We’ve compiled a sortable list of all the AP Calculus past prompts! The AP Calc FRQs are 60% of the exam including 2 long questions and 4 short questions. It’s important that you understand the rubrics and question styles going into the exam. Use this list to practice!

By practicing with previously released free-response questions (FRQs), you’ll build critical-thinking and analytical skills that will prepare you for the exam. These past prompts have been designed to help you connect concepts and ideas to each other while applying your knowledge to real-life scenarios. You’ll also learn how to tackle the exam in a better format, and you won’t be surprised come test day with certain questions.

Find the 2020 exam schedule, learn tips & tricks, and get your frequently asked questions answered on Fiveable’s Guide to the 2020 AP Exam Updates.

### A Note On AB vs. BC

The vast majority of the FRQs on this table are technically from AP Calculus **BC** FRQ documents. However, it is essential to remember that the majority of Calc BC material is AB! There are only 2 BC specific units: Parametric Equations, Polar Coordinates, and Vector-Valued Functions and Infinite Sequences and Series.

**Notes for the 2020 New AP Calc Exam Format**

The AP Calculus AB and BC exam will take place on **May 12th at 2PM Eastern.**

**What units will be covered on the ****AP Calculus AB and BC ****exam? **

The exam will only cover the following topics:

- Unit 1: Limits and Continuity
- Unit 2: Differentiation: Definition and Fundamental Properties
- Unit 3: Differentiation: Composite, Implicit, and Inverse Functions
- Unit 4: Contextual Applications of Differentiation
- Unit 5: Analytical Applications of Differentiation
- Unit 6: Integration and Accumulation of Change
- Unit 7: Differential Equations

For BC Only: All of the above plus:

- Unit 8: Applications of Integration
- 10.2, 10.5, 10.7, 10.8, and 10.11 (See the Course and Exam Description (CED) for specifics)

Note: While the exam won’t be testing on Units 8 (for AB) and 9-10, we strongly recommend that you study the course in full as the content is especially relevant and may still prove useful on the exam! **Click the + to see the prompt in both College Board and plain English!**

Exam Year | Type | Question | Unit Title | Topic | Mean Score | Prompt | Link |
---|---|---|---|---|---|---|---|

2019 | Calculator | 1 | Integration and Accumulation of Change | Accumulation | 3.7/9 | Fish enter a lake at a rate modeled by the function E given byE(t) = 20 + 15 sin( pi*t/6 ). Fish leave the lake at a rate modeled by the function L given by L(t) = 4 2^0.1(t^2). Both E(t) and L(t) are measured in fish per hour, and t is measured in hours since midnight (t = 0). (a) How many fish enter the lake over the 5-hour period from midnight (t = 0) to 5 A.M. (t = 5) ? Give your answer to the nearest whole number. (b) What is the average number of fish that leave the lake per hour over the 5-hour period from midnight (t = 0) to 5 A.M. (t = 5) ? (c) At what time t, for 0 <= t <= 8, is the greatest number of fish in the lake? Justify your answer. (d) Is the rate of change in the number of fish in the lake increasing or decreasing at 5 A.M. (t = 5) ? Explain your reasoning. | PDF Link |

2019 | Calculator | 2 | Contextual Applications of Differentiation | Position, Velocity, Acceleration | 2.83/9 | The velocity of a particle, P, moving along the x-axis is given by the differentiable function vp, where vp(t ) is measured in meters per hour and t is measured in hours. Selected values of vp(t) are shown in the table above. Particle P is at the origin at time t = 0. (a) Justify why there must be at least one time t, for 0.3 <= t <= 2.8, at which vp’(t), the acceleration of particle P, equals 0 meters per hour per hour. (b) Use a trapezoidal sum with the three subintervals [0, 0.3], [0.3, 1.7], and [1.7, 2.8] to approximate the value of the integral from 0 to 2.8 of vp(t) dt . (c) A second particle, Q, also moves along the x-axis so that its velocity for 0 <= t <= 4 is given by vq(t) = 45sqrt(t)cos(0.063t^2) meters per hour. Find the time interval during which the velocity of particle Q is at least 60 meters per hour. Find the distance traveled by particle Q during the interval when the velocity of particle Q is at least 60 meters per hour. (d) At time t = 0, particle Q is at position x = −90. Using the result from part (b) and the function vQ from part (c), approximate the distance between particles P and Q at time t = 2.8. | PDF Link |

2019 | Non-Calculator | 3 | Integration and Accumulation of Change | Fundamental Theorem of Calculus, | 2.70/9 | The continuous function f is defined on the closed interval −6 <= x <= 5. 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). It is known that the point (3, 3 − 5 ) is on the graph of f. If the integral from -6 to 5 of f(x) dx = 7 , find the value of integral from -6 to -2 of f(x) dx. Show the work that leads to your answer. Evaluate the integral from 3 to 5 of 2f’(x) + 4 dx. The function g is given by g(x) = the integral from -2 to x of f(t) dt. Find the absolute minimum value of g on the interval -2 <= x <= 5. Justify your answer. Find lim x→1 (10^x - 3f’(x)/f(x)-arctan(x)) | PDF Link |

2019 | Non-Calculator | 4 | Contextual Applications of Differentiation, Differential Equations | Related Rates, separable differential equations | 1.81/9 | A cylindrical barrel with a diameter of 2 feet contains collected rainwater, as shown in the figure above. The water drains out through a valve (not shown) at the bottom of the barrel. The rate of change of the height h of the water in the barrel with respect to time t is modeled by dh/dt = -1/10 * sqrt(h), where h is measured in feet and t is measured in seconds. (The volume V of a cylinder with radius r and height h is V = pir^2h.) (a) Find the rate of change of the volume of water in the barrel with respect to time when the height of the water is 4 feet. Indicate units of measure. (b) When the height of the water is 3 feet, is the rate of change of the height of the water with respect to time increasing or decreasing? Explain your reasoning. (c) At time t = 0 seconds, the height of the water is 5 feet. Use separation of variables to find an expression for h in terms of t. | PDF Link |

2019 | Non-Calculator | 5 | Applications of Integration | Volumes of revolution, area between two curves | 3.21/9 | Let R be the region enclosed by the graphs of g(x) = -2 + 3cos(pi/2 * x) and h(x) = 6 - 2(x-1)^2, the y axis, and the vertical line x=2, and shown in the figure above. (a) Find the area of R. (b) Region R is the base of a solid. For the solid, at each x the cross section perpendicular to the x-axis has area A(x) = 1/x+3. Find the volume of the solid. (c) Write, but do not evaluate, an integral expression that gives the volume of the solid generated when R is rotated about the horizontal line y = 6. | PDF Link |

2019 | Non-Calculator | 6 | Analytical applications of differentiation, Limits | Analysis of functions, L'Hopitals Rule, Squeeze Theorem | 2.84/9 | Functions f, g, and h are twice-differentiable functions with g(2) = h(2) = 4. The line y = 4 + 2/3(x-2) is tangent to both g at x = 2 and h at x = 2. (a) Find h’(2) (b) Let a be the function given by a(x) = 3x^3h(x). Write an expression for a’(x). Find a’(2). ©. The function h satisfies h(x) = x^2-4/1-(f(x))^3 for x != 2. It is known that lim x→2 h(x) can be evaluated using L’Hopital’s rule. Use lim x→2 h(x) to find f(2) and f’(2). Show your work. (d) It is known that g(x) <= h(x) for 1 < x < 3. Let k be a function satisfying g(x) <= k(x) <= h(x) for 1 | PDF Link |

2018 | Calculator | 1 | Contextual Applications of Differentiation, Analytical Applications of Differentiation | Extrema, connections between f(x) and f'(x) | 2.82/9 | (a) How many people enter the line for the escalator during the time interval 0 <= t <= 300? (b) During the time interval 0 <= t <= 300, there are always people in line for the escalator. How many people are in line at time t = 300? (c) For t>300, what is the first time t what there are no people in line for the escalator? (d) For 0 <= t <= 300, at what time t is the number of people in line a minimum? To the nearest whole number, find the number of people in line at this time. Justify | PDF Link |

2018 | Calculator | 2 | Contextual Applications of Differentiation, Applications of Integration | Position, Velocity, Acceleration | 4.21/9 | A particle moves along the x-axis with v(t) = 10sin(0.4t^2)/(t^2-t+3) for time 0<=t<=3.5. The particle is at position x=-5 at time t=0. (a) Find the acceleration of the particle at time t=3 (b) Find the position of the particle at time t=3 (c) Evaluate the integral from 0 to 3.5 of v(t) dt and the integral from 0 to 3.5 of |v(t)| dt. Interpret the meaning of each in the context of the problem (d) A second particle moves along the x-axis with position given by x2(t) = t^2 - t for 0<=t<=3.5. At what time t are the two particles moving with the same velocity? | PDF Link |

2018 | Non-Calculator | 3 | Analytical applications of differentiation, Integration and Accumulation of Change | Integration, Concavity, Increasing and Decreasing Functions | 3.36/9 | (a) Find f(1) = 3, what is the value of f(-5)? (b) Evaluate the integral from 1 to 6 of g(x) dx (c) From -5 < x < 6 on open intervals, if any, is the graph of f both increasing and concave up? Give a reason for your answer. (d). Find the x coordinate of each point of inflection on the graph of f. Give a reason | PDF Link |

2018 | Non-Calculator | 4 | Integration and Accumulation of Change, Analytical Applications of Differentiation, Contextual Applications of Differentiation | Related Rates, Trapezoidal Sum, Mean Value Theorem | 2.77/9 | (a) Use the data in the table to estimate H'(6). Using correct units, interpret the meaning of H'(6) in the context of the problem (b) Explain why there must be at least one time t, for 2 < t < 10, such that H'(t) = 2 (c) Use a trapezoidal sum with 4 subintervals indicated by the data to approximate the average height of the tree over time time interval 2<=t<=10. (d) The height of the tree can also be modeled by G, given by G(x) = 100x/1+x, where x is the diameter of the base in meters. When the tree is 50 meters tall, the diameter of the base is increasing at a rate of 0.03 meters per year. According to this model, what is the rate of change of the height of the tree with respect to time, in meters per year, at the time when the tree is 50 meters tall? | PDF Link |

2018 | Non-Calculator | 5 | Applications of Integration, Differentiation | Average rate of change, absolute extrema, limits | 2.59/9 | Let f be defined by f(x) = e^x cos(x). (a) Find the average rate of change of f on the interval 0 <= x <= pi (b) Whe is the slope of the line tangent to the graph of f at x = 3pi/2? (c) Find the absolute minimum of the function on the interval [0, 2pi]. Justify (d) Let g be a differentiable function such that g(pi/2) = 0. The graph of g' is shown below. Find the value of lin x-->pi/2 f(x)/g(x) or state that it does not exis. Justify. | PDF Link |

2018 | Non-Calculator | 6 | Differential Equations | Slope Fields, Separable Differential Equations | 3.65/9 | (a) A slope field for the given diff. equation is shown bellow. Sketch the solution curve that passes through (0, 2) and the solution curve that passes through (1, 0). (b) Let y=f(x) be the particular solution to the given diff eq with the initial condition f(1) = 0. Write an equation for the line tangent to the graph of y=f(x) at x=1. Use this equation to approximate f(0.7) (c) Find the particular solution y=f(x) to the given differential equation with initial condition f(1) = 0 | PDF Link |

2017 | Calculator | 1 | Integration and Accumulation of Change, Applications of Integration, Contextual Applications of Differentiation | Riemann Sum, Cross Section Volumes, Related Rates | 3.32/9 | A tank has a height of 10 feet. 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 function A is continuous and decreases as h increases. Selected values for A(h) are given in the table above. (a) Use a left Riemann sum with the three subintervals indicated by the data in the table to approximate the volume of the tank. Indicate units of measure. (b) Does the approximation in part (a) overestimate or underestimate the volume of the tank? Explain your reasoning. (c) The area, in square feet, of the horizontal cross section at height h feet is modeled by the function f given by f(h) = 50.3/e^0.2h + h, based on this model, find the volume of the tank. Indicate units of measure. (d) Water is pumped into the tank. When the height of the water is 5 feet, the height is increasing at the rate of 0.26 foot per minute. Using the model from part (c), find the rate at which the volume of water is changing with respect to time when the height of the water is 5 feet. Indicate units of measure. | PDF Link |

2017 | Calculator | 2 | Integration and Accumulation, Contextual Applications of Differentiation | Adding and subtracting rates, accumulation of change | 4.01/9 | (a) How many pounds of bananas are removed from the display table during the first 2 hours the store is open? (b) Find f'(7) . Using correct units, explain the meaning of f'(7) in the context of the problem. (c) Is the number of pounds of bananas on the display table increasing or decreasing at time t = 5 ? Give a reason for your answer. (d) How many pounds of bananas are on the display table at time t = 8 ? | PDF Link |

2017 | Non-Calculator | 3 | Analytical Applications of Differentiation | Extrema, increasing and decreasing functions | 3.63/9 | The function f is differentiable on the closed interval [−6, 5] and satisfies f (-2) = 7. The graph of f' consists of a semicircle and three line segments. (a) Find f(-6) and f(5) (b) On what intervals is f increasing? Justify (c) Find the absolute minimum value of f on the closed interval [-6, 5]. Justify (d) For each of f''(-5) and f''(3), find the value of explain why it doesn't exist | PDF Link |

2017 | Non-Calculator | 4 | Differential Equations, Analytical Applications of Differentiation | Separable differential equations, concavity, linear approximation | 1.54/9 | At time t = 0, a boiled potato is taken from a pot on a stove and left to cool in a kitchen. The internal temperature of the potato is 91 degrees Celsius (deg C) at time t = 0, and the internal temperature of the potato is greater than 27 C for all times t > 0. The internal temperature of the potato at time t minutes can be modeled by the function H that satisfies the differential equation dH/dt = -1/4(H-27), where H(t) is measured in deg. C and H(0) = 91. (a) Write an equation for the line tangent to the graph of H at t = 0. Use this equation to approximate the internal temperature of the potato at time t = 3. (b) Use d^2H/dt^2 to determine whether your answer in part (a) is an underestimate or an overestimate of the internal temperature of the potato at time t = 3. (c) For t < 10, an alternate model for the internal temperature of the potato at time t minutes is the function G that satisfies the differential equation dG/dt = -(G-27)^2/3, where G(t) is measured in degrees Celsius and G(0) = 91. Find an expression for G(t). Based on this model, what is the internal temperature of the potato at t=3? | PDF Link |

2017 | Non-Calculator | 5 | Contextual Applications of Differentiation | Position, Velocity, Acceleration | 3.59/9 | Two particles move along the x-axis for 0<=t<=8, the position of particle P at time t is given by xp(t) = ln(t^2 - 2t + 10), while the velocity of particle Q at time t is given by vQ(t) = t^2-8t+15. Particle Q is at position x=5 at time t=0. (a) For 0<=t<=8, when is particle P moving to the left? (b) For 0<=t<=8, find all times t where the two particles travel in the same direction (c) Find the acceleration of Q at t=2. Is the speed of the particle Q increasing decreasing or neither at time t=2? Justify (d) Find the position of particle Q the first time it changes direction | PDF Link |

2017 | Non-Calculator | 6 | Differentiation | Chain rule, product rule, Mean value theorem | 3.32/9 | (a) Find the slope of the line tangent to the graph of f at x = pi. (b) Let k be the function defined by k(x) = h(f(x)). Find k'(pi). (c) Let m be the function defined by m(x) = g(-2x) * h(x). Find m'(2). (d) Is there a number c in the closed interval [-5, 3] such that g'(c) = -4? Justify your answer. | PDF Link |

2016 | Calculator | 1 | Contextual Applications of Differentation, Integration and Accumulation | Riemann Sums, Accumulation of Change, adding and removing functions | 5.70/9 | (a) Estimate R'(2). Show your work and indicate units (b) Use a left Riemann sum with four subintervals to estimate the total amount of water removed from the rank. Is this an over or underestimate? Explain (c) Use your answer from part (b) to find an estimate of the total amount of water in the tank, to the nearest liter, at the end of 8 hours (d) For 0 <= t <= 8, is there a time t where the rate at which water is pumped into the tank at the same rate it is removed? Explain why or why not | PDF Link |

2016 | Calculator | 2 | Contextual Applications of Differentiation, Parametric Equations, Polar Coordinates, and Vector-Valued Functions | Position, Velocity, Acceleration, Parametric functions | 4.14/9 | At time t, the position of a particle moving in the xy-plane is given by the parametric functions (x(t), y(t)) where dx/dt = t^2 + sin(3t^2). The graph of y is shown. At t=0, the particle is at position (5, 1). (a) Find the position of the particle at t=3 (b) Find the slope of the line tangent to the path of the particle at t=3 (c) Find the speed of the particle at t=3 (d) Find the total distance traveled by the particle from t = 0 to t = 2 | PDF Link |

2016 | Non-Calculator | 3 | Integration, Analytical Applications of Differentiation | Fundamental Theorem of Calculus, Extrema, Increasing and Decreasing | 4.15/9 | (a) Does g have a relative minimum, relative maximum, or neither at x=10? Justify your answer (b) Does the graph of g have a point of inflection at x=4? Justify (c) Find the absolute minimum and absolute maximum value of g on the interval -4 <= x <= 12. Justify (d). For -4 <= x <= 12, find all intervals for which g(x) <= 0 | PDF Link |

2016 | Non-Calculator | 4 | Differential Equations | Euler's Method, Separable differential equations | 4.68/9 | Consider the differential equation dy/dx = x^2 - 1/2y (a) Find d^2y/dx^2 in terms of x and y (b) Let y = f(x) by the particular solution to the given differential equation whose graph passes through the point (-2, 8). Does the graph of f have a relative minimum, relative maximum, or neither at the point (-2, 8)? Justify your answer (c) Let y = g(x) be the particular solution to the given diff eq with g(-1) = 2. Find lim x-->1 (g(x)-2/3(x+1)^2). Show your work (d) Let y = h(x) be the particular solution to the given diff eq with h(0) = 2. Use Euler's method, starting at x = 0 with 2 steps of equal size to approximate h(1) | PDF Link |

2016 | Non-Calculator | 5 | Applications of Integration, contextual applications of differentiation | Average Value, related rates | 4.19/9 | (a) Find the average value of the radius of the funnel (b) Find the volume of the funnel (c) The funnel contains liquid that is draining from the bottom. At the instant when the height of the liquid is h=3 inches, the radius of the surface of the liquid is decreasing at a rate of 0.2 inch/second. At this instant, what is the rate of change of the height of the liquid with respect to time? | PDF Link |

2016 | Non-Calculator | 6 | Infinite Sequences and Series | Taylor Series | 3.82/9 | The function f has a taylor series about x = 1 that converges to f(x) for all x in the interval of convergence. It is known that f(1) = 1, f'(1) = 1/2, and the nth derivative of f at x=1 is given by f^(n)(1) = (-1)^n * (n-1)!/2^n for n>=2. (a) Write the first four nonzero terms for the taylor series (b) The taylor series about x = 1 has a radius of convergence of 2. Find the interval of convergence and show your work (c) The Taylor series for f about x = 1 can be used to represent f(1.2) as an alternating series. Use the first 3 nonzero terms of the alternating series to approximate f(1.2) (d) Show that the approximation found in part (c) is within 0.001 of the exact value of f(1.2) | PDF Link |

2015 | Calculator | 1 | Integration and Accumulation, Application of Integration | Accumulation, extrema | 5.10/9 | The rate at which rainwater flows into a drainpipe is modeled by the function R, where R(t) = 20sin(t^2/35) cubic feet per hour, t is measured in hours, and 0 <= t <= 8. The pipe is partially blocked, allowing water to drain out the other end of the pipe at a rate modeled by D(t) = -0.04t^3 + 0.4t^2 + 0.96t cubic feet per hour for 0 <= t <= 8. There are 30 cubic feet of water in the pipe at time t = 0. (a) How many cubic feet of rainwater flow into the pipe during the 8-hour time interval 0 <= t <= 8? (b) Is the amount of water in the pipe increasing or decreasing at time t = 3 hours? Give a reason for your answer. (c) At what time t, 0 <= t <= 8, is the amount of water in the pipe at a minimum? Justify your answer. (d) The pipe can hold 50 cubic feet of water before overflowing. For t > 8, water continues to flow into and out of the pipe at the given rates until the pipe begins to overflow. Write, but do not solve, an equation involving one or more integrals that gives the time w when the pipe will begin to overflow. | PDF Link |

2015 | Calculator | 2 | Parametric Equations, Polar Coordinates, and Vector-Valued Functions | Position, Velocity, Acceleration, Parametric functions | 4.87/9 | At time t >= 0, a particle moving along a curve in the xy-plane has position (x(t), y(t)) with velocity vector v(t) = (cos(t^2), e^0.5t). At t=1, the particle is at the point (3, 5). (a) Find the x-coordinate of the position of the particle at time t = 2. (b) For 0 < t <1, there is a point on the curve at which the line tangent to the curve has a slope of 2. At what time is the object at that point? (c) Find the time at which the speed of the particle is 3. (d) Find the total distance traveled by the particle from time t = 0 to time t = 1. | PDF Link |

2015 | Non-Calculator | 3 | Applications of Integration, Applications of Differentiation | Position, Velocity, Acceleration | 5.71/9 | Johanna jogs along a straight path. For 0<=t<=40, Johanna’s velocity is given by a differentiable function v. Selected values of v(t) , where t is measured in minutes and v(t) is measured in meters per minute, are given in the table above. (a) Use the data to estimate v'(16) (b) Using correct units, explain the meaning of the definite integral from 0 to 40 of |v(t)| dt in context of the problem. Approximate using a right riemann sum (c) Bob is riding his bicycle along the same path. For 0 <= t <= 10, Bob’s velocity is modeled by B(t) = t^3 - 6t^2 + 300, where t is measured in minutes and B(t) is measured in meters per minute. Find Bob's acceleration at time t = 5. (d) Based on the model B from part (c), find Bob's average velocity during the interval 0 <= t <= 10. | PDF Link |

2015 | Non-Calculator | 4 | Differential Equations | Slope Fields, Separable Differential Equations | 4.40/9 | Consider the differential equation dy/dx = 2x - y (a) On the axes, sketch a slope field for the differential equation (b) Find d^2y/dx^2 and determine the concavity of all solution curves for the given differential equation in Quadrant II. Give a reason (c) Let y = f(x) be the particular solution to the differential equation with the initial condition f(2) = 3. Does f have a relative minimum, a relative maximum, or neither at x = 2 ? Justify your answer. | PDF Link |

2015 | Non-Calculator | 5 | Integration and Accumulation, Analytical Applications of Differentiation | Partial Fraction Decomposition, Tangent lines, extrema | 6.21/9 | Consider the function f(x) = 1/(x^2 - kx) where k is a non-zero constant. The derivative of f is given by f'(x) = k-2x/(x^2 - kx)^2 (a) Let k=3. Write an equation for the line tangent to the graph of f at the point whose x-coordinate is 4 (b) Let k=4. Determine whether f has a relative minimum, relative maximum, or neither at x = 2. Justify (c) Find the value of k for which f has a critical point at x = -5. (d) Let k = 6. Find the partial fraction decomposition and find the antiderivative for f(x) with respect to x | PDF Link |

2015 | Non-Calculator | 6 | Infinite Sequences and Series | Maclaurin Series | 3.96/9 | The Maclaurin series for a function f is given by the sum from n = 1 to infinity of (-3)^n-1/n and converges to f(x) for |x| < R where R is the radieus of convergence (a) Use the ratio test to find R (b) Write the first four nonzero terms of the Maclaurin series for f'. Express f' as a rational function for |x| < R (c) Write the first four nonzero terms of the Maclaurin series for e^x. Use the Maclaurin series for e^x to write the third-degree Taylor polynomial for g(x) = e^x f(x) about x = 0. | PDF Link |

2014 | Calculator | 1 | Contextual Applications of Differentiation | Linearization, interpreting rates of change | 4.26/9 | Grass clippings are placed in a bin, where they decompose. For 0 <= t <=30 , the amount of grass clippings remaining in the bin is modeled by A(t) = 6.687(0.931)^t, where A(t) is measured in pounds and t is measured in days. (a) Find the average rate of change of A(t) over the interval 0 <= t <=30. Indicate units of measure. (b) Find the value of A'(15). Using correct units, interpret the meaning of the value in the context of the problem (c) Find the time t for which the amount of grass clippings in the bin is equal to the average amount of grass clippings in the bin over the interval 0 <= t <= 30. For t > 30, L(t), the linear approximation to A at t = 30, is a better model for the amount of grass clippings remaining in the bin. Use L(t) to predict the time at which there will be 0.5 pound of grass clippings remaining in the bin. Show the work that leads to your answer. | PDF Link |

2014 | Calculator | 2 | Parametric Equations, Polar Coordinates, and Vector-Valued Functions | Polar Coordinates, integrating and differentiating over polar coordinates | 3.95/9 | The graphs of the polar curves r = 3 and 3 = 3 - 2sin(2*theta) are shown in the figure about for 0<=theta<=pi. (a) Let R be the shaded region that is inside the graph of r = 3 and inside the graph of r = 3 - 2sin(2*theta). Find the area of R. (b) The distance between the two curves changes for 0 | PDF Link |

2014 | Non-Calculator | 3 | Integration and Accumulation, Analytical Applications of Differentation | Fundamental Theorem of Calculus, Integration, Tangent Lines, Concavity | 5.24/9 | The function f is defined on the closed interval [-5, 4]. The graph of f consists of three line segments and is shown in the figure above. Let g be the function defined by the integral from -3 to x of f(t) dt. (a) Find g(3). (b) On what open intervals contained in -5 < x < 4 is the graph of g both increasing and concave down? Give a reason for your answer (c) The function h is defined by h(x) = g(x)/5x. Find h'(3). (d) The function p is defined by p(x) = f(x^2 - x). Find the slope of the line tangent to the graph of p at the point where x = -1 | PDF Link |

2014 | Non-Calculator | 4 | Integration, applications of differentiation | Position, Velocity, Acceleration, trapezoidal sum | 3.97/9 | Train A runs back and forth on an east-west section of railroad track. Train A’s velocity, measured in meters per minute, is given by a differentiable function Va(t) where time t is measured in minutes. Selected values for Va(t) are given in the table above. (a) Find the average acceleration of train A over the interval 2 ≤ t ≤ 8 (b) Do the data in the table support the conclusion that train A’s velocity is −100 meters per minute at some time t with 5 < t < 8? Give a reason for your answer. (c) At time t = 2, train A’s position is 300 meters east of the Origin Station, and the train is moving to the east. Write an expression involving an integral that gives the position of train A, in meters from the Origin Station, at time t = 12. Use a trapezoidal sum with three subintervals indicated by the table to approximate the position of the train at time t = 12. (d) A second train, train B, travels north from the Origin Station. At time t the velocity of train B is given by vB(t) = -5t^2 + 60t + 25 and at time t = 2 the train is 400 meters north of the station. Find the rate, in meters per minute, at which the distance between train A and train B is changing at time t = 2. | PDF Link |

2014 | Non-Calculator | 5 | Applications of Integration | Area between 2 curves, volume of revolution, perimeter | 4.45/9 | Let R be the shaded region bounded by the graph of y = xe^x^2, the time y = -2x, and the vertical line x = 1. (a) Find the area of R. (b) Write, but do not evaluate, an integral expression that gives the volume of the solid generated when R is rotated about the horizontal line y = −2. (c) Write, but do not evaluate, an expression involving one or more integrals that gives the perimeter of R. | PDF Link |

2014 | Non-Calculator | 6 | Infinite Sequences and Series | Taylor Series | 3.10/9 | The Taylor series for a function f about x = 1 is given by the sum from 1 to infinity of (-1)^n+1 * 2^n/n * (x-1)^n and converges to f(x) for |x+1| < R, where R is the radius of convergence. (a) Find the value of R. (b) Find the first three nonzero terms and the general term of the Taylor series for f ′, the derivative of f, about x = 1. (c) The Taylor series for f ′ about x = 1, found in part (b), is a geometric series. Find the function f ′ to which the series converges for |x+1| < R . Use this function to determine f for |x+1| < R. | PDF Link |

2013 | Calculator | 1 | Integration and Accumulation of Change, Analytical Applications of Differentiation, Contextual Applications of Differentiation | Integration, fundamental theorem of calculus, Increasing or Decreasing functions w/ 2 rates | 2.57/9 | 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 + 45cos(t^2/18), where t is measured in hours and 0 ≤ t ≤ 8 At the beginning of the workday, t = 0 , the plant has 500 tons of unprocessed gravel. During the hours of operation, 0 ≤ t ≤ 8 the plant processes gravel at a constant rate of 100 tons per hour (a) Find G′(5). Using correct units, interpret your answer in the context of the problem. (b) Find the total amount of unprocessed gravel that arrives at the plant during the hours of operation on this workday. (c) Is the amount of unprocessed gravel at the plant increasing or decreasing at time t = 5 hours? Show the work that leads to your answer. (d) What is the maximum amount of unprocessed gravel at the plant during the hours of operation on this workday? Justify your answer. | PDF Link |

2013 | Calculator | 2 | Parametric Equations, Polar Coordinates, and Vector-Valued Functions | Polar coordinates, Position, Velocity, Acceleration | 2.55/9 | The graphs of the polar curves r = 3 and r = − 4 2sin θ are shown in the figure above The curves intersect when θ = π/6 and θ = 5π/6. (a) Let S be the shaded region that is inside the graph of r = 3 and also inside the graph of r = 4 − 2sin . θ Find the area of S. (b) A particle moves along the polar curve r = 4 − 2sin θ so that at time t seconds, θ = t^2. Find the time t in the interval 1 <= t <= 2 for which the x-coordinate of the particle's position is -1. (c) For the particle described in part (b), find the position vector in terms of t. Find the velocity vector at time t = 1.5. | PDF Link |

2013 | Non-Calculator | 3 | Contextual Applications of Differentation, Integration and Accumulation | Approximating derivatives, MVT, average value, derivatives | 3.65/9 | Hot water is dripping through a coffeemaker, filling a large cup with coffee. The amount of coffee in the cup at time t, 0 <= t <= 6, is given by a differentiable function C, where t is measured in minutes (a) Use the data in the table to approximate C'(3.5). Show the computations that lead to your answer, and indicate units of measure. Is there a time t, 2 <= t <= 4, at which C'(t) = 2? Justify. (c) Use a midpoint sum with three subintervals of equal length indicated by the data in the table to approximate the value of 1/6 * the integral from 0 to 6 of C(t) dt. Interpret with correct units (d) The amount of coffee in the cup, in ounces, is modeled by B(t) = 16 - 16e^-0.4t. Using this model, find the rate at which the amount of coffee in the cup is changing when t = 5 | PDF Link |

2013 | Non-Calculator | 4 | Analytical applications of differentiation | Extrema, extreme value theorem, tangent lines | 2.62/9 | The figure above shows the graph of f ′, the derivative of a twice-differentiable function f, on the closed interval 0 <= x <= 8. The graph of f ′ has horizontal tangent lines at x = 1, x = 3, and x = 5. The areas of the regions between the graph of f ′ and the x-axis are labeled in the figure. The function f is defined for all real numbers and satisfies f(8) = 4 (a) Find all values of x on the open interval 0 < x< 8 for which the function f has a local minimum. Justify your answer (b) Determine the absolute minimum value of f on the closed interval 0 <= x <= 8. Justify your answer. (c) The function g is defined by g(x) = (f(x))^3. If f(3) = -5/2, find the slope of the line tangent to the graph of g at x = 3 | PDF Link |

2013 | Non-Calculator | 5 | Differential Equations | L'Hopitals, Euler's Method, Separable DiffEqs | 4.14/9 | Consider the differential equation dy/dx = y^2(2x+2). Let y = f(x) be the particular solution to the differential equation with initial condition f(0) = -1. Find lim x-->0 f(x)+1/sinx. Show your work (b) Use euler's method, starting at x = 0 with 2 steps of equal size, to approximate f(1/2) (c) Find the particular solution with initial condition f(0) = -1 | PDF Link |

2013 | Non-Calculator | 6 | Infinite Sequences and Series | Taylor series | 3.17/9 | A function f has derivatives of all orders at x = 0. Let Pn(x) denote the nth-degree Taylor polynomial for f about x = 0. (a) It is known that f(0) = -4 and that P1(1/2) = -3. Show that f'(0) = 2. (b) It is known that f''(0) = -2/3 and f'''(0) = 1/3. Find P3(x). The function h has first derivative given by h'(x) = f(2x) It is known that h(0) = 7. Find the third-degree Taylor polynomial for h about x = 0. | PDF Link |

2012 | Calculator | 1 | Applications of Integration, Differentiation, Integration and Accumulation | Integrals of a derivative, Average value | 5.88/9 | 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. At time t = 0, the temperature of the water is 55°F. The water is heated for 30 minutes, beginning at time t = 0. Values of W(t) at selected times for t for the first 20 minutes are given in the table above (a) Use the data in the table to estimate W′(12). Show the computations that lead to your answer. Using correct units, interpret the meaning of your answer in the context of this problem. (b) Use the data in the table to evaluate the integral from 0 to 20 of W'(t) dt. Interpret with correct units (c) For 0 <= t <= 20, the average temperature of the water in the tub is 1/20 * the integral from 0 to 20 of W(t) dt. Use a left Riemann sum with the four subintervals indicated by the data in the table to approximate this integral. Does this approximation overestimate or underestimate the average temperature of the water over these 20 minutes? Explain your reasoning. For 20 <= t <= 25, the function W that models the water temperature has first derivative given by W'(t) = 0.4 * sqrt(t) * cos(0.06t) Based on the model, what is the temperature of the water at time t = 25 ? | PDF Link |

2012 | Calculator | 2 | Parametric Equations, Polar Coordinates, and Vector-Valued Functions | Parametric functions, Position, velocity, acceleration | 5.07/9 | For t ≥ 0, a particle is moving along a curve so that its position at time t is (x(t), y(t)). At time t = 2, the particle is at position (1, 5). It is known that dx/dt = sqrt(t+2)/e^t and dy/dt = sin^2(t). (a) Is the horizontal movement of the particle to the left or to the right at time t = 2 ? Explain your answer. Find the slope of the path of the particle at time t = 2. (b) Find the x-coordinate of the particle’s position at time t = 4. (c) Find the speed of the particle at time t = 4. Find the acceleration vector of the particle at time t = 4. (d) Find the distance traveled by the particle from time t = 2 to t = 4. | PDF Link |

2012 | Non-Calculator | 3 | Integration and accumulation | Fundamental Theorem of Calculus | 4.29/9 | Let f be the continuous function defined on [−4, 3] whose graph, consisting of three line segments and a semicircle centered at the origin, is given above. Let g be the function given by g(x) = the integral from 1 to x of f(t) dt. (a) Find the values of g(2) and g(−2 ) . (b) For each of g′(−3) and g′′(−3) , find the value or state that it does not exist. (c) Find the x-coordinate of each point at which the graph of g has a horizontal tangent line. For each of these points, determine whether g has a relative minimum, relative maximum, or neither a minimum nor a maximum at the point. Justify your answers. (d) For −4 < x < 3, find all values of x for which the graph of g has a point of inflection. Explain your reasoning | PDF Link |

2012 | Non-Calculator | 4 | Infinite Sequences and Series, Differential Equations, Applications of Integration | Euler's Method, integrating derivatives, taylor series | 3.24/5 | The function f is twice differentiable for x > 0 with f(1) = 15 and f''(1) = 20. Values of f' are given for selected values of x. (a) Write an equation for the line tangent to the graph of f at x = 1. Use this line to approximate f(1.4). (b) Use a midpoint Riemann sum with two subintervals of equal length and values from the table to approximate the integral from 1 to 1.4 of f'(x) dx. Use this approximation to estimate the value of f(1.4). (c) Use Euler’s method, starting at x = 1 with two steps of equal size, to approximate f (1.4). Show the computations that lead to your answer. (d) Write the second-degree Taylor polynomial for f about x = 1. Use the Taylor polynomial to approximate f(1.4). | PDF Link |

2012 | Non-Calculator | 5 | Differential Equations, Analytical Applications of Differentiation | Separable differential equations, concavity | 2.19/4 | The rate at which a baby bird gains weight is proportional to the difference between its adult weight and its current weight. At time t = 0, when the bird is first weighed, its weight is 20 grams. If B(t) is the weight of the bird, in grams, at time t days after it is first weighed, then dB/dt = 1/5(100 - B). Let y = B(t) be the solution to the differential equation above with initial condition B(0) = 20. (a) Is the bird gaining weight faster when it weighs 40 grams or when it weighs 70 grams? Explain your reasoning. (b) Find d^2B/dt^2 in terms of B. Use d^2B/dt^2 to explain why the graph of B cannot resemble the following graph. (c) Use separation of variables to find y = B(t), the particular solution to the differential equation with initial condition B(0) = 20. | PDF Link |

2012 | Non-Calculator | 6 | Infinite Sequences and Series | Maclaurin Series | 4.75/9 | The function g has derivatives of all orders, and the Maclaurin series for g is the sum from n = 0 to infinity of (-1)^n * (x^2n+1)/(2n+3) = x/3 - x^3/5 + x^5/7... (a) Using the ratio test, determine the interval of convergence of the Maclaurin series for g. (b) The Maclaurin series for g evaluated at x = 1/2 is an alternating series whose terms decrease in absolute value to 0. The approximation for g(1/2) using the first two nonzero terms of this series is 17/120. Show that this approximation differs from g(1/2) by less than 1/200. (c) Write the first three nonzero terms and the general term of the Maclaurin series for g'(x) | PDF Link |

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