AP^® Calculus

This problem involves vehicles arriving at a toll plaza at a given rate (in vehicles per hour), and asks for: a definite integral representing the total number of vehicles arriving over a time interval, the average rate over that same time interval, and an interpretation of whether that rate is increasing or decreasing at a particular time. Given a condition under which vehicles would accumulate in line, the problem also asks for the greatest number of vehicles in line at any time.

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This problem presents graphs of the functions f and g and asks for: the area of the region enclosed by the graphs, an interpretation of whether the vertical distance between the two graphs is increasing or decreasing at a specific value, the volume of a solid whose base lies in the region, and a calculation of the rate of change of a cross-sectional area moving through the solid over time.

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This problem presents the first derivatives of the x and y coordinate positions of a particle moving along a curve along with the position of the particle at a specific time, and asks for: the slope of a tangent line at a specific time, the speed, and the acceleration vector of the particle at that time as well as the y-coordinate of the particle at another time, and the total distance traveled by the particle over a time interval.

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Particle motion along a coordinate axis (rectilinear motion): Given the velocities and initial positions of two particles moving along the x-axis, this problem asks for positions of the particles and directions of movement of the particles at a later time, as well as calculations of the acceleration of one particle and total distance traveled by the other.

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Particle motion in the coordinate plane: Given the vector-valued velocity and initial position of a particle moving in the coordinate plane, this problem asks for calculations of speed and the acceleration vector at a given time, the total distance traveled over a given time interval, and the coordinates of the particle when it reaches its leftmost position.

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Working with a piecewise (line and circle segments) presented function: Given a function whose graph is made up of connected line segments and pieces of circles, students apply the Fundamental Theorem of Calculus to analyze a function defined by a definite integral of this function.

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Working with a table of velocity values: Given a table of velocity values for a particle moving along a vertical line, students calculate or approximate associated derivative and integral values, interpreting them in the context of the problem (for example; position, acceleration, etc.).

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Working with a graphically presented derivative: Given the graph of a function’s derivative, along with some initial values, students analyze limits, derivatives and integrals. They will also compute the perimeter, area and volume associated with a given region bounded by the graph.

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Working with a table of parametric velocity values: Given a table of derivative values for one parametric coordinate and an algebraic expression for the other parametric coordinate, students calculate and approximate associate values, interpreting them in the context of the problem. Euler’s Method is one approximation technique used, and the tabular values are also used in creating a Taylor polynomial approximation.

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Fish Lake: This problem has fish entering and leaving a lake and poses questions involving accumulation, averages and optimization.

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Velocity table: This problem involves two particles’ motion along the x-axis. Velocities are presented in tabular and algebraic forms with questions about rectilinear motion (position, velocity and acceleration).

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Function graph and FTC: Given the graph of a function f (continuous, defined piecewise by line segments and a circle arc), questions require evaluating derivatives and definite integrals using the graph. Fundamental Theorem of Calculus is used to find the maximum of a related function g.

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Rain barrel: A cylindrical barrel collects rainwater, with questions relating the rates of the water height and volume, and a separable differential equation to solve explicitly for the height as a function of time t.

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Area/Volume: A region R is bounded between two graphs and two vertical lines. Questions include finding the area of R, the volume of a solid with R as base and known cross-sections, and the volume of a solid of revolution generated by rotating R.

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Take it to the limit: Two twice-differentiable functions share a tangent line. Questions on limits, continuity, and derivatives of these and three other related functions are asked, and both L’Hopital’s Rule and Squeeze Theorem are used.

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Polar regions: A polar curve problem involving areas enclosed by the curve as well as subregions defined by intersections with lines and circles. (Average distance also appears in one part.)

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Rational function family: Tangent line, proper and improper integrals arise in these questions about a rational function family with a parameter, k, in the denominator.

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Taylor polynomials and Maclaurin series: Given the graphs of a function f and its tangent line at x=0, as well as some higher derivative values at 0, this problem asks for terms of Taylor polynomials and Maclaurin series of several functions, as well as approximations using them.

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This problem has people on an escalator and poses questions involving accumulation and optimization.

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This problem involves two particles with given velocities moving along a straight line.

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Given the derivative (presented graphically and algebraically), this problem involves the analysis of the original function.

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The tree problem — with heights at selected times presented in a table — involves approximations of both derivatives and integrals and includes a related rates question.

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A function involving a product of exponential and trig functions is presented. Questions include average rate, slope of tangent line and absolute extremum. Problem concludes with a L’Hopital’s rule application.

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This problem involves a separable differential equation along with its slope field.

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This problem involves a research vessel investigating plankton density in the ocean, with questions involving rates of change, accumulation and approximation. There is also a question about the distance traveled by the vessel given its parametric velocity components.

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Polar curve problem involving the area between two polar curves and a tangent to a polar curve. It even includes a related rates problem for a particle moving along a polar curve.

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A power series question: Given the Maclaurin series for a function, one is asked to find the series for a related function, an interval of convergence, and an upper bound on a function value approximation using alternating series bound.

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This problem involves a cross-sectional area, volume, a Riemann sum and related rates.

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This problem involves bananas, rate functions and accumulation.

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