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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 x5 5. y 3 x6 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 x4 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 x2 4 x x2 2.) x 5 3 x 1 5 5.) 3.) 60 60 2 x x 5 x x 3 x2 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