I
HE
"Quali,! Stu{h! rs To(!ay, Dynanic ltaders Toniorrow"
Jame-!
Birch, Principsl
Dear Future Calculus Student,
BCI I am excited to have you as a student, and I am excited to
to this
you on your path to success in this class. You will leam that your math education
Welcome to AP Calculus
start
calculus'
point in your life has prepared you to be able to have a culminating experience in
That is why I had
Many students find that every day they are able to say something like, "oH!
to leam how to do
that!" In order for you to enjoy these
experiences, you need to start the year
feeling like your "pre-calculus" math skills are sharp'
assignments
In this packet you will find a wonderful set of review assignments. These
year' The
will help you prepare for the brief review quiz we will have the first week of next
I am providing it for you
homework may be easy for you if you complete it now, but remember,
of class next year' You are
now so that your "pre-calculus" skills will be sharp on the first day
the first day of school next
free to decide to do it any time between now and the night before
and fill in the
year. You will need to complete the old final exams from a pre-calculus course
to complete either of these
blank unit-circle chart. you should not need noies or a calculator
exams with an 80% or
assignments. It is HIGHLY recommended that you pass the pre-calculus
information from the "10
higher in order to be successful in calculus. If you don't have the
by the beginning of the
Basic Functions" pages memorized, you will want to have it memorized
for the quiz' If you are
year. You may also want to look over the rules of logarithms/exponents
just in case you have forgotten something
wondering how to check your work or where to look
mathisfun.com, and
over the summer, you can check wolframalpha.com, khanacademy.org,
many other reliable math websites.
T
Hr
scHoo
"
lJ
Quality Stude,ts Todnf, Dynamic Leaders Tonarmw'
James Birch, Principal
It will be obvious to me on the quiz if you have not taken the assignment seriously'
Please be prepared to succeed on this
success,
quiz. Starting the year on a high note, by experiencing
will help you to enjoy the class
and continue to be successful throughout the year'
Also included in this packet is a letter from last year's class which includes some advice
you will leam. I
for how to succeed this year. I have also provided a general outline for what
class at:
encourage you to visit the AP website and do some research about the
will notice that all
Again, I am excited for you to start this experience with calculus. You
this class, we will examine
the mathematics you have learned in your life will culminate during
earn a fairly large
many practical applications of this information, and hopefully you will
amount of college credit for your efforts. See you in Augustl
Siryretv,
I //
/,za'k/4-
Mrs. Teagan Sweet
Calculus Teacher
Herriman High School
T
Non-Calculalor
ANSWERS
Final Exam - PreCalculus
Name:
Period:
F'ind the value:
-l.
3tt
tan
4
-=
)
I
I
I
^J.
cos
lltt
-:6
4,
I
I
I
s. n'na e, cose=f $2t4
6.
Fittd
0,tanq=l
7.
Find
d, sin-'1-1;=9
8.
Solve sin2 r-sinx=0 [02n]
9.
Findthedomain ot
.
10
all solutions
f
(i=-?:7x2
-x-12
(-o,co)
(-3,4)
C.
E. (- -, -3) tt (-3, 4) u (4, co)
A.
B. G., -3)
D. (4,0o)
10. A trigonometric representation of -1 +,'f3r
lsin30")
C. 2(cos90" + isin90')
A.
2(cos30o +
E.
2(cos150" + isinl50")
is
B. 2(cos60' + isin60')
D. 2(cos120'+ tsini20')
ll
11. Find the amplitude
ofy
:
2 + 6cos3r
A.6
B. 3
c.2
D.
E. -2
12.
T
12. Find the period ofy = 2 + 6cos3x
A.
^
L.-
2tr
2tt
E.
none ofthese
B.
D.
3
13
'r*
13. solrr"
14. Graph
x+1
/(x) = -(r
15. Solve 4b-r =
16.
16. Given (x)
t7
17. Evaluate
c.2
E. -3
J
t+6-2.0
15
A.0
1t
+ 5)'?(.r
-
3)(.r + 1) (abel the roots on the graph)
8
:I
+
3.x
and g(r)
:
2x - 5, find (g(.r)).
I
'
los-'125
B. I
D.3
T
18. Which one ofthe following could represent a complete graph of
qx) = -ra + 13 + pr2 + qx + r where p, q, and r are real numbers?
18
,
,
I
,
c
v
-.-/l
t
o
E
\/
tl
V
19. The following three transforrnations
to the graph ofy = *'
19
I.
A vertical stretch by a factor of
II. A horizontal shift right 5 units
III. A vertical shift down 6 units
are applied (in the given order)
3.
Which ofthe following is an equation for the graph produced as a result
of applying these transformations?
B. y:3(x-5)2-6
D. y=:l- t
n. v=51 - I
C. y =:1x + S;'z- e
f,. y=3(x-6)2+5
x-int:=y-int:_, =H.A.:V.A.:-
20.
20. Identify.r
and y intercepts, horizontal and vertical asymptotes.
Sketch the graph. State the end behavior and vertical asymptote behavior.
5
x'-9
Behavior
y
65 -y -co,;f(r) -+as x -+co,./(.r) -+
End
Vertical Asymptote Behavior
as x-->-*,f(r)-+---..-.as
x-+--,f(x)-+---
as x -+
as n ---)
.
-'.f(x)-+ -+
--.jr(x)
T
Name:
Non-Calculator (No Notes)
Final Exam - PreCalculus
Period:
Write the partnt function of each gnph.
'r\l
T
THE CHART
'
m
R
m
o
cos 0
sin 0
tan 0
sec 0
csc 0
cot 0
0
xl6
n/4
JTi )
nl2
2nl3
3nl4
5n/6
1l
7ttl6
5nl4
4nl3
3nl2
5r./3
7
nl4
tht6
AP Cal(u us Sum.ner lnst.rJre
.la
y perersol
T
Ten Basic Functions
The Identity Function
Equatiou
The Quadratic Function
/(r)= r
Equarion:
o)
(-o, o)
Domain: (-"o,
Range:
This is the only function that acts on every real number by
leaving it alone
/(.r)
Domain: (---,
nange:
[0,
x2
"")
*)
The graph ofthis function, called a parabola, has a refleclion
property that is useful in rnaking flashlights and satellite
dishes.
The Cubic Function
Equation:
/(;r)=:rr
Donrain: (-":,"o)
Range:
-l
( *,
".')
he origin is called a "poinl of inflection" ibr this curve
because the graph changes curvature at that point.
The Square Root Function
tiq
rrarinn
r
Domain:
(r)- r/i
[0.
rc)
Range: [0, *)
Put any positive number into your calculator. Tak€ the
square root. Then take the sqLlare root again. Then take the
square root again. And so on. Eventually you will always
pel l.
The Rational Function
EqLration:
1(x)=
Domain; (-x,,0)u
I
(0,
nange: (-m,O)v(0,
"c)
o)
This curve, called a hyperbola, also has a reflection property
that is useful in satellite dishes.
The Exponential
Equation:
F
Lrnction
/('r)= e'
Domain: (--'o. "o)
nange:
(0,-)
The number e is an irrational number (like r) that shows up
in a variety ofapplications. The symbols e ancl I were both
brought into popular use by the great Swiss n]athematician
l,eonhald Euler ( 1707- I 783).
T
The Natural Logarithm Function
Equation: /(.r) = ln
Domain: (0, oo)
The Sine Function
I
Range: (--, *)
This function increases very slowly. Ifthe;r-axis andy-axis
were both scaled with unit lengths ofone inch, you would
have to travel more than two and a half miles along the
curve j ust to get a foot above the x-axis.
Equation: /(;r) = sin x
Domain: (-o, oo)
nange:
lt,t]
This function and the sinus cavities in your head derive their
names from a common root: the Latin word for "bay." This
is due to a l2'r'-centLrry mistake made by Robert ofbhester,
who translated a word incorrectly from an Arabic
manuscript.
The Cosine Function
Equation:
/(;r)=
cos
The Absolute Value Function
r
Equation:
Domain: (-"o, .o)
nange:
lt,t]
The local extrema ofthe cosine lunction occur exactly at the
zeros ofthe sine firnction, and vice-versa.
/(;r)= lxl
o)
Domain: (-oo,
Range: [o,o)
of direction (a ,,corner")
origin, while other functions are all ,.smooth" on their
l'his function
at the
domains.
has an abrupt change
I
sinLii=ffi
ODD
LUY}l =
ranE=S
= h'Jp'sin g
idj=h'Jp.cosg
adj
Ij]T
aPP
0n the Unit Ci rule
Sino =
-
,q) = (cosrt,sin.s)
U
q Sirr6
X LOs O
x Cos o f
=-=-
O
Positive
5lIto
(A,s,r,c)
fins.l Alt
r... =r==_L
U srr'ln
TanslGss
LOIO
g
-lt
:'eCO =-:;=-
IotalSec]t
-1
.3rtstn
1_-
!
J5
erccot
irctin
grcSec
Addition ldPntitiPe
sin(r 1b) = Sin a Cos b I Coss sin b
llos(! t b) = Cosa Cos b T:lin e Sirr b
Heqitive Angles
51h
(-3J = -Srn
l-ost-rJ
I + lilr .l lin
L-l
3rl
D
D
nr
1,4
ll/
/4
l
llrt,
1t
r'l
l
srl
?1rl
o
trt - It
1T
7
z.1t/
-t
E
240
27'D
300
f,15
trnt
-Js; ' 1t
-1
w/,
ltrt
/+
t\
0
..
!1'r
t)
1
L-t
-.I?
:irn LiIJJ =
2r,.
Triin'llF ldentitiPg
5f-r
ttng+l-{'sJ:t=l
. _ 1_
i
I + I.in E = -ieD lJ
1+Dot?t4=csr?6
-
e.F
Cofunclion ldentilies
sin
lf
- rJ =
1165
11
LD:IF - IJJ =
e1r,
lJ
rrn
{f
- r) = r;6i s
r Iilu-
.t2
nxrnrit4 ll t.
_?
'J
- Jz
'a
1
'ti
1T
=
A Sin [B(x
- C)]+
22,1
D
,t=AcoslB(x-u)]+0
fl;
zE' v;
1
! 51'l' b Lqs b
_22
Lot\Z6)=Uo5H-lilrrtJ
tos\28) = 2tros'B col{?E)=1-?5,in24
- ?J7,
t
-l
f.l,rut,le AriglP Forrnui.!s
-
-l
-tE
1
11!r,
.12
- e.]j,
0
'L/
-2
I
1
4n.
-2
-l
rl
u",,:,
A.r,ll,r F,:,rnr!l!J
qiht{
tf 1-f:f,< n
,rnz=-sffi==Iiti;z
7
.Jl
2
-l
Hilf
-:-'lr'r i: fi---iiii-F
n; ^.
:
o
-tE
1
150
LI
EJ
Trni-s) = -T.rn(s)
lt,'
21,
t;
0
-k
120
rll
2
1
1
v;
n/
90
1
.li
ltJ
/j
r,D
0
F
(BJ
= Lirr
U
A = Anrplitudq
E = Fre,lupn.! iri ?fr
[ZnlB = Feriod)
E=x
lhift
I
or
355./1 13
L!',{s of Sines
and C'rsines
tlrr .:i tlng
Slli
crb-L
o_ = b_ + c_ - 2.b'.-,'for
y
d
AP CalcuLus Summer lnstitute, Larry Peterson
I
Dear Incoming Calcr-rlus BC Student,
You are about to embark on the most challenging and academic rewarding experience in high school as
yor-r tackle AP Calcr-rlus BC. As fomrer students we would like to offer you the following advice.
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.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
o
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.
.
Don't Panic!
Calculus is a difficult class. You shouldn't expect an easy'A'. An 'A' will be the by-product of
learning Calculus well and not necessarily from doing all ofyour HW assignments.
Life is OK and will still be great, even if this class wrecks your 4.00 GPA.
Don't be too hard on yourselfl- even when you bornb a test. Pick yourselfup and be determined to eam
higher. Excessive stress is the killer ofprogress.
When you are challenged, you are growing. It's OK to feel discouraged or not completely understand a
topic the first time you leam it.
Study daily- even on the weekends. Give Calculus adequate time to learn it well. Ask your teachers to
teach you good ways to study. You will get new knowledge, understanding, and capability further than
yor: ever imaginedl
The work load is manageable/ It's not too much to handle.
Everybody in the class is working for the same thing yor,r are. Learn to work well with your peers. fhey
can really teach you a lot. Yon may even make a lasting friendship with someone outside ofyour
regular social group.
Don't try to tackle Calculus alone. Work as much as you can in class and under the guidance ofyour
teachers. Dedicated Work Days are opportunities for learning. Don't let them be dedicated nap or goofoff days. PIan and attend extra study sessions.
DON'T PROCRASTINATE! Don't craml Do homework when assignment is given.
Manage youl time well. Take a study skills class or the AP research period.
Do EVERY assignment plus some extra problems for proficiency. Learn the concepts, don'tjust
nemorize the fblmulas. It's helpful to talk and explain your learning to others.
Stick with it and ENDURE to the end! It's worth it. Especially avoid the er.rd-of-year-senior-meltdown.
Ask iots olquestions to yoursell, to your peers, and to your teachers. Ilyou don't know, askl Flowever,
you can only ask questions ifyou've been studying. If you haven't been studying, then you'll have no
chre in heck what to ask abor:t!
Asking questions is not a sign of being dumb. It shows how much yor"r want to learn.
Review often, Review your triqonometry notes. the unit circle, Algebra, and your past tests. Every
topic in Calculus is an impofiant topic.
I{arely r"lse the solution manuals to get answers. Mistakes on HW are OK. Understand the concepts
firlly before searching for shorl-cr:t methods.
Remember the little nuances of Calculus- derivatives are rates of change and yor: r.reed (+ C) for
indefinite integrals.
Accept the arbitrary !
You will becon.ie addicted to the feelings of success and accomplishment. Embrace it.
Stay attenlive in class. ll you are sleepy, then start asking questions or volunteer to do a problem on the
board. PLtt away your phone and music devices. They sap brain energy needed for learning Calculus.
Learn to Llse your calculator very well, but be able to do most problems withor"rt it.
Your teacher loves you and will help you. Unless yor"r give up on vourself ancl then watch or"(. . .. !
Dor.r't Par.ric!
Besl ol Luck,
llen'inran High Former Calcuhrs Students
T
AP Calculus BC-Course Outline
Preparation for Calculus
.
o
.
.
.
Lines
o
Slopes (including parallel and perpendicular), equations oflines and applications
Functions and Graphs
Functions, domain and range, even and odd functions, symmetry, piecewise
functions, absolute value functions and composite functions
o
ExponentialFunctions
o Exponential growth, exponential decay, the number e
Functions and Logarithms
o One-to-one functions, inverses, logarithmic lunctions. properties of logarithms
Tr igonometric Functions
o Graphs oftrigonometric functions, periodicity, transformations, inverses of
trigonometric funclions
Limits and Continuity
. Rates of Change
o Average and instantaneous speed, definition ola limit. propefiies ollin.rits,
o
o
o
one-
sided and two-sided limits, sandwich theorem
Limits Involving Infinity
o Finite limits, infinite limits, end behavior models
Continr:ity
o Continuity at a point, continuous lunctions, algebraic conrbinations. composiles,
internrediate value theorem for continuous luneLions
Rates oi Cl.range and Tangent Lines
o Average rates olchange, tangent to a cnrve, slope ola cun,e, normal to a curve
Derivatives
Investigations: Applications olDifferential Calculus (AP Sr-rmmer Institute) and A 20 Minr-rte
Ride (AP Surnmer Institute)
. Derivatives o1'a Function
o Deflnition ola delivative, notatioli. rclalionship between 1he graphs otJ'and /',
glaphing the delivative from data, one-sided derivatives
r
.
.
.
Differentiability
o
o
How/'(a) might lail to exist, differentiability implies local linearity,
clerivatives
on a calculator, diffelentiability implies continr-rity, intermediate value theorem
for derivatives
Rules lor Differentiation
o Power rule (inclLrding multiples. sums and dil'ferences). prodr:ct rule. quotienl
mle, second and higher order derivatives
Velocity and Other Rates of Change
o Instantaneous rates ofchange, motion along a line, sensitivity to change
Derivatives of Trigonometric Functions
T
o
.
o
.
o
Derivatives ofthe sine and cosine functions, simple harmonic motion, jerk,
derivatives of other basic trigonometric functions
Chain Rule
o Derivative ofa composite ftrnction, "outside-inside" rule, repeated use ofthe
chain rule
ImplicitDifferentiation
o Implicitly defined functions, tangent and normal lines, derivatives ofhigher order
Derivatives of Inverse Trigonometric lunctions
o Derivatives of inverse functions
Derivatives ofExponential and Logarithmic Functions
o Derivative of e', derivative ofa{, derivative of lnr, derivative of log,:r
Applications of Derivatives
o
.
.
o
r
.
Extreme Values of Frrnctions
o Absolute (global) extreme values, local (relative extreme values, finding extreme
values
Mean Value 'fheorern
o Mean value theorern, physical interpretation, increasing and decreasing functions
Connecting,f ,f ',,f "
o First derivative test for Iocal extrema, concavity, points of inflection, second
delivative test for local extrema, colrnecting derivatives to functions
Modeling and Optimizarion
o Maximizing ancl minirnizing quantities lrom matl-rernatics, business, indr:stry and
economics
Linearization and Newtor.r's Method
o Linear approximation, NeMon's Method, diflerentials
Related Rates
o Related rates eq rrar io n s
The Definite Integral
Investigation: Fundamental Theorem ofCalculus Using the Rule ofThree (AP Summer Institute)
. Estimating with Finite Sums
o Distance traveled, rectangular approximation method
. Definite Integrals
o Riemann sutns, tcrminology and notation olintegratiorl. delinitc integral ancl area,
constant functiot.rs, integlals on a calculator, discontinr"rous integrable functions
. Definite Integrals and Antiderivatives
o Properlies of definite integrals, average value ofa function, mean value theorerr
lor definite integrals, connecting dilferential and integral calculus
. Fundanental Theorem of Calculus
o Fundamental Theorem (part 1 and 2), area connection, ar.ralyzing antiderivatives
graphically
T
.
Trapezoid Rule
Trapezoid Approximation, Simpson's Rule
o
Differential Equations and Mathematical Modeling
.
Investigation: Slope Fields and Differential Equations (AP Summer Institute)
Activity: Ultraviolet Voodoo (AP Fall Institute)
. Slope Fields and Euler's Method
o Differential equations, slope fields, Euler's method
. Antidifferentiationby Substitution
o Indefinite integral, Leibniz notation and Antiderivatives, substitution in indefinite
integrals, substitution in definite integrals
o Antidifferentiation by Parts
o Product rule in integral form, solving for the unknown integral, tabular method,
inverse trigonometric and logarithmic functions
. Exponential Growth and Decay
o Separable differential equations, law ofexponential change, continuously
compounded interest, radioactivity, Newton's Law of Cooling
o .Logistic Growth
o How populations grow, partial fraction decomposition, the logistic differential
equation, logistic groMh models
Applications of Defi nite Integration
.
o
o
.
.
o
Integrai as Net Change
o Linear motion, consumption over time, net change from data, work
Areas in the Plane
o Area between curves, area enclosed by intersecting curves, boundaries with
changing fnnctions, integrating with rcspect to y. saving time with geometry
formulas
Volume: The Disk and Washer Method
Volume ol solids with known cross sectiolrs
Volume: The Shell Method
Arc Length and Surfaces ofRevolution
Sequences,
.
L'Hopital's Rule, and Improper Integrals
Sequences
o
o
Defining a sequence, arithmetic and geometric sequences, gr.aphing a seqllence,
limit of a seqr-lence
L'Hoptial's Rule
o
.
lndererminare Forms of L'Hopital's Rrrlc
Improper Integrals
o
[0,*,-,--,1".00,*o.]
\o "
)
lnfinite limits olintegration, integrands with infinite discontinuities, test Ibr
convergence and divergence, applications
T
Infinite Series
. Geometric and Power Series
. Taylor and Maclaurin Series
. Series and Convergence
. Taylor Polynomials and Approximations
' Taylor's Theorem and Lagrange form of the Remainder
. Tests for Convergence and Divergence
o nth term Test
o The Integral Test
o Comparisons of Series
o Altemating Series
o The Ratio and Root Tests
. Radius of Convergence
. Testing Convergence at Endpoints
Parametric, Vector, and Polar Functions
o
Parametric Functions
o
r
.
.
.
Derivative at a point,
,l2
;i
,,
, leneth of a smooth curve, surface area
Vectors in the Plane
Component form, Zero vector, Vector operations, Angle between vectors,
Applications
Vector-ValuedFunctions
Standard unit vectors, Derivatives and motion, Differentiation rules, Integrals
Polar Coordinates and Polar Graphs
Polar coordinates, polar graphing, relating polar coordinates and Caftesian
coordinates
Calculus of Polar Curves
Slope, Area in the plane, Length ofa curve, Area ofa surface ofrevolution
o
o
o
o