Welcome To AP Calculus BC! - Erie High School

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If you have any questions, you can contact me over the summer atpanakhyo [email protected] I will be out of town part of the summer, but generally check thise-mail every 2-3 days. Please send me a brief e-mail introducing yourself and so I have a wayof contacting you over the summer.I am looking forward to an exciting year in AP Calculus BC next year and getting to know allof you!Have a good summer and see you in the fall,Anna PanakhyoSummer Homework Part 1: Kahn Academy1. Go to kahnacademy.com2. Create an account with your school e-mail address.3. Once you log-in, you will see your name in the upper right corner. Click on the arrow nextto your name. Click on “Profile.”3

4. Your Dashboard will now be open. Click on “Coaches.”5. Under “Add a Coach,” enter the AP Calculus BC 2016-2017 code . You will nowbe part of the course and Ms. Panakhyo can recommend you the following skills for youpractice. It is suggested that you practice the skills in this order, as they may not be in thisorder on your screen. This is the most logical order. There are 20 skills you must practice. Finding limits numerically One sided limits from graphs Two-sided limits from graphs Two sided limits using algebra Continuity Limits at infinity where f(x) is unbounded The formal and alternate form of the derivative Recognizing slope of curves Graphs of functions and their derivative Visualizing derivatives Power rule (basic) Special derivatives Chain rule on two functions Product Rule Quotient Rule Application of derivatives: tangent and normal lines Application of derivatives: motion along a line Extreme values from graphs Concavity and the second derivative Local linearizationYou are not limited to only practicing these skills. You are welcome to practice additionalCalculus skills as well. These will be the skills I will check for in August.4

6. On the Dashboard bar, click on “Progress.” Then click on “Coach Recommendations.” Atfirst you will see a grey/empty box under the “Mastery Status” column. Once you answer 5questions in a row correctly you will reach the “Practiced” level. You will need to meet the atleast the “Practiced” level for these 20 skills by the first day of school. Every so often you willget a Mastery Mission, where you can move up a level or reach the Mastery level. Completethese Mastery Missions.7. For each skill, you will complete the exercises indicated with a star on the timeline on theleft side of the screen. For the exercises, you can get a hint by clicking the “I’d like a hint.”You can also watch a video pertaining to the question’s topic. If this is a new topic or conceptfor you, I suggest you watch the videos and tutorials before the exercise indicated by anarrow on the timeline on the left-side of the screen.8. If it has been a while since you logged into Kahn Academy to check your progress, makesure on your Progress screen on the top right of the chart you change “Activity from” to“Last 30 days” or “All time.” The automatic default is “Last 7 days.”5

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Summer Homework Part 2: Problem SetYou will find an attached problem set that reviews limits, derivatives and applications. Theseproblems are due the first day of class. Use your notes from previous math classes andCalculus AB to help you complete these problems. You may also work with others, butremember each person must turn in their own work. All work must be shown. If you use acalculator, you must set-up the problem on your paper and indicate what the calculatorproduced.NOTE: Problems with an asterisks (**) are required for students who have taken APCalculus AB. Students who have not taken AP Calculus AB may need to research how tocomplete theses problems. These problems are not required for students who have not takenAP Calculus AB, but recommended to try.For #1-17, consider the graph of f below.1) lim 𝑓(𝑥) 9) lim 𝑓(𝑥) 2) lim 𝑓(𝑥) 10)3) lim 𝑓(𝑥) 11)4) lim 𝑓(𝑥) 12) lim 𝑓(𝑥) 5) lim 𝑓(𝑥) 13)6) lim 𝑓(𝑥) 14)7) lim 𝑓(𝑥) 15) 𝑓 2 8) lim 𝑓(𝑥) 16) 𝑓 3 &' &- & .- .' . /' /lim 𝑓(𝑥) 0'lim 𝑓(𝑥) 0 0lim 𝑓(𝑥) 1lim 𝑓(𝑥) '1 /-7

Evaluate the following limits algebraically: '417) lim 521)18) lim (2𝑥 / 3𝑥 4) 22) 4 '4 619)20)lim '/lim : -/ 5 '46'& 5 - '. '& 5 '; 23)24). 5 -6 lim '1 4' / '& lim cos 𝑥 1lim '1lim/ 56 5 '; '/ 5 - '1 &' 𝑥 / 𝑎/ 𝑥 𝑖𝑓 𝑥 24 2𝑥 / 𝑖𝑓 𝑥 2a) Find the lim 𝑓 𝑥25) **Let 𝑓 𝑥 /'b) Find the lim 𝑓 𝑥 /-c) Find all values of a that make f continuous at 2. Justify your answer.Find the derivative of the following526) 𝑓 𝑥 𝑥 'E33) **𝑓 𝑥 27) 𝑓 𝑥 𝑥 & 𝑥 / 2𝑥34) 𝑓 𝑥 28) **𝑓 𝑥 5𝑒 329) 𝑓 𝑥 30) 𝑓 𝑥 31) 𝑓 𝑥 :𝑥JK 5 L'& 35) 𝑓 𝑥 sin 𝑥 cos 𝑥36) 𝑓 𝑥 𝑥 / cos 𝑥4H37) 𝑓 𝑥 2 cot 𝑥 𝑥 sec 𝑥 I 5 -0 -&38) **𝑓 𝑥 𝑥 & 14HH 32) **𝑓 𝑥 𝑥 / 𝑒 39) **𝑓 𝑥 𝑥 / 7𝑥8

40) ** If 𝑓 𝑥 𝑥 / 2𝑒 , find the value of 𝑓′(1)41) If 𝑓 𝑥 2𝑥 / 𝑥 & , find 𝑓′(𝑥), 𝑓′′(𝑥), and 𝑓′′′(𝑥)42) Find the slope of the tangent line to the graph of 𝑔 𝑥 𝑥 / 4 at the point (1, 3).&43) Find the equation of the tangent line to the hyperbola 𝑦 at the point (3, 1). 44) ** FindVW45) ** FindVWV V by implicit differentiation: 𝑥 / 𝑦 / 25by implicit differentiation: 𝑥 / 𝑥𝑦 𝑦 & 89

46) ** A particle moves according to a law of motion 𝑠 𝑓 𝑡 𝑡 & 12𝑡 / 36𝑡, 𝑡 0, wheret is measure in seconds and s in meters.a) Find the average velocity on the interval [2, 7].b) Find the velocity at time t.c) What is the velocity after 3 seconds?d) When is the particle at rest?e) Find the total distance traveled during the first 8 seconds.f) Find the acceleration at time t.g) Find the acceleration after 3 seconds.10

47) Suppose that a function f and its first derivative have the following values at 𝑥 0 and𝑥 1.𝑥𝑓(𝑥)𝑓′(𝑥)019 3 24Find the first derivative of the following functions at the given values at x. a) 3𝑓 𝑥 , 𝑥 1d),𝑥 1 b) 𝑥𝑓 𝑥 , 𝑥 1c) 𝑥 / 𝑓 𝑥 , 𝑥 1e) 5 -/,𝑥 0f) 𝑓 𝑥 𝑓(𝑥), 𝑥 048) The accompanying figure shows thegraph of the derivative of a function f. Thedomain of f is the closed interval [-3, 3].a) For what values of x in the openinterval (-3, 3) does f have a relativeminimum? Justify your answer.b) For what values of x in the openinterval (-3, 3) does f have a relativemaximum? Justify your answer.c) ** For what values of x is the graph off concave up? Justify your answer.d) ** Suppose 𝑓 3 0. Sketch a possible graph of f on the domain [-3, 3].11