Download Topic 1: Fractional and Negative Exponents - Parkway C-2

Honors AP Calculus BC
Name_______________________________
Thrill-a-Minute Summer Opportunity 2013
Favorite Pre-Calculus Topic _______
Your summer assignment is to basically 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). You are to do this by preparing a
packet of the homework assigned below, which will exhibit your efforts related to the
chapters. The homework packet will be due on the first day of class. 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 partially on quality completion, and partially on correct answers.
Much of the two chapters is very readable, IF YOU READ IT!!! Read everything 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
August 1st (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 look at each graph as an opportunity to synthesize what you know about it, i.e.
domain, range, zeros, asymptotes, max-min, whether increasing or decreasing and on what
intervals.
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 the
third or fourth day of class. This will set the stage for your calculus grade (test and
homework combined  130 points), so DO NOT skimp on the effort. Be thoughtful about
every step and use this opportunity to become familiar with your author’s style of
presentation.
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 (see further instructions next page)
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, 23,29,32,34,35,40,56,57
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, 31,46,50,59, 79,84,100
Section 1.5: Infinite Limits pp. 88-90: 13,17,26,36,41,50,62
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, 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, 30, 64,70,78,85,94,98,100,111
Section 2.4: The Chain Rule p.137-140: 14, 22, 32, 46, 52, 56, 62, 65, 70, 78, 82, 86, 114, 116
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.
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.
I assume you have basic skills in algebra. 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 I see that students consistently do not have down in their basic skill set
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. 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 sets should be done without it (except, possibly, to
check you answers). 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.
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.
I will be in my room the 3 days before school starts too if you want to stop by for help, or
you can e-mail me and I’ll respond as soon as I check my e-mails.
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