AP Calculus
Summer Review Packets A & B
Instructions: Do not write your answers on this packet. Print it out, however, and turn it in with your work.
Packet A: The answers to Packet A will be posted on the website in mid-August but you still must show the work required to answer the problems. On
packet A, if a problem can be done in your head, you may simply write down the answer. But if you had to do ANY work, write it down. This packet
does not have to be “tidy.” It is intended to warm up your thinking after a summer break, and identify areas in which you need help.
Packet B: Show all work and your solutions on your own paper, then staple your work to the back of the assigned problems. I want to see how you
organize your work and your thinking. I also want to see clear handwriting that is large enough to read (it matters on AP exams!). Math and science use
a lot of paper; it makes me feel badly, but ya gotta do it anyway.
BOTH Packets: If you need help on some problems, please highlight or clearly make a note so that I can help you! Yes, the packets will be graded, but
do your own work. We want to identify your weak areas so that I can help you... not your friends weaknesses that you copied. I can help them
separately! :) Both packets are due the first day of class in September. We will be taking an exam on this material on the 5th day of class.
Summer Review Packet Contents. Problems include, but are not limited to:
I.
Algebra Skills
A.
Notations: ( ), { }, [ ], ∆, f ⁰ g |
B.
Strong multi-step skills (solving radicals, power functions, quadratics, etc.)
C.
Factoring, Completing the Square (1, 2 and 3 Dimensional)
D.
Long and synthetic Division of Polynomials
E.
Horizontal and Vertical Line Tests
F.
Finding inverse functions
G.
Re-expression (logarithmic, or substitution)
H.
Conversion between polar, rectangular coordinates and equations
II.
Lines
A.
B.
C.
D.
Point Slope Form
Slope-Intercept Form
Parallel and Perpendicular Slopes
TI-83/84/89 Regressions, Scatter Plots, Line Plots, Zoom, Window Settings
III.
Functions and Graphs
A.
Literal Equations
B.
Domain, Ranges (with, without calculators)
C.
Odd/Even Symmetry
D.
Piecewise: graphing and writing equations from graphs
E.
Parent Functions and Transformations (Know your parent functions!)
Polynomials (linear through nth degree)
Exponential & Logarithmic
Trig/Inverse Trig
Periodic
Rational
Piecewise
F.
Discontinuities
G.
Asymptotes and Limits (horizontal, vertical, end behavior)
IV.
Exponential & Logarithmic Functions
A.
Negative Exponents
B.
Solving Exponential & Log Equations with/out a calculator
C.
Converting between Exp & Logarithmic Form
D.
Natural Logs
E.
Log Properties
F.
TI-83/84: All of the above, with/out calculators
G.
TI-83/84: Regressions, graphing
V.
Trigonometric Functions
A.
Convert between radians and degrees (get comfortable with radians!)
B.
Finding Trig functions from given information
C.
Inverse Trig Functions
D.
Convert between coordinate and polar forms
E.
Trig Identities
VI.
Limits
A.
B.
C.
D.
Limits of function as x  ±∞
Left-handed and Right-handed (with/out calculators!)
Asymptotes (with/without calculators!): Horizontal, Vertical, and End-behavior
TI-83/84: All of the above with and without graphers
Summer Review Packet
Basic Functions:
Multiple Choice Questions: NO Calculator should be used for these question! If you cannot answer them without a
calculator, you need math help... ask a friend, refer to a text or online resource before doing Packet B.
Free Response: A Calculator is OK or required for these questions, unless otherwise noted.
𝑎𝑥
15) The rational function y =
has a vertical asymptote at x = 2 and a horizontal asymptote at y = 3.
𝑏𝑥+𝑐
Find a and c in terms of b. Then, express y in terms of ONLY x.
16) Factor the trig equation 4 sin2 (x) – cos(x) = 1 in terms of cos(x) ONLY. the trig equation 4 sin2 (x) – cos(x) =
1 in terms of cos(x) ONLY. Then solve for cos(x). No calculator (notice we're not trying to find 'x').
Lines:
Write an equation through the specified line, in point-slope form:
17) Through (1, -6) with slope 3
18) Through (3,1) and parallel to 2x – y = -2
19) Through (-2,-3) and perpendicular to 3x – 5y = 1
Write an equation through the specified line, in slope-intercept form:
20) With x-intercept 3 and y-intercept -5
Write an equation through the specified line:
21) The vertical line through (0, -3)
22) The horizontal line through (0, 2)
Functions & Properties (even/odd/composite/VA/HA)
Determine whether the graph of the function is even, odd, both, or neither. Support your work both algebraically AND
graphically.
23) Y = x2 + 1
24) Y = 1 – cos(x)
25) Y = sin(x)
26) Y = √𝑥 4 − 1
Find the domain and range of each function. Try without a calculator first, then verify with your calculator.
27) Y = |x| - 2
28) Y = √16 − 𝑥 2
29) Y = ln(x – 3) + 1
−𝑥
30) Y = { √
√𝑥
−4 ≤ 𝑥 ≤ 0
0 ≤ 𝑥 ≤ 4
1
Composites: Using f(x) = 2 – x and g(x) = Find the following:
𝑥
31) f(g(x))
32) g(f(2))
33) (g ͦ g) (x)
Find the inverse of the following:
34) The curve: F(x) = 2 – 3x
35) The curve: G(x) = (x + 2) 3
36) The point: (5, 7)
Piecewise Functions:
37) Write a piecewise formula for the function graphed here:
Trig Functions
38) Find the measure of sin –1 (0.6) in radians AND in degrees
39) Find the six basic trig values of θ = cos–1 (3/7). Give EXACT answers.
40) Solve sin x = -0.2 on the interval [0, 2𝜋 ]
Transformations. The graph of F(x) is shown. Draw the graph of each transformation in Questions #41-43
41) F(-x)
42) -2 F(x + 1) + 1
43) –F(x)
Logs and Exponents
44) Solve without a calculator: log 7 49
45) Solve without a calculator:
5
3
log 2 8
46) Solve with a calculator: log 6 15
47) Solve without a calculator: 𝑒 3𝑥 = 1
48) Solve without a calculator: 6 ln 𝑒 4𝑥 = 18
Graphing Calculator Skills
-3
-130
-2.4
-75
-0.8
-3
1.5
16
3.5
208
Use the data set to find:
49) Quadratic regression & squared correlation coefficient
50) Cubic regression & squared correlation coefficient
51) Now, use the best fitting curve to estimate the y value when x = 1
52) Now, use the best fitting curve to estimate the x value when y = 1
Limits and Asymptotes
Using the function F(x) =
53) What is the
100
𝑥2 − 5
lim 𝐹(𝑥) ?
𝑥 →∞
54) Identify any vertical asymptotes of F(x) if they exist.
55) Identify any horizontal asymptotes of F(x) if they exist.
56) What is the
57) What is the
lim
30 𝑥
𝑥 →∞ 𝑥 − 5
lim
𝑥2
𝑥 →∞ 2𝑥+1
?
?
Summer Review Packet B
Name: _____________________________
Math Skills:
3𝑥 − 1
1) Find f(2) for y = { 2
𝑥 −1
𝑥<2
𝑥 ≥2
2) Write the inequality in the form a < x < b:
|x| < 4
3) Find the quotient, q(x), and the remainder, r(x) when f(x) is divided by g(x) and
f(x) = 2x5 – x3 + x – 1 and g(x) = x3 – x2 + 1
4) Write a formula for f(
1
𝑥
) for the function: f(t) = ( t +
1
𝑡
) sin(t)
5)
Complete the square: y2+ 3y – 4 = 8
6)
Complete the square: 4x2 – 20x + 17 = 0
7)
Complete the square in both x and y: 4x2 – 20x + 17 + y2+ 3y – 4 = 8
8)
Without a calculator, find the exact roots of: 2x2 – 6x - 2 = 0
9)
Factor in terms of y:
9y2 + 3y – 6
10) Use the quadratic formula to solve: 3x2 – 6x = 5
11) Solve without a calculator: |x – 4| < 8
12) Solve without a calculator: x2 – x – 12 > 0
13) Solve WITH a calculator. Write down the type of calculator you used and the steps used (did you use a 'solve'
feature, 'graphing', or something else?) : 6x3 - 11x2 - 10x = 0
Lines:
14) Write the equation of the line parallel to the graph of 4x + 3y = 8 that passes through the point (2, -1)
15) Write the equation of the line perpendicular to the graph of 4x + 3y = 8 that passes through the point (2, -1)
16) If the point with coordinates (3, k) is on the line 2x – 5y = 8, find the value of k.
Functions & Properties (even/odd/composite/VA/HA)
17) If f(x) = x2 – x + 1, then f(x + 1) =
18) Find the domain of g(x) = √ 𝑥 3 − 𝑥 2 . No Calculator.
19) Find the domain of h(x) = ln(tan(x)) on the interval [-π , π ]. Calculator OK.
20) Identify all holes, vertical asymptotes and horizontal asymptotes of: M(x) =
( 𝑥−1)2
𝑥2 − 1
21) If f(x) is an odd function and the graph of f contains the point (6, 5), name another point on the graph of f. Explain.
22) If g(x) is an even function and the graph of g contains the point (4, -7), what are two points on the INVERSE of g?
Explain.
23) If f(x) = x – 2, find
𝑓(𝑥+ℎ)−𝑓(𝑥)
ℎ
24) WITHOUT A CALCULATOR: Find all the zeroes (real and complex) of: N(x) = x 3 + 2x – 3. Remember to show all
your work!
25) WITHOUT A CALCULATOR: sketch f(x) =
𝑥−1
𝑥 2 − 3𝑥+2
Show the algebraic work needed prior to graphing.
26) Write a piecewise formula for the function:
27) No calc: What is the
28) No calc: What is the
250 𝑥
lim
𝑥 → − ∞ 𝑥+500
lim
?
7𝑥 2
𝑥→∞
3𝑥 2 +4𝑥 + 1
?
29) Identify which of THREE functions are identical. Do NOT use a calculator.
Y1 = 3(2x + 4)
Y2 = 3(2x) + 4
Y3 = 9(x + 2)
Y4 = 3(2x) 34
Y5 = (3(2x) )4
Y6 = (9(2x) )4
1
30) Do NOT use a calculator. Solve: y = log 7 ( ) for y.
7
31) Do NOT use a calculator. Solve: log (2x + 1) + log (4) = log(16)
32) No Calculator: Use f(t) =
15
3 + 5 (0.8)𝑥
.
a) Find the y-intercept. Write it as a point (not just the y-value).
b) Find all asymptotes.
c) Graph by hand using your answers for parts (a) and (b), and your knowledge of the general shape for this
parent function (think back to precalculus days). Then, verify your answers using a calculator.
33) Do NOT use a calculator. Find all the angles in the domain [0, 2π ] for cos (Θ ) =
√2
2
34) Evaluate the six basic trig functions which correspond to the point (-1, 2), where Θ is taken from the standard
position (the positive x-axis). (Hint: draw an x-y graph, with the coordinate point (-1, 2) marked, and the angle
shown from the positive x-axis).
35) Point P is on the terminal side of the angle, θ. Find the measure of θ for [0, 360°] if Point P = (5, 9)
36) Point P is on the terminal side of the angle, θ. Find the measure of θ for [0, 360°] if Point P = (5, -9)
37) Point P is on the terminal side of the angle, θ. Find the measure of θ for [0, 360°] if Point P = (-5, -9)
38) Solve for θ in degrees: sin-1 (
39) Solve for y in terms of x:
3
√29
)
xy + x = 4
40) Solve the system algebraically (no calculator):
3x – 2y = 8
5x + 4y = 28