and range of the parent function. Then
on the same coordinate plane, graph the following
translations and identify the domain and range of each
translations.
In BLACK: 𝑦 = 𝑒 K
Domain:
Range:
K
In RED: 𝑦 = 𝑒 − 4
Domain:
Range:
KL%
In BLUE: 𝑦 = 𝑒
Domain:
Range:
In GREEN: 𝑦 = −𝑒 K Domain:
Range:
K
In ORANGE: 𝑦 = 2𝑒 Domain:
Range:
L. Trigonometry
- Do you know the values of sin, cos, and tan for the unit circle?
- Can you graph the sin, cos, and tan graph and translations?
- Can you graph the sinL! 𝑥 , cos L! 𝑥 , 𝑎𝑛𝑑 tanL! 𝑥 graphs and their translations?
- Can you complete proofs involving trigonometric identities?
11
62. Complete the following chart for the trigonometric values. Convert each of the degree measures to radians. Leave
answers as radicals in simplest form (no decimals!). You MUST know these values for Calculus.
Degrees
Radians
sin
cos
tan
0
0
30
𝜋
6
45
60
90
𝜋
2
1
0
undefined
120
135
150
180
210
225
240
270
300
5𝜋
3
−
3
2
315
330
360
12
#61-67 Should be completed WITHOUT a calculator. You MUST know the graphs of #61-63 for Calculus.
63. Graph 𝑦 = tan 𝑥 and 𝑦 = cot 𝑥 on the interval
−2𝜋 ≤ 𝑥 ≤ 2𝜋 on the same graph. Include all
asymptotes. State the domain and range of each.
64. Graph 𝑦 = cos 𝑥 and 𝑦 = sec 𝑥 on the interval
−2𝜋 ≤ 𝑥 ≤ 2𝜋 on the same graph. Include all
asymptotes. State the domain and range of each.
65. Graph 𝑦 = sin 𝑥 and 𝑦 = csc 𝑥 on the interval
−2𝜋 ≤ 𝑥 ≤ 2𝜋 on the same graph. Include all
asymptotes. State the domain and range of each.
13
66. Consider 𝑦 = 2sin(x)
a) Graph
b) What is the domain?
c) What is the range?
67. Consider 𝑦 = sin x + 3
a) Graph
b) What is the domain?
c) What is the range?
68. Consider 𝑦 = cos x − 2
a) Graph
b) What is the domain?
c) What is the range?
14
69. Consider 𝑦 = cos(𝑥 + 𝜋)
a) Graph
b) What is the domain?
c) What is the range?
70. A kite is 100 ft above the ground. If there are 200 ft of string out, what is the angle between the string and the
horizontal? (Assume that the string is perfectly straight).
71. Verify that sin 𝑥 − cos 𝑥
$
= 1 − 2 sin 𝑥 cos 𝑥 is an identity.
72. Verify that cos $ 𝑥 = sec $ 𝑥 − tan$ 𝑥 − sin$ 𝑥is an identity.
73. Verify that cot 𝑥 =
fgf K
ghf K
is an identity.
M. Geometry
- Do you know the formulas for the following: circle, square, parallelogram, trapezoid?
- Do you know the volume formulas for the following: cone, cylinder, prism, pyramid?
There are no problems for this section. Please review the listed formulas.
15
N. Function Operations and Inverses
- Can you add/subtract/multiple/divide functions?
- Can you compose functions?
i K-j Li(K)
-
Can you simplify
-
Can you find the inverse of a function?
Can you evaluate function notation?
74. Simplify
75. Simplify
j
i K-j Li(K)
j
i K-j Li(K)
j
?
, where 𝑓 𝑥 = 2𝑥 + 3.
, where 𝑓 𝑥 = 𝑥 $
!
For #76-80: Perform the following operations given 𝑎 𝑥 = 3𝑥 $ − 1, 𝑏 𝑥 = 2𝑥 + 7, 𝑐 𝑥 = . State the domain and
K
range.
76. 𝑎 + 𝑏 𝑥 79. 𝑏 𝑐 𝑥
77. 𝑎 𝑥 𝑏 𝑥
80.
m(K)
n(K)
78. 𝑐 𝑏 𝑥 81. Find the inverse of 𝑓 𝑥 = 3𝑥 − 4
82. Graph 𝑓 𝑥 = 3𝑥 − 4 and its inverse (#81)
16
83. Consider functions f and g shown on the graph below. Compute the following.
a) 𝑓 4 + 𝑔(0)
b) 𝑓 𝑔 −2 c) 𝑓 3 − 𝑔 −4
d) 𝑔 0 + 3
e) Sketch the graph of 𝑔 𝑥 + 3 on the graph above in GREEN.
f) Sketch the graph of 𝑓(𝑥 + 2) on the graph above in ORANGE.
O. Derivatives
- Can you find the derivative of functions using the power rule, quotient rule, and product rule?
- Can you find the derivate of trigonometric functions?
84. Find the derivative of the following functions.
a) 𝑓 𝑥 = 3𝑥 $ + 7 𝑥 $ − 2𝑥 + 3 b) 𝑔 𝑥 =
$
%K M
c) ℎ 𝑥 = 2𝜃 − 3𝑠𝑖𝑛𝜃
d) 𝑘 𝑥 = 𝑥 sin 𝑥
e) 𝑚 𝑥 = K M -KL!
K M L!
f) 𝑐 𝑥 = 𝑥 % − 3𝑥 $ U
g) 𝑑 𝑥 = 6 𝑥 + 3 𝑥 17
P. Calculator
- Can you use SOLVER on your graphing calculator to find the zeroes of a function?
- Can you use your graphing calculator to find the zeroes of a function?
- Can you use your graphing calculator to find the intersection of two function?
- Can you use your graphing calculator to find the minimum/maximum of a function?
- Can you use the table function of your graphing calculator?
- Can you set an appropriate viewing window for a graph?
85. For the function 𝑓(𝑥) = 𝑥 % − 0.5𝑥 $ − 10𝑥 + 2 (Use the graphing features of your calculator to complete these
questions)
a) Sketch the graph
b) What is an appropriate viewing window for this function?
c) Determine the relative minimum(s). (Round to the nearest
thousandth, if necessary)
d) Determine the relative maximum(s). (Round to the
nearest thousandth, if necessary)
e) Determine the interval(s) when the function is increasing.
f) Determine the interval(s) when the function is decreasing.
g) Graph 𝑔 𝑥 = 𝑥 $ − 6 and 𝑓(𝑥) = 𝑥 % − 0.5𝑥 $ − 10𝑥 + 2. How many times do the functions intersect? What
are the points of intersection?
86. Find all real roots to the nearest 0.001 using the SOLVER function of the graphing calculator. The following
websites can help you understand the SOLVER function of the graphing calculator:
http://www2.ohlone.edu/people2/joconnell/camt/ti/8384solver/
http://www.youtube.com/watch?v=6fHKfcR51X8
a) 𝑓 𝑥 = 𝑥 " − 3𝑥 % + 2𝑥 $ − 7𝑥 − 11
b) 𝑓 𝑥 = 3 sin 2𝑥 − 4𝑥 + 1𝑓𝑟𝑜𝑚[−2𝜋, 2𝜋] (in radian mode)
c) 𝑓 𝑥 = 𝑥 − 3 + 𝑥 − 6
18