!
Summer Practice
Welcome to Calculus! The purpose of this packet is to review
the most important mathematics you learned in past math classes
so you are well prepared for taking calculus this year.
This packet is a mix of content and practice problems, so it is
important that you read through it completely (don’t just skip to
the problems). The contents on the next page will guide you
through the packet.
All I ask is that you try every problem and turn in your work on
the first day of class. You will not be given a grade for your work
on this packet, but you are required to complete it.
Working in groups is encouraged, but remember—don’t simply give
answers, help your friends understand the material.
If you have any questions, or need a hint on a problem, email me
at vbonina@btfiloh.org.
Good luck!
!
Contents
Part I: Algebra Review (12 Problems)
This section contains practice problems on linear equations,
inequalities, polynomials, rational expressions, and functions.
This is material you learned in your math classes in 9th and
10th grade. Complete each practice problem on a separate
sheet of paper and show all your work.
Part II: Trigonometry Review (10 Problems)
This section contains practice problems on trigonometry. It
also contains some reminders of material you should review
before calculus. Complete each practice problem on a
separate sheet of paper and show all your work.
Part III: The Most Important Graphs to Know for
Calculus
This section contains a review of the most important graphs
you should know for calculus. You should practice identifying
these graphs and also sketching them. Have a friend quiz
you or make flash cards.
Part IV: Resources for Calculus
This section contains some suggestions I have for additional
resources to obtain before or during your first calculus class.
Part I: Algebra Review
1.
Graph the inequality −2 ≤ ! < 3.
2.
Solve and graph the inequality −7 ≤ 1 − 2! < 11.
3.
Solve and graph the inequality 2 − 3 ! + 1 < −5.
4.
Simplify the expression:
5.
Simplify the expression:
6.
Simplify the expression:
7.
Solve:
8.
Solve:
9.
Solve:
!
−8! ! ! ! ! !
! !! ! ! !
3!! !! !!
!+1
5
− !
!
14!
! − 3! !
2! − 7 2! − 1
=
!+6
!−2
4
!+5
−3=
2! + 1
!
−
!!
5
3
1
+ != !
− 4! + 4 !
! − 2!
!
!!!
10.
Given ! ! = ! ! , ! ! = !!!!, and ℎ ! =
11.
Identify the vertical asymptotes to the graph:
−3
!= !
3! − 23! − 36
12.
Expand the logarithmic expression:
log !
2! !
!−!
!
!!
, find !(! ℎ ! .
Part II: Trigonometry Review
1.
!
Evaluate the following trigonometric functions: sin ! , cos
!
!!
!
, and
cos − ! .
!!!
2.
Identify the reference angle for
3.
If tan ! = − !", and
4.
Evaluate tan!! 1.
5.
Evaluate sin!! (sin
6.
Calculate the period of ! ! = −5 sin 2! + 1 − 6.
7.
Graph the equation in question 5.
8.
Show that
!
!!
!
!!!
!
!
.
< ! < 2!, evaluate sin !.
).
cot ! + tan ! =
1
cos ! sin !!
9.
Find all solutions to the equation 3 sin ! + 1 = sin ! + 4 on the
interval [0,2!).
10.
Calculate the exact solutions to the equation tan! ! ! = tan !.
Remember the unit circle? It’s back and you will need it to
evaluate trigonometric functions in calculus problems. Take some
time and review it. Have a friend quiz you or make some flash
cards.
Part III: The Most Important Graphs to Know
for Calculus
There are some very important graphs that you should memorize.
You’ll want to be able to identify them and also sketch them
without using your calculator.
Linear Function:
y=x
Parabola:
y = x2
Cubic Function:
y = x3
Absolute Value
Function:
y = |x|
Square-Root:
y=
!
Inverse Function:
y = 1/x
Inverse-Square
Function:
y = 1/x2
Logarithmic Function:
y = ln(x) or
y = log(x)
Notice the function
crosses point (1, 0).
Exponential Function:
y = ex or y = 10x
Notice the function
crosses at point (0, 1).
Sine and Cosine
Functions:
y = sin(x) or
y = cos(x)
Tangent Function:
y = tan(x)
Part IV: Resources for Calculus
Calculus is a challenging course and it always helps to have a few
extra resources to use throughout the year.
1. Consider obtaining a calculus quick reference
sheet. This is a laminated pamphlet that
summarizes all of calculus in just a few pages
and can be easily carried in any binder. If you
get good at using it, you can save a lot of time
rather than flipping through a textbook or
your notes. You can buy one for around $5.00
in the study guide section of a bookstore or
online.
2. As the course progresses consider visiting Khan Academy and
using his quick 10-minute mathematics videos. Below is a link for
the calculus videos, but you can access any level of mathematics
for review.
https://www.khanacademy.org/math/calculus
3. If you are intending on taking the AP Calculus AB exam after
our course (not required), I highly recommend the book 5 Steps to
a 5: AP Calculus AB.
This is the most recent edition, available July
19, 2013. If you obtain this book, I
encourage you to read p. 3 – 13 over the
summer, which will familiarize you with the
exam. You do not need to start the content.