AP Calculus AB Summer Homework - Weebly

AP Calculus AB Summer HomeworkNameDear Future Calculus Student,I hope you are excited for your upcoming year of AP Calculus! This branch of mathematics is extremelyexciting and is unlike any other branch of mathematics you’ve studied thus far. In a nutshell, calculus isdescribed as ‘the mathematics of change’ – how fast things change, how to predict change, and how to useinformation about change to interpret the world around us. As is true with each new branch ofmathematics, calculus takes what you already know a step further.Going into AP Calculus, there are certain algebraic skills that have been taught to you over the previousyears, which we must assume you have. If you do not have these skills, you will find that you willconsistently get problems incorrect next year, even though you understand the calculus concepts. As youcan imagine, it is extremely frustrating for students to be tripped up by the algebra and not the calculus.This summer homework is intended to help you review and/or get reacquainted with the algebraconcepts needed to be successful in calculus.Please submit this packet during your second calculus class period this fall. It will be graded. Workneeds to be shown in a neat and organized manner, and it is perfectly acceptable to complete the packeton separate sheets of paper. Just be sure to staple any extra papers to the packet. Also, do not rely on acalculator. Half of your AP exam next year will be taken without a calculator; paper and penciltechniques only.Best,Mrs. CavicchiNeed help with your Summer math packet?Assistance will be available from 10:30AM to 12PM on August 8th, 10th, and 15th at Wareham HighSchool. Feel free to stop by with any questions you might have. Additionally, you may email yourquestions to Mrs. Cavicchi at kcavicchi@wareham.k12.ma.us. To ensure the fastest response, pleaseinclude your name, summer assignment name, and (if possible) a picture of the problem and youraccompanying work.Directions: Before answering any questions, read through the given notes and examples for each topic.This packet is to be submitted during your second calculus class period.All work must be shown in the packet or on a separate sheet of paper stapled to the packet.To avoid a penalty on your grade, final answers MUST BE BOXED or CIRCLED.1

Part 1 - Functions:Let f(x) 2x 1 and g(x) 2x2 – 11. f(2) 2. g(-3) 3. f(t 1) 4. f[g(-2)] 5. g[f(m 2)] 6. [f(x)]2 – 2g(x) 2

Let f(x) sin(2x). Find each exactly.𝜋2𝜋7. 𝑓( ) 8. 𝑓( ) 43Let 𝑓(𝑥) 𝑥 2 , 𝑔(𝑥) 2𝑥 5, ℎ(𝑥) 𝑥 2 1.9. h[f(-2)] 11.10.f[g(x – 1)] g[h(x3)] 3

Part 2 - Intercepts of a GraphFind the x- and y-intercepts for each of the following.12.y 2x – 513.y x2 x - 214.𝑦 𝑥 16 𝑥 215.𝑦 2 𝑥 3 4𝑥4

Part 3 - Points of IntersectionFind the point(s) of intersection of the graphs for the given equations.16.𝑥 𝑦 8{4𝑥 𝑦 718.𝑥 3 𝑦2{𝑦 𝑥 117.𝑥2 𝑦 6{𝑥 𝑦 45

Part 4 - Domain and RangeFind the domain and range of each function. Write your answer in interval notation.19.𝑓(𝑥) 𝑥 2 520.𝑓(𝑥) 𝑥 321.𝑓(𝑥) 3 sin 𝑥22.𝑓(𝑥) 2𝑥 16

Part 5 - InversesFind the inverse for each function.23.25.𝑓(𝑥) 2𝑥 1𝑔(𝑥) 5𝑥 2𝑥224.𝑓(𝑥) 26.𝑦 4 𝑥 1327.If the graph of 𝑓(𝑥) has the point (2, 7), then what is one point that will beon the graph of 𝑓 1 (𝑥)?28.Explain how the graphs of 𝑓(𝑥) and 𝑓 1 (𝑥) compare.7

Part 6 - Equation of a Line29.Determine the equation of a line passing through the point (5, -3) with anundefined slope.30.Determine the equation of a line passing through the point (-4, 2) with aslope of 0.31.Use point-slope form to find the equation of the line passing through the2point (0, 5) with a slope of .38

32.Use point-slope form to find the equation of the line passing through the5point (2, 8) and parallel to the line 𝑦 𝑥 1.633.Use point-slope form to find a line perpendicular to y -2x 9 passingthrough the point (4, 7).34.Find the equation of the line passing through the points (-3, 6) and (1, 2).35.Find the equation of the line with an x-intercept (2, 0) and a y-intercept(0, 3).9

Part 7 - Unit Circle36.You must have these memorized or know how to calculate their valueswithout the use of a calculator.3𝜋5𝜋𝜋a. sin 𝜋 b. cos d.sin() c.sin ( )22𝜋e. cos 4i. cos2𝜋m. cos4𝜋33𝜋4f. cos( 𝜋) g. cos h. sin( ) 𝜋k. tan 𝜋 l. tan j. tan n. sin(411𝜋6) 3o. tan7𝜋4 5𝜋6𝜋3𝜋p. sin ( ) 610

Part 8 - Trigonometric EquationsSolve each of the equations for 0 𝑥 2𝜋.37.sin 𝑥 1239.4𝑠𝑖𝑛2 (𝑥) 3*Recall 𝑠𝑖𝑛2 (𝑥) (𝑠𝑖𝑛𝑥)2*Recall if x2 25, then x 5.38.2 cos 𝑥 340.2𝑐𝑜𝑠 2 (𝑥) 1 𝑐𝑜𝑠𝑥 0Part 9 - Transformation of Functions41.Given 𝑓(𝑥) 𝑥 2 and 𝑔(𝑥) (𝑥 3)2 1. How does the graph of g(x) differfrom the graph of f(x)?42.Write an equation for the function that has the shape of 𝑓(𝑥) 𝑥 3 , butmoved 6 units to the left and reflected over the x-axis.43.If the ordered pair (2, 4) is on the graph of 𝑓(𝑥), find one ordered pair thatwill be on the following functions:a. f(x) – 3b. f(x – 3)c. 2f(x)d. f(x – 2) 1e. –f(x)11

Part 10 – Vertical AsymptotesFind the vertical asymptote for each of the following problems:44.𝑓(𝑥) 47.𝑓(𝑥) 1𝑥24 𝑥𝑥 2 1645.𝑓(𝑥) 48.𝑓(𝑥) 𝑥2𝑥 2 4𝑥 1𝑥 2 𝑥 246.𝑓(𝑥) 49.𝑓(𝑥) 2 𝑥𝑥 2 (1 𝑥)5𝑥 20𝑥 2 1612

Part 11 – Horizontal AsymptotesDetermine all horizontal asymptotes in the following problems:50.𝑓(𝑥) 53.𝑓(𝑥) 𝑥 2 2𝑥 1𝑥 3 𝑥 7(2𝑥 5)2𝑥 2 𝑥51.𝑓(𝑥) 5𝑥 3 2𝑥 2 84𝑥 3𝑥 3 554. 𝑥52.𝑓(𝑥) 3𝑥 1 𝑥 2 𝑥𝑓(𝑥) 4𝑥 23𝑥 2 7*Remember 𝑥 2 This page is VERY important for our first unit!13

Part 12 – Exponential FunctionsSolve for x:55.33𝑥 5 92𝑥 156.( )𝑥 272𝑥 419157.( )𝑥 2166Part 13 – LogarithmsEvaluate the following logarithms:58.log 7 7 59.log 3 27 60.log 261.log 25 5 62.log 9 1 63.log 4 8 132The statement 𝑦 𝑏 𝑥 can be written as 𝑥 log 𝑏 𝑦.They mean the SAME thing. Remember: Alogarithm is just an exponent! Recall ln 𝑥 log 𝑒 𝑥. The value of e is 2.718281828 or1 𝑥lim (1 𝑥) .𝑥 14

Part 14 – Properties of LogarithmsUse properties of logarithms to evaluate the following:67.ln 𝑒 368.66.log 2 2570.2log2 1071.74.log10 25 log10 476.log 2 ( 2)5𝑒 ln 8log 2 8369.log 3 972.9 ln 𝑒 273.log 9 9375.log 2 40 log 2 5515

Part 15 – Even and Odd FunctionsState whether the following graphs and functions are even, odd, or neither.77.78.79.𝑓(𝑥) 2𝑥 4 5𝑥 280.𝑔(𝑥) 𝑥 5 3𝑥 3 𝑥81.ℎ(𝑥) 2𝑥 2 5𝑥 382.𝑗(𝑥) 2 cos 𝑥83.𝑘(𝑥) sin 𝑥 484.𝑙(𝑥) cos 𝑥 316

Part 16 – Unit CircleFill in the unit circle below with the appropriate exact values (degrees and radians).The Unit Circle17

AP Calculus AB Summer Homework Name _ Dear Future Calculus Student, I hope you are excited for your upcoming year of AP Calculus! . concepts needed to be successful in calculus. Please submit this packet during your second calculus class period this fall. It will be graded. . final answers MUST BE BOXED or CIRCLED. 2 Part 1 - Functions .