Download Topic 1: Fractional and Negative Exponents

Honors AP Calculus BC
Thrill-a-Minute Summer Opportunity 2016
Name_______________________________
Favorite Pre-Calculus Topic _______
Your summer assignment is to have the review packet (a review of Algebra 2 / Trig.
and Pre-Calculus), Chapter 1, and 2.1-2.4 (all of which you learned from Mr. Kozeny) of the
Calculus book mastered by the time school begins in the fall (this includes working problems
from the book and the attached worksheets). For the first day of class, you need to prepare a
packet of the homework assigned below, which will exhibit your efforts related to the
chapter. Work all indicated problems and as many more as you need to work in order to fully
understand each topic presented. Neatly show your work on all the problems assigned.
Your homework grade will be based mostly on quality completion, and partly on correct
answers. If there are any topics you don’t remember or don’t understand the directions to,
give it your best effort, Google it, ask your friends, and then know we will be going over it for
a couple of weeks before we take the first test. However, you need to try all the problems on
your own so you get the points for the summer homework, and then I will help you fill in the
gaps and answer your questions.
Much of the chapter is very readable. If you need help, read the sections slowly and with
great care. Much of what you read will be review and should be well within your grasp. At
all cost, give everything an honest attempt. I suggest you begin working by July 27th (at the
latest) and continue to do about two sections or worksheets per day. That way you won’t be
laboring on it all summer and you won’t have to cram in the last few days before school
starts. Don’t procrastinate. Let’s get the year off to a flying start (then we can find its
related rate)!
It will be expected that you have a good working knowledge of your graphing utility
including max-min, intersect, zero, window... You will do quite a bit of “analysis” of graphs
in calculus where you work exclusively with the graph of the function without ever knowing
or needing to know the definition of that function. This may be new to you. To prepare I
suggest you try to look at the graphs of some of the functions in your calculator, check the xintercepts that you found, etc. If you need any help with your graphing calculator, let me
know when we get back and I will help you with using it.
Have fun doing this work - make it good - anything worth doing is worth doing well. BE
WARNED: this assignment will be worth points and we will take a test on this on about
the fifth day of class. This will set the stage for your calculus grade (test and homework
combined  125 points, maybe more), so DO NOT skimp on the effort. Be thoughtful about
every step.
AP Calculus Summer Homework Worksheet Instructions
Going into AP Calculus, there are certain skills that have been taught to you over the previous
years that I assume you have. If you do not have these skills, you will find that you will consistently get
problems incorrect next year even though you understand the Calculus concepts. It is frustrating for
students when they are tripped up by the algebra and not the Calculus. This summer packet is intended for
you to brush up and possibly relearn these topics. Being able to solve equations, work with algebraic
expressions, and basic factoring, for example, should now be a part of you. So only the topics that I see
students consistently struggle with are included here. These are skills that are used continually in AP
Calculus.
On the following 8 pages, you have 15 problem sets. Each problem should be done in the space
provided and all work should be shown. DO NOT rely on the calculator to think for you. You can use it to
analyze graphs and check your answers, but not to do all your work. Half of your AP exam next year is no
calculator, so use pencil, paper, and brain techniques to solve these problems. We will definitely work a lot
with the calculator next year, but these review sets should be done without it. Realize that certain concepts
are interrelated. Domain, for example, may require you to be expert at working with inequalities. Solving
quadratic equations may involve techniques used in solving fractional equations.
When I talk about “appropriate sign charts” in the directions to a section, I mean to set the
problem equal to zero, find the points where it equals zero, put them on a number line, and then check each
interval to see if it’s positive or negative. Like this:
If you learned to solve these with a different technique, go ahead and use that on this packet and then let me
know and I’ll explain sign charts in our review days. They become important later on in Calculus so I want
to start you on them early.
If you have questions, a good website for algebra review topics is Purple Math at
http://www.purplemath.com/modules/index.htm or Khan Academy at www.khanacademy.org.
Otherwise, Google search a topic and lots of sites will pop up.
So, here it is. The assignment you’ve been waiting for:
1. Complete the ODD problems on topics 1-8 and ALL problems on topics 9-15 on
the Worksheets.
2. Complete the following problems from Chapter 1 and 2 in your text:
Section 1.2: Finding Limits Graphically and Numerically p. 54-57: 3, 11- 17 odd,
20, 23, 29, 32, 34, 35, 37, 40, 56, 57
Section 1.3: Evaluating Limits Analytically p. 67-69: 9, 37, 50, 53, 58, 60, 69, 77, 87, 89, 91
Section 1.4: Continuity and One-Sided Limits p. 78-81: 12,16, 21, 31, 40, 46, 50, 59, 76, 78,
84, 86, 100
Section 1.5: Infinite Limits pp. 88-90: 13, 17, 26, 36, 39, 41, 50, 62
Section 2.1: The Derivative and the Tangent Line Problem (NOTE: BE SURE to use the
Definition of the Derivative and NOT the power rule to differentiate these
problems) p. 103-106 : 21, 23, 26, 37-40all, 42, 57, 71, 74, 82, 93, 95
Section 2.2: Basic Differentiation Rules and Rates of Change pp. 115-118:
26, 32, 44, 52, 55, 62, 63, 91, 94, 97, 103, 106
Section 2.3: Product and Quotient Rules and Higher-Order Derivatives pp. 126-129:
3, 6, 7, 18, 20, 22, 43, 48, 70, 78, 82, 85, 94, 98, 100, 115
Section 2.4: The Chain Rule p.137-140: 14, 22, 32, 46, 52, 56, 62, 65, 70, 82, 86, 105, 114
Thank you for your efforts and I look forward to a fun and rigorous year of Calculus.
Never forget…there is no rest in the quest for Calculus knowledge.
Topic 1: Fractional and Negative Exponents
Simplify using only positive Exponents
 2   2 
1. 3x 3
2. 2 

2
 2  x   (2  x) 
1
3.
3
 2 x  1   2(2 x  1)  2(2 x  1) 
4. 4 
 

(2 x  1)2
 2x 1  

x2

sin x
2
3
1
( 2 x  5) 2
5. 2
3
2
1
 1
4
1 2
6.  2  1 1  2 
x y
y 
x
Topic 2: Domain
Find the domain of the following functions
1. y 
3x  2
4x 1
x2  5x  6
2. y  2
x  3x  18
4. y 
x 2  8 x  12
4
x5
5. y 
3
x6
x 2  x  30
3. y  x  3  x  3
6. y  tan x
Topic 3: Solving inequalities (absolute value)
Write the following absolute value expressions as piecewise expressions
2. y  4 x 1  2 x  3
1. y  2 x 4
Solve the following absolute value inequalities
3. x 3  4
4. 3x 4  2
5. x 1  x 3
Topic 4: Solving inequalities (quadratic)
Write the following absolute value expressions as piecewise expressions
1. x 2 1
2. x 2 4 x  4
Solve the following by factoring and making appropriate sign charts to check the critical
points.
3. x 2  6 x  16  0
5. 2sin 2 x  sin x
4. 2 x 2  4 x  3
0  x  2
Topic 5: Special Factorization
Factor Completely
1. x3  8
2. 27 x3  125 y 3
4. x 2  12 x  36  9 y 2
3. ac  dc  ab  bd
5. ( x  3)2 (2 x  1)3  ( x  3)3 (2 x  1)2
Topic 6: Function Transformation
If f ( x)  x 2 1 , describe in words what the following would do to the graph of f(x):
1.) f ( x)  4
2.) f ( x  4)
3.) -f(x+2)
4.) 5 f ( x)  3
5.) f (2 x)
6.) f ( x)
Here is f(x):
f(x)…
7.) y  2 f ( x)
10.) y  f ( x  2)
Now graph the following based on
8.) y   f ( x )
11.) y  f ( x)
9.) y  f ( x  1)
12.) y  f x
Topic 7: Factor theorem ( p over q method/synthetic division)
Use the p over q method and synthetic division to factor the polynomial P (x). Then solve P (x) =0
1.) P( x)  x3  4 x 2  x  6
2.) P( x)  x3  6 x 2  3x  10
3.) P( x)  x 4  5x3  6 x 2  4 x  8
Topic 8: Even and Odd functions
Show work to determine if the relation is even, odd, or neither.
1.) f ( x)  2 x 2  7
4.) f x  x  x2  1
2.) f ( x)  4 x 2  4 x  4
5.) y  e x 
1
ex
3.) f ( x)  x 
6.) 3x  y
1
x
Topic 9: Solving quadratic equations and quadratic formula
Solve each equation (show your work).
1.) 7 x 2  3x  0
4.) x 4  9 x 2  8  0
2.) x 2  6 x  4  0
3.) 2 x 2  ( x  2)( x  3)  12
5.) x  10 x  9  0
6.)
1 1
 6
x2 x
Topic 10: Asymptotes
For each function, find the equations of both the vertical asymptote(s) and horizontal asymptotes (if they
exist)
1.) y 
x
x 3
2 x2  6 x
4.) y  3
x  3x 2  4 x
2.) y 
x4
x2  1
3.) y 
5.) y 
x
2 x  10
2
x2  9
x3  3x 2  18 x
Topic 11: Complex Fractions
1
x
1.)
1
x
x
x2  y2
xy
3.)
x y
y
2
3x
2.)
4
x
9x
x
1
4
2

5.) x  5 x  2
2x
3
2
x  3 x  10
x 3  x
4.) 2
x 1
Topic 12: Composition of Functions
If f ( x)  x , g ( x)  2 x  1,and h( x)  2 x , find the following:
2
   1 
   2 
1.) f(g(2))
2.) f(h(-1))
3.) g  f  h    
4.) g(f(x))
5.) g(g(x))
6.) f(h(x))
Topic 13: Solving Rational Equations
1.)
x 1 x 1

1
3
2
4.)
x
2x
5


2
x2 4 x
x2
2.)
x 5 3

x 1 5
5.)
3.)
60 60
2


x x 5 x
x
3
x2
 2

2 x  6 x  6 x  9 3x  9
Topic 14: Basic Right Triangle Trigonometry
1.) If the point (2,-4) is on the terminal side of  , find all 6 trig. functions of  . Include a picture.
2.) If cos  
5
, and  in quadrant II, find sin  and tan 
13
Find the exact value of the following without using a calculator:
5.) ( 4 cos 30  6 sin120 )2
4.) sin 2 225  cos 2 300
Solve the triangle (round to 3 decimal places – get used to it )
6.) A =
B=
C = 90
a = 6 feet
b=
c = 95 inches
Topic 15: Solving Trigonometric Equations
Solve each equation on the interval [0, 2 )
1.) cos 2 x  cos x
2.) 2 cos x  3  0
3.) 2 sin 2 x  sin x  1
4.) 2sin x cos x  sin x  0
5.) 8 cos 2 x  2 cos x  1
6.) sin 2 x  cos 2 x  0