Topic 1: Fractional and Negative Exponents

Honors AP Calculus AB Name______Thrill-a-Minute Summer Opportunity 2013 Favorite Pre-Calc Topic ______

Your summer assignment is to basically have the review packet (a review of Algebra 2 / Trig. and Pre- Calculus), Chapter 1, and 2.1-2.2 (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 chapter. 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 140 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 ALL problems on the Worksheets (see further instructions next page) 2. Complete the following problems from Chapter 1 and 2 in your text: Section 1.1: A Preview of Calculus p. 47: 1-6 all Section 1.2: Finding Limits Graphically and Numerically p. 54-57: 3,11- 17 odd 20,21,23,29,32,34,35,40,55,56,57 Section 1.3: Evaluating Limits Analytically p. 67-69:9,13,28,37,44,50,53,58,60,69,77,87,89,91 Section 1.4: Continuity and One-Sided Limits p. 78-81: 12,16,21,30,31,46,50,59,60,79,84,100 Section 1.5: Infinite Limits p. 88-90: 13,17,22,26,36,41,50,62 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 : 16,21,23,26,34, 37-40all,42,57,71,74,82,89,95,96 Section 2.2: Basic Differentiation Rules and Rates of Change pp. 115-118: 9,21,26,29,32,44,52,55,62,63,91,94,97,103,106

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 骣 2轾 - 2 -3 2 1. -3x 2. 琪犏 2 桫2-x臌 (2 - x ) -1 -3 2 骣2x- 1轾 2(2 x + 1) - 2(2 x - 1) x -4 3. - sin x 4. 琪犏 2 2 桫2x+ 1臌 (2 x + 1)

1 -3 (2x + 5 ) 2 -1 2 5. 2 6. 骣1 4 1 3 琪 -2+ - 1 - 1 + - 2 桫x x y y 2

Topic 2: Domain

Find the domain of the following functions

3x - 2 x2 -5 x - 6 1. y = 2. y = 3. y= x -3 - x + 3 4x + 1 x2 -3 x - 18

x2 +8 x + 12 3 x - 6 4. y = 5. y = 6. y= tan x 4 x + 5 x2 - x - 30

Topic 3: Solving inequalities (absolute value)

Write the following absolute value expressions as piecewise expressions 1. y=2 x - 4 2. y=4 x + 1 + 2 x - 3

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. x2 -1 2. x2 +4 x + 4

Solve the following by factoring and making appropriate sign charts to check the critical points.

3. x2 +6 x - 16 > 0 4. 2x2 + 4 x 3

5. 2sin2 x sin x 0�x 2p

Topic 5: Special Factorization Factor Completely 1. x3 + 8 2. 27x3- 125 y 3 3. ac+ dc - ab - bd 4. x2+12 x + 36 - 9 y 2 5. (x- 3)2 (2 x + 1) 3 + ( x - 3) 3 (2 x + 1) 2

Topic 6: Function Transformation

If f( x )= x2 - 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.) 5f ( x )+ 3 5.) f(2 x ) 6.) f( x )

Here is f(x): Now graph the following based on f(x)…

7.) y= 2 f ( x ) 8.) y= - f( x ) 9.) y= f( x - 1)

10.) y= f( x + 2) 11.) y= f( x ) 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 + 3 x + 10 3.) P( x )= x4 + 5 x 3 + 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 1.) f( x )= 2 x2 - 7 2.) f( x )= 4 x2 - 4 x + 4 3.) f( x ) = x - x

1 4.) f x= x - x2 +1 5.) y= ex - 6.)3x= y ex

Topic 9: Solving quadratic equations and quadratic formula

Solve each equation (show your work).

1.)7x2 - 3 x = 0 2.) x2 +6 x + 4 = 0 3.) 2x2 - ( x + 2)( x - 3) = 12 1 1 4.) x4-9 x 2 + 8 = 0 5.) x-10 x + 9 = 0 6.) - = 6 x2 x

Topic 10: Asymptotes

For each function, find the equations of both the vertical asymptote(s) and horizontal asymptotes (if they exist)

x x + 4 x2 - 9 1.) y = 2.) y = 3.) y = x - 3 x2 +1 x3+3 x 2 - 18 x

2x2 + 6 x x 4.) y = 5.) y = x3-3 x 2 - 4 x 2x2 - 10

Topic 11: Complex Fractions 1 2 x2- y 2 x - 1- xy 1.) x 2.) 3x 3.) 1 4 x+ y x + x - x 9x y

4 2 + x-3 - x 4.) 5.) x-5 x + 2 -2 2x x -1 + 3 x2 -3 x - 10

Topic 12: Composition of Functions If f( x )= x2 , g ( x ) =2 x - 1 , and h ( x ) = 2x , find the following: 骣骣骣1 1.) f(g(2)) 2.) f(h(-1)) 3.) g琪 f琪 h琪桫桫桫2

4.) g(f(x)) 5.) g(g(x)) 6.) f(h(x))

Topic 13: Solving Rational Equations x+1 x - 1 x - 5 3 60 60 2 1.) - =1 2.) = 3.) - = 3 2 x +1 5 x x- 5 x

x2 x 5 x3 x - 2 4.) + = 5.) - = x-2 4 - x2 x + 2 2x- 6 x2 - 6 x + 9 3 x - 9

Topic 14: Basic Right Triangle Trigonometry

1.) If the point (2,-4) is on the terminal side of q , find all 6 trig. functions of q . Include a picture.

5 2.) If cosq = - , and q in quadrant II, find sinq and tan q 13

Find the exact value of the following without using a calculator: 4.) sin2225°- cos 2 300 ° 5.) (4 cos 30°- 6 sin 120 ° )-2

Solve the triangle (round to 3 decimal places – get used to it )

6.) A = a = 6 feet B = b = C = 90 c = 95 inches

Topic 15: Solving Trigonometric Equations

Solve each equation on the interval [0 , 2p )

1.) cos2 x= cos x 2.) 2cosx + 3 = 0 3.) 2sin2 x+ sin x = 1

4.) 2sinx cos x+ sin x = 0 5.) 8cos2 x- 2 cos x = 1 6.) sin2x- cos 2 x = 0