Ap Calc 2013 Frq - AP Calculus AB Free Response.

Craigslist is a platform for selling everything from bikes to cameras to cars. 2 Part b If the vertical line x= hdivides Rinto two regions of equal area, then Z h 0 p xdx= 8 3;or (2) 2 3 h3. The World Health Organization reportedly spent more money last year on traveling around the globe, than they di. Explanation of 2013 AP Calculus AB Free Response Question 2. Teachers may reproduce them, in whole or in part, in limited quantities, for face-to-face. Apartment Therapy suggests tackling a different area of your home every day for 20 minutes and offers. AP® Calculus AB/BC 2021 Scoring Commentary. I have a lot more free problem solving videos on my website for AP Calculus as we. The figure above shows a portion of the graph of f, consisting of two line segments and a quarter of a circle centered at the point (5, 3). Theorem of Calculus to calculate the signed area of the semicircle, ∫4 f ′( x) dx = −2π , and subtracts. The average number of fish that leave the lake per hour from midnight to 5A. After overseeing franchises like 'World of Warcraft' and 'Candy Crush,' Activision COO Daniel Alegre will try to lead Yuga Labs into gaming. trevor casteel chattanooga f is defined on the closed interval. AP Calculus 2010 BC FRQ Solutions Louis A. 119 ) After the store has been open 7 hours, the rate at which bananas are being removed from the display table is decreasing by 8. The response earned 6 points: 1 point in part (a), 3 points in part (b), 2 points in part (c), and no point in part (d). Get in the habit of using parentheses when plugging things in. at 5:16 the audio got corrupted for a couple of seconds so make sure CC is on. This FRQ involves a table of values. Part B (AB): Graphing calculator not allowed Question 6 9 points. 4 x , as shown in the figure above. AP Classroom is a online platform that offers teachers and students access to AP courses, exams, and resources. feet per hour, t is measured in hours, and 0 ≤ t ≤ 8. trout lady twitter video In part (a) the student earned the first point for the definite integral in the first line. 2023 AP Calculus BC Exam Free Response Question #6Taylor Series Problem!Topics: product rule with function notation; evaluating derivatives; constructing Tay. Teaching AP Calculus is challenging but the 48 hours between the exam and the free response release are probably the hardest! We love seeing the new questions and working through them, always amazed at the original ways the College Board incorporates multiple concepts into a single question. Overview In this problem a particle P moves along the x-axis with velocity given by a differentiable function vP, where vP t is measured in meters per hour and t is measured in hours. For Part A, fill in only the circles for numbers 1 through 28 on page 2 of the answer sheet. Here’s the direct link to the 2017 AP Calculus BC free response questions. The difference between −1 2 and (−1) 2 can cost you points. com/playlist?list=PL6iwkLfBjZixEcq …. Unit 12: AP Calculus AB solved free response questions from past exams. harley davidson cvo saddlebag decals Find the volume of the solid generated when R is rotated about the horizontal line y = - 3. We're excited to once again share with you our early solutions to this year's AP Calc AB Free Response Questions. Each spinning toy is in the shape of the solid generated when such a region is. 96 t cubic feet per hour, for 0 ≤ t ≤ 8. Founded in 1900, the College Board is composed of more than 5,700 schools, colleges, universities and other educational organizations. The internal temperature of the potato is 91 degrees Celsius (°C) at time t = 0, and the internal temperature of the potato is greater than 27°C for all times t > 0. Watch as Sal solves free response questions from past AP Calculus exams. We solve all free response questions from the 2023 AP Calculus AB exam. AP Calculus AB 2014 Free Response Question 2. hardest deathrun codes A customer at a gas station is pumping gasoline into a gas tank. Jan 29, 2024 · We can refer to the AP® Student Score Distributions, released by the College Board annually. amber quinn reddit AP Calculus 2013 AB FRQ Solutions Louis A. The sample question packet from the College Board includes some samples of the updated AP Calculus AB free response questions. The graph of f consists of two quarter circles and one line segment, as shown in the figure above. The graph of the function f above consists of three line segments. Within each individual free-response question, at. Emeritus Professor of Mathematics Metropolitan State University of Denver August 9, 2017 1 Problem 1 1. The response earned 7 points: 1 point in part (a), 2 points in part (b), 1 point in part (c), and 3 points in part (d). the multiple-choice questions in this exam. The velocity of particle P at time t is given by v t P () = sin t1. Let R be the region bounded by the x-axis and the graphs of f and g, as shown in the figure above. There are a total of 51 questions. The materials included in these files are intended for use by AP® teachers for course and exam preparation in the classroom; permission for any other use must be sought from the Advanced Placement Program. AP Central is the official online home for the AP Program and Pre-AP: apcentral. The electrical load of a home basically tells you how much electricity your home is using. Here’s the direct link to the 2016 AP Calculus BC free response questions. Use correct notation, language, and mathematical conventions to communicate results or solutions. f and g are shown in the figure above. If you are using assistive technology and need help accessing these PDFs in another format, contact Services for Students with Disabilities at 212-713-8333 or by email at ssd@info. What was the last common ancestor of apes and humans? Learn more about new primate research that could answer the question at HowStuffWorks. 2021 AP Calculus AB and AP Calculus BC Exam Free Response #3The Spinning Toy problem!Full playlist: https://www. The pipe is partially blocked, allowing water to drain out the other end of the pipe at a rate modeled by D ( t ) = − 0. A water tank at Camp Newton holds 1200 gallons of water at time t = hours, water is pumped into the tank at the rate. This chapter contains solutions and notes for the free-response questions from past AP Calculus exams. Let fbe a function with f(1) = 4 such that for all points (x;y) on the graph of fthe slope is given by 3x2+ 1 2y. AP® Calculus AB 2008 Scoring Guidelines The College Board: Connecting Students to College Success The College Board is a not-for-profit membership association whose mission is to connect students to college success and opportunity. The radius of the balloon, in feet, is modeled by a twice-differentiable function r of time t, where t is measured in minutes. The figure above shows the graph of f′, the derivative of a twice-differentiable function f, on the closed interval 0 ≤ x ≤ 8. Welcome to Mathwired! I go over the released 2008 AP Calculus: AB FRQ. AP® Calculus AB 2012 Scoring Guidelines. AP Central is the official online home for the AP Program: apcentral. AP classes prepare learners to take tests on college-level knowledge in 38 subjects. AP Calculus 1998 AB FRQ Solutions Louis A. For f to be decreasing, f ′ must be negative. Derivatives and properties of functions. Bigfoot or an Ancient Ape? - Sasquatch researchers have no conclusive evidence that sasquatches exist. AP Calculus BC Scoring Guide Unit 4 Progress Check: FRQ Part B Copyright © 2017. The College Board is a mission-driven not-for-profit organization that connects students to college success and opportunity. You can leave the answer unsimplified. A graphing calculator is required for some problems or parts of problems. Video tutorial on the AP calculus FRQ question number 1 from 2013. AP Calculus AB 2013 Free Response Question 4. Even if you aren't taking the exam, these are very useful problems for making sure you understand your calculus (as always, best to pause the videos and try them yourself before Sal does. Arkansas School of Mathematics, Sciences and the Arts Prepared by L. Here are the resources you will need for the AP Calculus AB free response questions. AP Calculus BC 2000 Scoring Guidelines. Understanding how many multiple-choice questions and free response points you’ll need to score a 3, 4, or …. The concepts involved in this question are power series, Macla. Because the pass rate of the AP Calculus AB exam is lower than the average pass rate across all AP classes, you could infer AP Calculus AB is more difficult than some others. If the velocity and acceleration have the same sign (both positive or both negative), then speed is increasing. (a) Write an equation for the line tangent to the graph of f at x = 1/2. AP Calculus 1999 AB FRQ Solutions Louis A. Teachers may reproduce them, in whole or in part, in limited quantities, for face-to-face teaching purposes but. It's not just allowed; you absolutely need a calculator to solve those questions. Scoring guidelines and notes contain examples of the most common approaches seen in student responses. These simple yet powerful ideas play a major role in all of calculus. AP® Calculus AB 1999 Scoring Guidelines. Best 2024 ap score calculator with rubric points | fiveable. University of Illinois at Chicago. (b) Write an equation for the line tangent to the graph of fat x= 1 and use it to approximate f(1:2). breaking news moberly mo When you have this condition, your body's immune system makes abnormal pr. AP Classroom also provides feedback, practice, and support for AP exams, as well as video lessons, assignments, and quizzes for various AP subjects. free response problem for AP test. 1 Part a We must first locate the right-hand corner of the region R—that is, the smallest positive value x= afor which 1+sin2x= ex=2. Part B: 15 questions; 45 minutes (graphing. The content of the ap calculus bc exam is pulled straight. For ≤ t ≤ 80, the position of particle P at time t is given by x P (t) = ln(t 2 - 2t + 10 ), while the velocity of particle Q at time t is given by v Q = t 2 - 8t …. This will help you quickly find zeros, extrema, intersection points, etc. Only the solutions are provided. This is a good preparation for your upcoming exam! AP Calculus AB 2017 free response. The area of the cross sections as a function of x was: A = (ln (x+3) - (x 4 + 2x 3))2 Now plugging this into the calculator as a derivative with respect to x would be easy, but we want to derivative to be with respect to TIME. 6 0 ounces, over the time interval 0 ≤ t ≤ 6 minutes. Continuity requires that the behavior of a function around a point matches the function's value at that point. The region R is shown in the figure above. and the nth derivative of f at x — I is given by f (n) (1) It is known that f (1). Note that the AP Calculus AB exam has had small changes made to its format. A few of these free response questions were also on the 2023 AP Calc BC exam. Emeritus Professor of Mathematics Metropolitan State University of Denver. Section I Total Time: 1 hour 45 minutes Number of Questions: 45* Percent of Total Score: 50% Writing. 1 2 : : integral 1 : answer (b) 1 5 50 Lt d t 6. CALCULUS BC 2017 AP ® CALCULUS BC FREE-RESPONSE QUESTIONS SECTION II, Part A Time—30 minutes Number of questions—2. On a certain workday, the rate, in tons per hour, at which unprocessed gravel arrives at a gravel processing plant is modeled by G (t) = 90. 5 By the Mean Value Theorem, there is a value c, 0. If you would like to quickly see all of the course content aligned to a particular standard, the Standards aligned to content page. What makes humans human? How are we different from the other great apes? Find out what makes humans human in this article from HowStuffWorks. The function f is twice differentiable with. Mighty Cable Company manufactures cable that sells for $120 per meter. 1 Part a According to the data in the table, W(15) = 67:9 F, while W(9) = 61:8 F. The AB level is worth the equivalent of one semester of college credit upon successful completion of the AP examination offered in May. Let R be the shaded region bounded by the graph of y = ln x and the line y = x - 2, as shown above. Advertisement Scientists speculated in. Calculating the raw average for the prior seven years allows us to determine a mean score of 2. If you are planning to invest, there’s no better place to look for high-quality growth stocks than the portfolio of Cathie Wood, the founder of Ark Invest. I share my bedroom with at least one person, sometimes two or three depending on whether any children migrate during the night. 2000 yamaha gas golf cart On AP Central, we release the free-response questions for exams. I have a lot of students who are preparing for AP Calculus exams right now, so I thought I'd cover some helpful tips to prepare. com/ye26cc54Stay motivated and keep. independent landlord rentals AP Classroom also provides feedback, practice, and support for AP exams, as well as video lessons, assignments, and quizzes for …. In the figure above, f ( t ) = - 1. Find Bob’s acceleration at time t 5. Fortunately, there are many more official free-response Calc BC questions available, so you'll have a lot to practice with. We have links to the best online AP Calculus practice exams. Let f and g be the functions given by f x ) 1 + sin ( 2 ) x and g x ) e 2. Watch as Thinque Prep's Master Math instructor walks through the AP Calculus AB Sample Questions (2020) and breaks down the questions step by step. AP Calculus AB Multiple Choice 2013 Practice Exam Questions and Solutions Part B Try the free Mathway calculator and problem solver below to practice various math topics. Emeritus Professor of Mathematics Metropolitan State University of Denver May 10, 2016 1 Problem 1 1. Emeritus Professor of Mathematics Metropolitan State University of Denver May 28, 2017 1 Problem 1 1. AP Calculus 2013 BC FRQ Solutions Louis A. 45 Questions | 1 Hour 45 minutes | 50% of Exam Score. By definition, the annual percentage rate (APR) is the percent of your loan balance that you pay per year as a cost of borrowing money. n 1 en second point by correctly evaluating the limit resulting in 1, referencing that 1 0 , and presenting a …. The term going private refers to a company&aposs departure from listing shares on any ex. AP Calc AB Cram Unit 1: Limits and Continuity. Created Date: 5/14/2013 1:10:22 PM. AP® Calculus AB 2009 Free-Response Questions The College Board The College Board is a not-for-profit membership association whose mission is to connect students to college success and opportunity. We love seeing the new questions and working through them, always amazed at the original ways the College Board incorporates multiple concepts into a single question. Part A (AB or BC): Graphing calculator required Question 1 9 points. Numerical solution gives a= 1:13569. AP Calculus BC exams: 2008 1 c&d. Emeritus Professor of Mathematics Metropolitan State University of Denver September 3, 2017 1 Problem 1 1. Similarly, Z 5 2 f0(x)dx= f(5) 7, while the value of this integral is the area of a triangle of base three, altitude two, less the. The area of the horizontal cross section of the tank at height h feet is given by the function A, where A h is measured in square feet. The derivative of r with respect to θ is. Here is a rundown of the format of the AP Calculus AB free-response section: Six questions total; One hour 30 minutes total; Worth 50% of your total. You’re given 30 minutes to complete the first two—Part A. Founded in 1900, the association is composed of more than 5,600 schools, colleges, universities and other educational organizations. AP Calculus 2005 AB (Form B) FRQ Solutions Louis A. t The internal temperature of the potato at time t =3 minutes is approximately 43 91 −16 3 = ⋅ degrees Celsius. AP Calculus BC Scoring Guide Unit 10 Progress Check: FRQ Part A Copyright © 2017. The graph of the function f shown above consists of two line segments and a semicircle. Even a scientific discipline like biology can use calculus to study rates of diffusion etc. Topics: Derivative of a function, tangent line, definition of continuity, u-substitution, definite integral. When you set up your wireless connection, the router or device that enables the connection is called an access point. 931 , where A ( t ) is measured in pounds and t is measured in days. Here are the task verbs you’ll see on the exam: Approximate: Use rounded decimal values or other estimates in calculations, which require writing an expression to show work. The region R is bounded by the. 1 At t = 5, the rate at which gravel arrives at the processing plant is decreasing at the rate of 24. The number of people in line is a minimum at time t = 33. A jewelry company uses a machine to apply a coating of gold on a certain style of necklace. 1997 ap® calculus ab free-response questions calculus ab section ii a graphing calculator is required for some problems or parts of problems on this section of the examination. x Let g ( x ) = 2 x f ( t ) dt. AP Calculus AB Scoring Guidelines from the 2019 Exam Administration Keywords: AP Calculus AB Scoring Guidelines from the 2019 Exam Administration; scoring guidelines; 2019 AP exam administration; teacher resources; exam resources; exam practice; exam scoring information; exam preparation; course prepaÇX Û/HßoÂò ´ Created Date. aphmau my inner demons x reader Students cultivate their understanding of differential and integral calculus through engaging with real-world problems represented graphically, numerically, analytically, and verbally and using definitions and theorems to build arguments and justify conclusions as they explore concepts like change, limits, and the analysis of. Cross sections of the solid taken perpendicular to the x -axis are squares. 1 Part a The area of the region Ris Z 4 0 p xdx= 2 3 x3=2 = 16 3: (1) 1. Emeritus Professor of Mathematics Metropolitan State University of Denver September 19, 2017 1 Problem 1 1. AP® CALCULUS AB/CALCULUS BC 2014 SCORING GUIDELINES. percy and annabeth fanfiction truth or dare 3 the definition u = 4 - x 2 is correct and earned the second point. The response earned 5 points: 2 points in part (a), no points in part (b), 1 point in part (c), and 2 points in part (d). Question 1 (continued) In part (d) the response earned the first point for the correct value of g′ ( 140 ). AP® Calculus AB 2010 Scoring Guidelines The College Board The College Board is a not-for-profit membership association whose mission is to connect students to college success and opportunity. Learn how to make the most of the platform and successfully sell your stuff. A 529 plan offers a way to head off college debt by giving families a tax-advantaged way to save and invest for college, or even pay some costs in advance. q13 bus route map Solving numerically for b, we find that the smallest positive value. 2017 AP ® CALCULUS AB FREE-RESPONSE QUESTIONS CALCULUS AB SECTION II, Part A Time—30 minutes Number of questions—2. Today, the membership association is made up of more than 6,000 of the world’s leading educational. The global point was earned in part (a) with the statement G x f x. From 1995-1999, the calculator could …. compared to physics c, it was a first grader …. 1 6 C ( t ) dt is the average amount of coffee in the cup, in. 1 Part a If G(t) = 90+45cos t2 18; (1) then G0(t) = 5tsin t2 18: (2) Thus G0(5) = 25sin 25 18 tons per hour per hour. For Part B, fill in only the circles for numbers 76 through 92 on page 3 of the answer sheet. AP® CALCULUS AB 2011 SCORING GUIDELINES. Emeritus Professor of Mathematics Metropolitan State University of Denver September 2, 2017 1 Problem 1 1. Solution to the 2013 AP Calculus BC Free-Response Question 5. Right from the start, you will identify the. Choose from the listing below to get started with your AP Calc AB test prep!. The rate, in calories per minute, at which a person using an exercise machine burns calories is modeled by the function f. Just the other day I found myself in the waiting room of an automotiv. Founded in 1900, the association is composed of more than 5,400 schools, colleges, universities, and other educational organizations. Be proficient with the Trace tool on graphs. Answers given as a decimal approximation should be correct to three places after the decimal point. Write your responses to this question only on the designated pages in the separate Free Response. 2014 answers to released free response AP questions 2013 answers to released free response AP questions 2012 answers to free response AP questions Links to what your AP Calculus score will grant you in credit at universities and colleges. A particle moving along a curve in the xy-plane is at position ( x ( t ) , y ( t ) ) at time t 0. Emeritus Professor of Mathematics Metropolitan State University of Denver May xx, 2023 1. (a) Write an equation for the line tangent to the graph of f at x 1. AP® Calculus AB 2010 Free-Response Questions Scoring Statistics Question Mean. Calculus I (MATH 180) 294 Documents. Cleaning your pad is a chore, but it doesn't have to suck up your entire day. AP Calculus AB 2019 Free Response Question 1. Congratulations on your new home! Unsure what to do next? Our 8 essential tips will guide you through the next steps after buying a house. AP® Calculus BC 2002 Scoring Guidelines Form B. Note that the response continues by simplifying and rounding correctly to the boxed answer of 1. Within each individual free-response question, at most one point is not earned for inappropriate rounding. Description: AP Calculus AB - 2013: #5 Link: https://secure-media. High School Math Teacher explains FRQ #4 from the AP Calculus AB & BC Exams!See the entire AP Calculus AB 2023 FRQs here: https://apcentral. Try the free Mathway calculator and problem solver below to practice various math topics. 2022 AP Calculus AB Free Response Questions Louis A. The amount of gold applied to a necklace is approximately normally distributed. Teachers may reproduce them, in whole or in part, in limited quantities for noncommercial, face-to-face. A Walt Disney World annual pass is the cheapest way to visit Disney World theme parks multiple times a year plus discounts! Save money, experience more. lowes fabric Limits describe the behavior of a function as we approach a certain input value, regardless of the function's actual value there. Emeritus Professor of Mathematics Metropolitan State University of Denver May 16, 2023 1. From 1995 to 1999, the calculator could be used on all six free-response questions. Share this page to Google Classroom. Unit 8: AP Calculus practice questions. AP® Calculus AB Scoring Statistics 2017 Free-Response Questions Question Mean. Total Time: 1 hour and 45 minutes Part A: Time: 1 hour. The ap calculus ab exam for 2024 will. 10 − 7 at the beginning of line. Region S is the base of a solid whose cross sections perpendicular to the x-axis are squares. Click here to see the solutions to the 2013 AP Released Free Response questions. AP Calculus 2016 BC FRQ Solutions Louis A. 25 million this week, according to OpenSea data. People enter a line for an escalator at a rate modeled by the function r given by ⎧ ⎪ t t. Emeritus Professor of Mathematics Metropolitan State University of Denver June 1, 2017 1 Problem 1 1. Students cultivate their understanding of differential and integral calculus through engaging with real-world problems represented graphically, numerically, analytically, and verbally and using definitions and theorems to build arguments and justify conclusions as they explore …. For the solid, each cross section perpendicular to the x-axis is a square. 01 Visit the College Board on the Web: www. AP®︎ Calculus AB Standards mappings. Write, but do not evaluate, an expression involving an integral that gives the perimeter of R in terms of k. But the value of this integral is the area of a triangle whose base is four and whose altitude is two, so 7 f( 6) = 4, and f( 6) = 3. Concise resources for the AP College Board Calculus BC course. Score Higher on AP Calculus 2024: FRQ Tips from Students. AP® CALCULUS AB/CALCULUS BC 2015 SCORING GUIDELINES. 1 Part a During the time interval 0 t 300, Z 300 0 r(t)dt= 44 Z 300 0 t 100 3 1 t 300 7 dt= 270 (1) people enter the line for the escalator. According to the model, 270 people enter the line for the escalator during the time interval 0 ≤ t ≤ 300. AP Calculus BC: 2013 #1 ; 2015 AP Calculus AB BC Free Response question 1. com: Title: AP® Calculus AB 2010 Free-Response Questions Scoring Statistics Author: …. The student does not provide an interpretation of this value, so the second point was not earned. 3 AP® CALCULUS AB/CALCULUS BC 2017 SCORING GUIDELINES Question 4 (a) H′(0 )− = 1 4 (91 16−27 )− =H ( )0 =91 An equation for the tangent line is y =91 −16. Walkthrough of the 2023 AP Calculus AB FRQ #6Website: http://www. com/playlist?list=PL3Gnjw2fQSnH_Okp7aPcH71yC8SqHKc-oWebsite: http://www. If you're looking for help in AP Physics 1, check out a new course I've put together here I'm going to put together my own solutions for the 2019 AP Physics 1, AP Physics 2, AP Physics C, AP Calculus AB, AP Calculus BC, and AP Statistics Free Response questions here, …. AP Calculus BC Scoring Guide Unit 3 Progress Check: FRQ Part A Copyright © 2017. Use the menu above to start your test prep with the best AP Calculus BC practice exams, free response questions, notes, videos, and study guides. Updated March 21, 2023 • 5 min rea. The position of the particle at time t = 2 is (1. Let R and S be the regions in the first quadrant shown in the figure above. Each one directs you to complete a specific type of response. All the FRQs are available in a library (for free) at . cost of producing a portion of the cable varies with its distance from the beginning of the cable. Founded in 1900, the association is composed of more than 4,700 schools, colleges, universities, and other. 🌶️ AP Calc Cram Review: Unit 1: Limits and Continuity. 1 Part a We must first find the intersection nearest the y-axis of the curve y= f(x) with the nega-tive x-axis. The functions f and g are differentiable for all real numbers, and g is strictly increasing. Download questions here: https://tinyurl. (c) The rate of change in the number of fish in the lake at time t is given by E(t. While it might seem obvious that you should know how to work your calculator, knowing exactly how and. In part (a) the student presents a correct value for. For 0 ≤ t ≤ 30, the amount of grass clippings remaining in the bin is modeled by A ( t ) = 6. AP® Calculus BC 2003 Scoring Guidelines. AP 2013 Released Free Response Solutions. Time—45 minutes Number of problems—3. Selected derivatives of f at x = 0 are given in the table above. Write, but do not evaluate, an integral expression that can be used to find the volume of the solid generated when R is rotated about the y-axis. Spreadsheets are computer files that have the appearan. As always, your corrections and suggestions are welcome--> e-mail your corrections or suggestions by clicking here. correct to three places after the decimal point. Write an equation of the line tangent to the graph of f at the point (2,–28). Two particles move along the x-axis. I only do AP Calculus, Statistics, and Physics. You can see this year's AB questions here. (b)Use a left Riemann sum with the four subintervals indicated by the table to estimate the total amount of. Show the analysis that leads to your conclusion. On a certain workday, the rate, in tons per hour, at which unprocessed gravel arrives at a gravel processing plant. 2013 AP Calculus AB FRQ 4 YouTube, Check out our complete collection of ap calculus ab practice exams to find all the prep material you need plus helpful study tips. During the second timed portion of the free-response section (Part B), students are permitted to. All released free response questions and answer keys are online right here. The function h is given by h x. FRQ 1: Function Concepts (Calculator). The temperature of water in a tub at time t is modeled by a strictly increasing, twice-differentiable function W, where W t is measured in degrees Fahrenheit and t is measured in minutes. Solution to 2013 AP Calculus AB Free-Response Question 1.