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AP Calculus BC Free Response Questions

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  • Unit 1: Limits and Continuity (27)
  • Unit 2: Differentiation: Definition and Fundamental Properties (22)
  • Unit 3: Differentiation: Composite, Implicit, and Inverse Functions (23)
  • Unit 4: Contextual Applications of Differentiation (32)
  • Unit 5: Analytical Applications of Differentiation (29)
  • Unit 6: Integration and Accumulation of Change (33)
  • Unit 7: Differential Equations (23)
  • Unit 8: Applications of Integration (32)
  • Unit 9: Parametric Equations, Polar Coordinates, and Vector-Valued Functions (29)
Unit 1: Limits and Continuity

Analyzing a Function with a Removable Discontinuity

Consider the function $$r(x)=\frac{x^2-9}{x-3}$$ for $$x\neq3$$ and $$r(3)=2.$$ Answer the follow

Easy

Complex Rational Function and Continuity Analysis

Let $$R(x)= \frac{x^3-8}{x-2}$$ for $$x \neq 2$$.

Easy

Continuity Analysis Using a Piecewise Defined Function

Let $$f(x)=\begin{cases} x^2, & x \leq 1 \\ 2x-1, & x> 1 \end{cases}$$.

Easy

Continuity in a Parametric Function Context

A particle moves such that its coordinates are given by the parametric equations: $$x(t)= t^2-4$$ an

Easy

Continuity of a Trigonometric Function Near Zero

Consider the function defined by $$ f(x)= \begin{cases} \frac{\sin(5*x)}{x}, & x \neq 0 \\ L, & x =

Easy

Continuity of Log‐Exponential Function

Consider the function $$f(x)= \frac{e^x - \ln(1+x) - 1}{x}$$ for $$x \neq 0$$, with $$f(0)=c$$. Dete

Easy

End Behavior Analysis of a Rational Function

Consider the function $$f(x)=\frac{2 * x^3 - 5 * x + 1}{x^3+4 * x^2-x}$$. Answer the following:

Extreme

Environmental Pollution Modeling

In a lake, a pollutant is added every year at a constant amount of 5 units. However, due to natural

Medium

Epsilon-Delta Style (Conceptual) Analysis

Consider the function $$f(x)=\begin{cases} 3*x+2, & x\neq1, \\ 6, & x=1. \end{cases}$$ Answer the

Medium

Establishing Continuity in a Piecewise Function

Consider the piecewise-defined function $$p(x)= \begin{cases} \frac{x^2-4}{x-2} & x \neq 2, \\ k & x

Easy

Evaluating a Complex Limit for Continuous Extension

Consider the function $$ f(x)= \begin{cases} \frac{\ln(1+x+e^x) - (x+e^x-1)}{x^2}, & x \neq 0 \\ C,

Hard

Exploring Removable and Nonremovable Discontinuities

Consider the function $$f(x)=\frac{(x-2)(x+3)}{(x-2)}$$ for $$x\neq2$$ and $$f(2)=7$$. Answer the fo

Easy

Exponential Inflow with a Shift in Outflow Rate

A water tank receives water at a rate given by $$R_{in}(t)=20\,e^{-t}$$ liters per minute. The water

Hard

Graph Analysis of Discontinuities

A function $$q(x)$$ is defined piecewise as follows: $$q(x)=\begin{cases} x+2, & x<1, \\ 4, & x=1,

Hard

Graphical Analysis of a Removable Discontinuity

Consider the function $$f(x)=\frac{x^2-1}{x-1}$$ for x \neq 1, with a defined value of f(1) = 3. Ans

Medium

Identifying and Removing a Discontinuity

Consider the function $$g(x)=\frac{x^2-4}{x-2}$$ for $$x\neq2$$, which is undefined at $$x=2$$.

Easy

Implicitly Defined Curve and Its Tangent Line

Consider the circle defined by the equation $$x^2+y^2=16$$. Answer the following:

Medium

Indeterminate Limit with Exponential and Log Functions

Examine the limit $$\lim_{x \to 0} \frac{e^{2x} - \cos(x) - 1}{\ln(1+x^2)}.$$

Medium

Limit at an Infinite Discontinuity

Consider the function $$g(x)= \frac{1}{(x-2)^2}$$. Analyze its behavior near the point where it is u

Easy

Limits and Continuity in Particle Motion

A particle moves along a straight line with velocity given by $$v(t)=\frac{t^2-4}{t-2}$$ for t ≠ 2 s

Extreme

Limits at Infinity and Horizontal Asymptotes

Consider the rational function $$g(x)= \frac{4*x^3-x+2}{2*x^3+3*x^2-5}$$.

Medium

Limits Involving Absolute Value Functions

Consider the function $$f(x)=\frac{|x-3|}{x-3}$$. Answer the following:

Easy

Limits with Composite Logarithmic Functions

Consider the function $$t(x)=x*\ln(x)$$ defined for x > 0.

Medium

Logarithmic Function Limits

Consider the function $$f(x)=\frac{\ln(1+3*x)}{x}$$ for $$x \neq 0$$. Answer the following:

Medium

Modeling with a Removable Discontinuity

A water reservoir’s inflow rate is modeled by the function $$R(t)=\frac{t^2-16}{t-4}$$ liters per mi

Easy

Rational Function with Removable Discontinuity

Consider the function $$f(x)= \frac{x^2-9}{x-3}$$ for $$x \neq 3$$.

Easy

Water Flow Measurement Analysis

A water flow sensor measures the flow rate $$Q(t)$$ (in cubic meters per second) of a stream at vari

Medium
Unit 2: Differentiation: Definition and Fundamental Properties

Analysis of a Quadratic Function

Consider the function $$f(x)=3*x^2 - 2*x + 5$$. Using the limit definition of the derivative, answer

Easy

Analyzing Car Speed from a Distance-Time Table

A car's position (in meters) is recorded at various times (in seconds) as shown in the table. Use th

Easy

Application of Derivative to Relative Rates in Related Variables

Water is being pumped into a conical tank, and the volume of water is given by $$V=\frac{1}{3}\pi*r^

Hard

Chemical Reaction Rate Analysis

The concentration of a reactant in a chemical reaction (in M) is recorded over time (in seconds) as

Medium

Composite Function and Chain Rule Application

Consider the function $$h(x)=\sin(2*x^2+3)$$. Using the chain rule, answer the following parts:

Medium

Cooling Model Rate Analysis

The temperature of a cooling object is modeled by $$T(t)=e^{-2*t}+\ln(t+3)$$, where $$t$$ is time in

Medium

Cooling Tank System

A laboratory cooling tank has heat entering at a rate of $$H_{in}(t)=200-10*t$$ Joules per minute an

Easy

Derivative from the Limit Definition: Function $$f(x)=\sqrt{x+2}$$

Consider the function $$f(x)=\sqrt{x+2}$$ for $$x \ge -2$$. Using the limit definition of the deriva

Medium

Derivative of Inverse Functions

Let $$f(x)=3*x+\sin(x)$$, which is assumed to be one-to-one with an inverse function $$f^{-1}(x)$$.

Hard

Derivative via the Limit Definition: A Rational Function

Consider the function $$f(x)=\frac{1}{x+2}$$. Use the limit definition of the derivative to find $$f

Hard

Error Bound Analysis for $$e^{2x}$$

In a study of reaction rates, the function $$f(x)=e^{2*x}$$ is used. Analyze the error in approximat

Hard

Exponential Growth and Its Derivative

A culture of bacteria grows according to the model $$P(t)= 100*e^{0.03*t},$$ where \(P(t)\) is th

Easy

Finding the Derivative of a Logarithmic Function

Consider the function $$g(x)=\ln(3*x+1)$$. Answer the following:

Medium

Higher Order Derivatives: Concavity and Inflection Points

Consider the function $$f(x)= x^4 - 4*x^3+6*x^2.$$ (a) Find the first derivative \(f'(x)\) and th

Medium

Implicit Differentiation with Exponential and Trigonometric Functions

Consider the curve defined implicitly by $$e^(y) + x*\cos(y) = x^2$$.

Hard

Oil Spill Containment

Following an oil spill, containment efforts recover oil at a rate of $$O_{in}(t)=40-2*t$$ (accumulat

Hard

Related Rates in a Conical Tank

Water is draining from a conical tank. The tank has a total height of 10 m and its radius is always

Medium

Secant and Tangent Lines: Analysis of Rate of Change

Consider the function $$f(x)=x^3-6*x^2+9*x+1$$. This function represents a model of a certain proces

Medium

Sine Function Analysis

Let $$g(x)=3*\sin(x)+2$$, where $$x$$ is in radians. Analyze its rate of change.

Easy

Tangent Line Approximation for a Combined Function

Consider the function $$f(x)= \sin(x) + x^2$$. Use the concept of the tangent line to approximate ne

Hard

Tangent Line Approximation for a Parabolic Arch

Engineers design a parabolic arch described by $$y(x)= -0.5*x^2 + 4*x$$.

Easy

Temperature Change Rate

The temperature in a chemical reactor is modeled by $$T(t)=\frac{\sin(2*t)}{t}$$ for \(t>0\), where

Hard
Unit 3: Differentiation: Composite, Implicit, and Inverse Functions

Analysis of a Composite Chemical Concentration Model

The concentration of a chemical in a reaction is modeled by the composite function $$C(t)= \ln(0.5*t

Medium

Analyzing a Composite Function from a Changing Systems Model

The displacement of an object is given by $$s(t) = \sqrt{t^2 + 4*t + 5}$$ (in meters), where $$t$$ i

Medium

Bacterial Culture: Nutrient Inflow vs Waste Outflow

In a bioreactor, the nutrient inflow rate is given by $$N(t)=\ln(2*t+1)$$ (in units/min). The waste

Hard

Chain Rule in the Context of Light Intensity Decay

The light intensity as a function of distance from the source is given by $$I(x) = 500 * e^{-0.2*\sq

Medium

Combined Differentiation: Inverse and Composite Function

Let $$f(x)= \ln(2*x+1)$$ and let $$g$$ be the inverse function of $$f$$. Answer the following parts:

Medium

Composite Chain Rule with Exponential and Trigonometric Functions

Consider the function $$f(x) = e^{\cos(x)}$$. Analyze its derivative and explain the role of the cha

Easy

Dam Water Release and River Flow

A dam releases water into a river at a rate given by the composite function $$R(t)=c(b(t))$$, where

Medium

Differentiation of a Log-Exponential Composition with Critical Points

Consider the function $$k(x)=x*\ln(e^{x}+3)$$. Answer the following parts.

Extreme

Differentiation of a Product Involving Inverse Trigonometric Components

Let $$m(x)=\arctan(x)+x*\arcsin\left(\frac{x}{2}\right)$$, where the domain of arcsin requires $$-2\

Medium

Enzyme Kinetics in a Biochemical Reaction

In an enzymatic reaction, the substrate concentration $$S(t)$$ and the product concentration $$P(t)$

Extreme

Ice Cream Storage Dynamics

An ice cream storage facility receives ice cream at a rate given by the composite function $$I(t)=d(

Easy

Implicit Differentiation Involving a Mixed Function

Consider the equation $$x*e^{y}+y*\ln(x)=10$$, where x > 0 and y is defined implicitly as a function

Hard

Implicit Differentiation Involving Exponential Functions

Consider the relation defined implicitly by $$e^{x*y} + x^2 - y^2 = 7$$.

Hard

Implicit Differentiation Involving Product and Logarithm

Consider the curve defined by $$x*y + \ln(y) = x^2$$. Answer the following parts:

Medium

Implicit Differentiation: Circle and Tangent Line

The equation $$x^2 + y^2 = 25$$ represents a circle. Use implicit differentiation to find the deriva

Medium

Inverse Function Derivative in an Exponential Model

Let $$f(x)= e^{2*x} + x$$. Given that $$f$$ is one-to-one and differentiable, answer the following p

Easy

Inverse Function Differentiation in a Logarithmic Context

Let $$f(x)= \ln(x+2) - x$$, and let $$g$$ be its inverse function. Answer the following:

Medium

Inverse Function Differentiation in a Trigonometric Context

Let $$f(x)= \sin(x) + x$$, defined on the interval $$[0, \frac{\pi}{2}]$$, and let $$g$$ be its inve

Hard

Logarithmic Differentiation of a Variable Exponent Function

Consider the function $$y= (x^2+1)^{\sqrt{x}}$$.

Medium

Particle Motion with Composite Position Function

A particle moves along a line with its position given by $$s(t)= \sin(t^2)$$, where $$s$$ is in mete

Easy

Second Derivative via Chain Rule

Let $$h(x)=(e^{2*x}+1)^4$$. Answer the following parts.

Hard

Tangent Line for a Parametric Curve

A curve is given parametrically by $$x(t)= t^2 + 1$$ and $$y(t)= t^3 - t$$.

Easy

Trigonometric Composite Inverse Function Analysis

Consider the function $$f(x)=\sin(x)+x$$ defined on the interval $$\left[-\frac{\pi}{2}, \frac{\pi}{

Easy
Unit 4: Contextual Applications of Differentiation

Applying L'Hôpital's Rule to an Exponential Function

Consider the function $$F(x)=\frac{e^{2*x}-1}{x}$$, which models the change in a certain system's qu

Easy

Applying L'Hospital's Rule to a Transcendental Limit

Evaluate the limit $$\lim_{x\to 0}\frac{e^{2*x}-1}{\sin(3*x)}$$.

Medium

Approximating Function Values Using Linearization

Consider the function $$f(x)=x^4$$. Use linearization at x = 4 to approximate the value of $$f(3.98)

Easy

Car Motion with Changing Acceleration

A car's velocity is given by $$v(t) = 3*t^2 - 4*t + 2$$, where $$t$$ is in seconds. Answer the follo

Medium

Chemistry: Rate of Change in a Reaction

In a chemical reaction, the concentration of a reactant is modeled by $$C(t)= 50e^{-0.2*t}+5$$, wher

Easy

Conical Tank Filling - Related Rates

A conical water tank has its volume given by $$V= \frac{1}{3}\pi*r^2*h$$, where \(r\) is the radius

Hard

Cooling Coffee Temperature Change

The temperature of a cup of coffee is modeled by $$T(t)=70+50e^{-0.1*t}$$ (°F), where $$t$$ is the t

Easy

Cycloid Tangent Line

A cycloid is defined by the parametric equations $$x(t)=2*(t-\sin(t))$$ and $$y(t)=2*(1-\cos(t))$$ f

Hard

Differentiating a Product: f(x)=x sin(x)

Let \(f(x)=x\sin(x)\). Analyze the behavior of \(f(x)\) near \(x=0\).

Easy

Expanding Circular Ripple

A stone is thrown in a pond, creating circular ripples. The area of the circle defined by the ripple

Easy

Exponential and Trigonometric Bounded Regions

Let the region in the xy-plane be bounded by $$y = e^{-x}$$, $$y = 0$$, and the vertical line $$x =

Medium

Implicit Differentiation on a Circle

Consider the circle defined by $$x^2+y^2=25$$, where both $$x$$ and $$y$$ are functions of time $$t$

Easy

Industrial Mixer Flow Rates

In an industrial mixer, an ingredient is added at a rate of $$I(t)=7t$$ (kg per minute) and is consu

Extreme

Instantaneous vs. Average Rate of Change in Temperature

A rod's temperature along its length is modeled by $$T(x)=20\ln(x+1)+e^{-x}$$, where x (in meters) i

Medium

Interpreting Position Graphs: From Position to Velocity

A graph of position (in meters) versus time (in seconds) is provided in the stimulus. The graph show

Medium

Interpreting the Derivative in Straight Line Motion

A particle moves along a straight line with velocity given by $$ v(t)= 3*t^2 - 4*t + 2 $$. Analyze a

Easy

Linearization in Engineering Load Estimation

In an engineering project, the load on a beam is modeled by $$L(x)=100*x^2+50*x+200$$, where $$L(x)$

Medium

Logarithmic Differentiation and Asymptotic Behavior

Let $$f(x)=\frac{(x^2+1)^3}{e^{2x}(x-1)^2}$$ for x > 1. Answer the following:

Hard

Minimum Time to Cross a River

A person must cross a 100-meter-wide river. They can swim at 2 m/s and run along the bank at 5 m/s.

Hard

Motion along a Curved Path

A particle moves along the curve defined by $$y=\sqrt{x}$$. At the moment when $$x=9$$ and the x-coo

Easy

Motion on a Straight Line with a Logarithmic Position Function

A particle moves along a straight line with its position given by $$s(t)=\ln(t+2)+t^2$$ (in meters),

Medium

Motion with Non-Uniform Acceleration

A particle moves along a straight line and its position is given by $$s(t)= 2*t^3 - 9*t^2 + 12*t + 3

Medium

Parametric Curve Motion

A particle’s trajectory is given by the parametric equations $$x(t)=t^2-1$$ and $$y(t)=2*t+3$$ for $

Medium

Parametric Motion in the Plane

A particle moves in the plane according to the parametric equations $$x(t)=t^2-2*t$$ and $$y(t)=3*t-

Hard

Parametric Motion with Logarithmic and Radical Components

A particle’s motion is described by the vector function $$\mathbf{r}(t)=\langle \ln(t+1),\sqrt{t} \r

Hard

Particle Motion Analysis Using Cubic Position Function

Consider a particle moving along a straight line with its position given by $$x(t)=t^3 - 6*t^2 + 9*t

Medium

Particle Motion with Measured Positions

A particle moves along a straight line with its velocity given by $$v(t)=4*t-1$$ for $$t \ge 0$$ (in

Medium

Polar Coordinates: Arc Length of a Spiral

Consider the polar curve defined by $$r(\theta)=1+\frac{\theta}{2}$$ for $$0 \le \theta \le \pi$$.

Hard

Related Rates: Expanding Circular Oil Spill

In a coastal region, an oil spill is spreading uniformly and forms a circular region. The area of th

Medium

Related Rates: Expanding Circular Ripple

A circular ripple in a pond expands such that its area, given by $$A=\pi r^2$$, is increasing at a c

Easy

Related Rates: Inflating Spherical Balloon with Exponential Volume Rate

A spherical balloon is being inflated so that its volume changes at a rate of $$\frac{dV}{dt}=8e^{0.

Hard

Savings Account and Interest Accrual

A student starts with an initial savings account balance of $$B_0=1000$$ dollars and makes monthly d

Medium
Unit 5: Analytical Applications of Differentiation

Analysis of a Function with Oscillatory Behavior

Consider the function $$f(x)=x+\sin(x)$$. Answer the following parts:

Medium

Analyzing Extrema for a Rational Function

Let $$f(x)= \frac{x^2+2}{x+1}$$ be defined on the interval $$[0,4]$$. Use calculus methods to analyz

Hard

Application in Motion: Approximate Velocity using Taylor Series

A particle’s position is given by $$s(t)=e^{-t}+t^2$$. Using Taylor series approximations near $$t=0

Hard

Application of Rolle's Theorem

Consider the function $$g(x)=x^3-3x$$ on the interval $$[-\sqrt{3},\sqrt{3}]$$. Answer the following

Medium

Application of the Mean Value Theorem

Let $$f(x)=\frac{x}{x^2+1}$$ be defined on the interval $$[0,2]$$. Answer the following questions us

Easy

Area and Volume of Region Bounded by Exponential and Linear Functions

Consider the functions $$f(x)=e^{x}$$ and $$g(x)=x+2$$. The region enclosed by these curves will be

Medium

Car Motion: Velocity and Total Distance

A car’s position along a straight road is modeled by $$s(t) = t^3 - 6*t^2 + 9*t + 15$$ (in meters),

Medium

Chemical Reaction Rate

During a chemical reaction, the concentration of a reagent (in M) is measured over time (in minutes)

Easy

Concavity Analysis in a Revenue Model

A company’s revenue (in thousands of dollars) is modeled by the function $$R(x) = -0.5*x^3 + 6*x^2 -

Medium

Determining the Meeting Point of Two Functions

Consider the functions $$f(x)= e^x$$ and $$g(x)= 3 + \ln(x)$$ representing two different processes.

Extreme

Economic Optimization: Maximizing Profit

The profit function for a product is given by $$P(x) = -2*x^3 + 27*x^2 - 108*x + 150$$, where \(x\)

Hard

Extreme Value Theorem in Temperature Variation

A metal rod’s temperature (in °C) along its length is modeled by the function $$T(x) = -2*x^3 + 12*x

Medium

Fractal Tree Branch Lengths

A fractal tree is constructed as follows: The trunk has a length of 10 meters. At each generation, e

Hard

Inverse Function and Critical Points in a Business Context

A company models its profit (in thousands of dollars) by $$f(x)= \ln(4*x+7)$$ for $$x \ge 0$$, where

Medium

Motion with a Piecewise-Defined Velocity Function

A particle travels along a line with a piecewise velocity function defined by $$ v(t)=\begin{cases}

Hard

Optimization in a Log-Exponential Model

A firm's profit is given by the function $$P(x)= x\,e^{-x} + \ln(1+x)$$, where x (in thousands) repr

Hard

Optimizing Material for a Container

An open-top rectangular container with a square base must have a fixed volume of $$32$$ cubic feet.

Hard

Parameter-Dependent Concavity Conditions

Consider the function $$ f(x)=x^3+a*x^2+2x,$$ where $$a$$ is a real parameter. Answer the following

Medium

Piecewise Function Discontinuities Analysis

Consider the piecewise function $$ f(x)= \begin{cases} \frac{x^2-4}{x-2} & \text{if } x \neq 2, \\

Medium

Planar Particle Motion with Time-Dependent Accelerations

A particle moves in the plane with its position given by $$\vec{s}(t)=\langle t^2-4*t+4,\; \ln(t+1)\

Medium

Profit Maximization in Business

A company’s profit function is given by $$P(x) = -2*x^3 + 30*x^2 - 100*x + 150$$, where x represents

Hard

Projectile Motion Analysis

A projectile is launched at a 45° angle with an initial speed of 20 m/s. Its motion is modeled by th

Medium

Second Derivative Test for Critical Points

Consider the function $$f(x)=x^3-9*x^2+24*x-16$$.

Medium

Ski Resort Snow Accumulation and Melting

At a ski resort, snow accumulates naturally at a rate given by $$S(t)=50*\exp(-0.1*t)$$ cm/hour due

Medium

Skier's Speed Analysis

A skier's speed (in m/s) on a slope was recorded at various time intervals. Use the data provided to

Medium

Stress Analysis in Engineering Structures

A beam experiences stress along its length given by $$S(x)=5*x^3-30*x^2+45*x$$ where x is the distan

Hard

Volume by Cross Sections Using Squares

A region in the xy-plane is bounded by $$y=x$$, $$y=0$$, and $$x=3$$. Perpendicular to the x-axis, c

Hard

Volume Using Cylindrical Shells

The region bounded by $$y=x$$, $$y=0$$, and $$x=4$$ is revolved about the y-axis to form a solid.

Hard

Water Tank Rate of Change

The volume of water in a tank is modeled by $$V(t)= t^3 - 6*t^2 + 9*t$$ (in cubic meters), where $$t

Medium
Unit 6: Integration and Accumulation of Change

Accumulated Displacement from Acceleration

A particle moving along a straight line has an acceleration of $$a(t)=6-4*t$$ (in m/s²), with an ini

Hard

Application of the Fundamental Theorem

Consider the function $$f(x)=x^2+2*x$$ defined on the interval $$[1,4]$$. Evaluate the definite inte

Easy

Area Estimation Using Riemann Sums for $$f(x)=x^2$$

Consider the function $$f(x)=x^2$$ on the interval $$[1,4]$$. A table of computed values for the lef

Medium

Area Estimation Using Trapezoidal Sums from Tabulated Data

The table below provides values of $$h(t)$$ over time for a process: | Time (t) | 0 | 2 | 5 | 8 | |

Hard

Area Estimation with Riemann Sums

A water flow rate function f(x) (in m³/s) is measured at various times. The table below shows the me

Easy

Area Under a Parametric Curve

A curve is defined parametrically by $$x(t)=t^2$$ and $$y(t)=t^3-3*t$$ for $$t \in [-2,2]$$.

Extreme

Bacteria Population Accumulation

A bacteria culture grows at a rate given by $$r(t)=0.5*t^2-t+3$$ (in thousand bacteria per hour) for

Medium

Biomedical Modeling: Drug Concentration Dynamics

A drug concentration in the bloodstream is modeled by $$f(t)= 5\left(1 - e^{-0.3*t}\right)$$ for $$t

Hard

Comparing Riemann Sums with Definite Integral in Estimating Distance

A vehicle's velocity (in m/s) is recorded at discrete times during a trip. Use these data to estimat

Medium

Continuous Antiderivative for a Piecewise Function

A function $$f(x)$$ is defined piecewise as follows: for $$x<2$$, $$f(x)=3*x^2$$, and for $$x\ge2$$,

Extreme

Convergence of an Improper Integral Representing Accumulation

Consider the improper integral $$\int_{1}^{\infty} \frac{1}{t^2}dt$$, which can be interpreted as th

Hard

Determining Constant in a Height Function

A ball is thrown upward with a constant acceleration of $$a(t)= -9.8$$ m/s² and an initial velocity

Medium

Distance vs. Displacement from a Velocity Function

A runner's velocity is modeled by $$v(t)=5-0.5*t$$ (in m/s) for $$0\le t\le10$$. The runner may chan

Medium

Economic Applications: Consumer and Producer Surplus

In a market, the demand function is given by $$D(p)=100-2p$$ and the supply function by $$S(p)=20+3p

Hard

Economics: Accumulated Earnings

A company’s instantaneous revenue rate (in dollars per day) is modeled by the function $$R(t)=1000\s

Medium

Estimating Area Under a Curve from Tabular Data

A function $$f(t)$$ is sampled at discrete time points as given in the table below. Using these data

Easy

Evaluating an Integral Using U-Substitution

Evaluate the indefinite integral $$\int (x-4)^{10}\,dx$$ using u-substitution.

Medium

Fuel Consumption Estimation with Midpoint Riemann Sums

A vehicle’s fuel consumption rate (in liters per hour) over a trip is recorded at various times. The

Medium

Fundamental Theorem of Calculus Application

Let $$F(x)=\int_{2}^{x} (t^{2} - 4*t + 3) dt$$. Answer the following:

Easy

Improper Integral and the p-Test

Determine whether the improper integral $$\int_1^{\infty} \frac{1}{x^2}\,dx$$ converges, and if it c

Extreme

Integration of a Rational Function

Consider the function $$f(x)=\frac{1}{x^2+4}$$ on the interval $$[0,2]$$. Evaluate the area under th

Hard

Integration Using U-Substitution

Evaluate the indefinite integral $$\int (4*x+2)^5\,dx$$ using u-substitution.

Medium

Net Displacement vs. Total Distance Traveled

A particle moving along a straight line has a velocity function given by $$v(t)= t^2 - 4*t + 3$$ (in

Medium

Population Growth: Rate to Accumulation

A population's growth rate (in thousands of individuals per year) is modeled by $$P'(t)=2*t - 1$$ fo

Easy

Probability Density Function and Expected Value

Let the probability density function (pdf) be defined by $$f(x)=k*x*e^{-x}$$ for $$x\ge0$$.

Hard

Reservoir Water Level

A reservoir experiences a net water inflow modeled by $$W(t)=40*\sin\left(\frac{\pi*t}{12}\right)-5$

Medium

Riemann Sum Estimation from Tabular Data

The following table lists values of a function $$f(x)$$ at selected points, which are used to approx

Easy

Solving for Unknowns using Logarithmic Properties in Integration

Consider the definite integral $$\int_(a)^(b) \frac{3}{x} dx$$ which is given to equal 6, where a is

Medium

Temperature Function Analysis with Inverses

A temperature profile over time is given by $$f(t)= \ln(2*t + 3)$$ for $$t \ge 0$$ (with temperature

Hard

Total Work Done by a Variable Force

A variable force $$F(x)$$ (in Newtons) is applied along a displacement, and its values are recorded

Medium

Transportation Model: Distance and Inversion

A transportation system is modeled by $$f(t)= (t-1)^2+3$$ for $$t \ge 1$$, where \(t\) is time in ho

Medium

Water Accumulation Using Trapezoidal Sum

A reservoir is monitored over time and its water level (in meters) is recorded at various times (in

Medium

Water Tank Inflow and Outflow

A water tank begins operation at t = 0 with an initial volume of 0 liters. Water flows in through an

Medium
Unit 7: Differential Equations

Analysis of a Nonlinear Differential Equation

Consider the nonlinear differential equation $$\frac{dy}{dx} = y^3-3*y$$.

Medium

Analysis of a Piecewise Function with Potential Discontinuities

Consider the piecewise defined function $$ f(x)=\begin{cases} \frac{x^2-4}{x-2} & \text{if } x < 2,

Easy

Bacteria Growth with Antibiotic Treatment

A bacterial culture has a population $$N(t)$$ that grows at a rate proportional to its size, given b

Medium

Capacitor Charging with Leakage

A capacitor is being charged by a constant current source of $$5$$ A, but it also leaks charge at a

Easy

Chemical Reaction Rate and Series Approximation

A chemical reaction is modeled by the differential equation $$\frac{dC}{dt} = -0.2 * C^2$$ with the

Medium

Coffee Cooling: Differential Equation Application

A cup of coffee is cooling according to Newton's Law of Cooling. The following table provides measur

Medium

Cooling Coffee Data Analysis

A cup of coffee cools down in a room according to Newton's Law of Cooling. The temperature $$T(t)$$

Hard

Differential Equation with Exponential Growth and Logistic Correction

Consider the modified differential equation $$\frac{dy}{dt} = a*y - b*y^2$$, where $$a$$ and $$b$$ a

Medium

Dye Dilution in a Stream

A river has dye added at a constant rate of $$0.5$$ kg/min, and the dye is removed at a rate proport

Easy

FRQ 16: Harvesting in a Predator-Prey Model

A prey population $$P(t)$$ in a marine ecosystem is modeled by the differential equation $$\frac{dP}

Extreme

Mixing Problem in a Tank

A tank initially contains a certain amount of salt dissolved in 100 L of water. Brine with a known s

Medium

Mixing Problem in a Tank

A tank initially contains 50 liters of pure water. A brine solution with a salt concentration of $$3

Medium

Modeling the Spread of a Disease Using Differential Equations

Suppose the spread of a disease in a population is modeled by the differential equation $$\frac{dI}{

Hard

Motion Under Gravity with Air Resistance

An object falling under gravity experiences air resistance proportional to its velocity. Its motion

Medium

Newton's Law of Cooling

A hot liquid is cooling in a room. The temperature $$T(t)$$ (in degrees Celsius) of the liquid at ti

Easy

Phase-Plane Analysis of a Nonlinear Differential Equation

Consider the logistic differential equation $$\frac{dy}{dt} = y(1-y)$$, which models a normalized po

Easy

Population Growth with Harvesting

A fish population in a lake is modeled by the differential equation $$\frac{dP}{dt}= rP - H$$, where

Medium

Population Growth with Logistic Differential Equation

A population $$P(t)$$ is modeled by the logistic differential equation $$\frac{dP}{dt} = r\,P\left(1

Medium

Radioactive Decay

A radioactive substance decays according to the differential equation $$\frac{dA}{dt}=-kA$$, where $

Easy

Separable and Implicit Solution for $$\frac{dy}{dx}= \frac{x}{1+y^2}$$

Consider the differential equation $$\frac{dy}{dx}= \frac{x}{1+y^2}$$, which is defined for all real

Hard

Separable Differential Equation with Initial Condition

Consider the differential equation $$\frac{dy}{dx} = \frac{4*x}{y}$$ with the initial condition $$y(

Easy

Tank Draining Problem

A tank with a variable cross-sectional area is being emptied. The height \(h(t)\) of the water satis

Medium

Traffic Flow on a Highway

A highway segment experiences an inflow of cars at a rate of $$200+10*t$$ cars per minute and an out

Medium
Unit 8: Applications of Integration

Analyzing Acceleration Data from Discrete Measurements

A vehicle’s acceleration (in m/s²) is recorded at discrete time intervals as given in the table. Use

Easy

Area Between a Parabola and a Line

Consider the curves given by $$y=5*x-x^2$$ and $$y=x$$. These curves intersect at certain $$x$$-valu

Medium

Area Between Curves: Park Design

A park layout is bounded by two curves: $$f(x)=10-x^2$$ and $$g(x)=2*x+2$$. Answer the following par

Medium

Area Between Two Curves: Parabola and Line

Consider the functions $$f(x)= 5*x - x^2$$ and $$g(x)= x$$. These curves enclose a region in the pla

Medium

Average Daily Temperature

The temperature during a day is modeled by $$T(t)=10+5*\sin((\pi/12)*t)$$ (in °C), where $$t$$ is th

Easy

Average Temperature Calculation

The temperature (in $$^\circ C$$) in City A is recorded at specific times over a 12-hour period. Est

Medium

Average Temperature Computation

Consider a scenario in which the temperature (in °C) in a region is modeled by the function $$T(t)=

Easy

Average Temperature Over a Day

A research team studies the variation in water temperature in a lake over a 24‐hour period. The temp

Easy

Average Value and Critical Points of a Trigonometric Function

Consider the function $$f(x)=\sin(2*x)+\cos(2*x)$$ on the interval $$\left[0,\frac{\pi}{2}\right]$$.

Hard

Average Velocity and Displacement from a Polynomial Function

A car's velocity in m/s is given by $$v(t)=t^2-4*t+3$$ for $$t\in[0,5]$$ seconds. Answer the followi

Easy

Center of Mass of a Lamina with Constant Density

A thin lamina occupies the region in the first quadrant bounded by $$y=x^2$$ and $$y=4$$. The densit

Hard

Center of Mass of a Thin Rod

A thin rod extends from $$x=0$$ to $$x=4$$ m and has a density function $$\lambda(x)=1+\frac{\ln(x+2

Hard

Comparing Average and Instantaneous Rates of Change

For the quadratic function $$f(x)= 3*x^2 - 4*x + 1$$ on the interval $$[1,3]$$, investigate both its

Medium

Designing a Bridge Arch

A bridge arch is modeled by the curve $$y = 10 - 0.25*x^2$$, where $$x$$ is measured in meters and $

Medium

Determining the Arc Length of a Curve

Consider the curve defined by $$y=\frac{1}{2}*e^{x/2}$$ over the interval $$[0,2]$$.

Hard

Drug Concentration Profile Analysis

The functions $$f(t)=5*t+10$$ and $$g(t)=2*t^2+3$$ (where t is in hours and concentration in mg/L) r

Medium

Integration in Cost Analysis

In a manufacturing process, the cost per minute is modeled by $$C(t)=t^2 - 4*t + 7$$ (in dollars per

Easy

Net Cash Flow Analysis

A company’s net cash flow is modeled by $$N(t)=50*\ln(t+1) - 2*t$$ (in thousands of dollars per mont

Medium

Particle Motion from Acceleration

A particle has an acceleration given by $$a(t)=3*t-6$$ (m/s²). With initial conditions $$v(0)=2$$ m/

Medium

Particle Motion with Variable Acceleration

A particle's acceleration is given by $$a(t)=4*e^{-t} - 2$$ for $$t$$ in seconds over the interval $

Medium

Particle Motion: Position, Velocity, and Acceleration

A particle moves along a straight line with acceleration given by $$a(t)=3*t-4$$ (in m/s²). At time

Medium

Shadow Length Related Rates

A 1.8-meter tall man is walking away from a 5-meter tall lamp post at a constant speed of $$1.5$$ m/

Medium

Volume of a Hollow Cylinder Using the Washer Method

A manufacturer designs a hollow cylindrical container. The outer surface is modeled by $$y=10-\sqrt{

Medium

Volume of a Solid via Shell Method

Consider the region bounded by $$y=x^2$$ and $$y=4$$ in the first quadrant. This region is revolved

Medium

Volume of a Solid via the Disc Method

The region bounded by the curve $$y=\sqrt{x}$$ and the x-axis for $$x\in[0,4]$$ is revolved about th

Medium

Volume of a Water Tank with Varying Cross-Sectional Area

A water tank has a cross-sectional area given by $$A(x)=3*x^2+2$$ in square meters, where $$x$$ (in

Easy

Work Done by a Variable Force

A variable force applied to move an object along a straight line is given by $$F(x)=3*x^2$$ (in newt

Easy

Work Done by a Variable Force

A variable force acting along the x-axis is given by $$F(x) = 2 * x + 3$$ (in Newtons). An object mo

Easy

Work Done by a Variable Force

A variable force acting along a straight line is given by $$F(x) = \frac{10}{x+2}$$ (in Newtons). Fi

Medium

Work Done by a Variable Force

A force acting along a straight line is given by $$F(x)=10 - 0.5*x$$ newtons for $$0 \le x \le 12$$

Medium

Work Done with a Discontinuous Force Function

A force acting on an object is defined piecewise by $$F(x)=\begin{cases} x^2, & 0 \le x < 4,\\ 16, &

Medium

Work to Pump Water from a Tank

A cylindrical tank of radius 2 ft and height 10 ft is partially filled with water to a depth of 8 ft

Hard
Unit 9: Parametric Equations, Polar Coordinates, and Vector-Valued Functions

Analyzing a Clock's Second Hand with Polar Coordinates

A clock's second hand rotates uniformly, and its tip traces a circle of radius 12 cm. Its position i

Medium

Arc Length of a Parametric Curve

Consider the curve defined by the parametric equations $$x(t)=3*\cos(t)$$ and $$y(t)=3*\sin(t)$$ for

Medium

Arc Length of a Polar Curve

Consider the polar curve given by $$r=2+\cos(\theta)$$ for $$0\le \theta \le \pi$$. Answer the follo

Medium

Arc Length of a Polar Curve

Consider the polar curve given by $$r = 2 + 2*\sin(\theta)$$ for $$0 \le \theta \le \pi$$.

Hard

Comparing Arc Lengths in Parametric and Polar Systems

Consider the curve given in parametric form by $$x(t)=\cos(2*t)$$ and $$y(t)=\sin(2*t)$$ for $$0\le

Extreme

Concavity and Inflection Points of a Parametric Curve

For the curve defined by $$x(t)=e^{t}-t$$ and $$y(t)=\ln(1+t^2)$$ for $$t \ge 0$$, answer the follow

Hard

Conversion and Differentiation of a Polar Curve

Consider the polar curve defined by $$ r=2+\sin(\theta) $$. Study its conversion to Cartesian coordi

Hard

Conversion and Tangents in Polar Coordinates

Consider the polar curve $$r=\sec(\theta)$$ for $$\theta \in \left[0, \frac{\pi}{4}\right]$$.

Medium

Conversion to Cartesian and Analysis of a Parametric Curve

Consider the parametric equations $$x(t)= 2*t + 1$$ and $$y(t)= (t - 1)^2$$ for $$-2 \le t \le 3$$.

Easy

Cycloid and Its Arc Length

Consider the cycloid defined by the parametric equations $$x(t)=t-\sin(t)$$ and $$y(t)=1-\cos(t)$$ f

Hard

Determining Curvature from a Vector-Valued Function

Consider the curve defined by $$\mathbf{r}(t)=\langle t, t^2, t^3 \rangle$$ for $$t \ge 0$$. Analyze

Hard

Differentiation and Integration of a Vector-Valued Function

Let $$\mathbf{r}(t)=\langle e^{-t}, \sin(t), \cos(t) \rangle$$ for $$t \in [0,\pi]$$.

Easy

Exponential Growth in Parametric Representation

A model for population growth is given by the parametric equations $$x(t)=t$$ and $$y(t)=e^{0.3t}$$,

Medium

Implicit Differentiation with Implicitly Defined Function

Consider the equation $$x^2+xy+y^2=7$$, which defines $$y$$ implicitly as a function of $$x$$.

Medium

Intersection of Parametric Curves

Consider the parametric curves $$C_1$$ given by $$x(t)= t^2,\; y(t)= 2t$$ and $$C_2$$ given by $$x(s

Easy

Kinematics in the Plane: Parametric Motion

A particle moves in the plane with its position given by the parametric equations $$ x(t)=t^2-2*t $$

Medium

Modeling Projectile Motion with Parametric Equations

A projectile is launched with an initial speed of \(20\) m/s at an angle of \(45^\circ\) above the h

Easy

Motion Along a Helix

A particle moves along a helix described by the vector-valued function $$\vec{r}(t)=<\cos(t),\, \sin

Medium

Parametric Curves and Concavity

Consider the parametric equations $$x(t)= \sin(t)$$ and $$y(t)= \cos(2*t)$$ for $$t \in [0, 2\pi]$$.

Hard

Particle Motion in the Plane

A particle moves in the plane with its position described by the parametric equations $$x(t)=3*\cos(

Medium

Particle Motion on an Elliptical Arc

A particle moves along a curve described by the parametric equations $$x(t)= 2*cos(t)$$ and $$y(t)=

Easy

Polar Coordinates: Area Between Curves

Consider two polar curves: the outer curve given by $$R(\theta)=4$$ and the inner curve by $$r(\thet

Hard

Spiral Intersection on the X-Axis

Consider the spiral defined by the parametric equations $$x(t) = e^{-t}\cos(2*t)$$ and $$y(t)= e^{-t

Medium

Spiral Motion with a Damped Vector Function

An object moves according to the spiral vector function $$\mathbf{r}(t)=\langle e^{-t}\cos(t),\; e^{

Extreme

Vector-Valued Functions: Tangent and Normal Components

A car’s motion on a plane is described by the vector-valued function $$\mathbf{r}(t)=\langle t^2, 4*

Hard

Vector-Valued Integrals in Motion

A particle's acceleration is given by the vector function $$\vec{a}(t)=<\ln(t),\, t^{-1},\, e^{t}>$$

Hard

Wind Vector Analysis in Navigation

A boat on a river is propelled by its engine giving a constant velocity of \(\langle 3, 4 \rangle\)

Hard

Work Done Along a Path in a Force Field

A force field is given by \(\mathbf{F}(x,y)=\langle x,\,y \rangle\), and a particle moves along a pa

Medium

Work Done by a Force along a Vector Path

A force field is given by $$\mathbf{F}(t)=\langle2*t,\;3\sin(t)\rangle$$ and an object moves along a

Hard

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Where can I find practice free response questions for the AP Calculus BC exam?
The free response section of each AP exam varies slightly, so you’ll definitely want to practice that before stepping into that exam room. Here are some free places to find practice FRQs :
  • Of course, make sure to run through College Board's past FRQ questions!
  • Once you’re done with those go through all the questions in the AP Calculus BCFree Response Room. You can answer the question and have it grade you against the rubric so you know exactly where to improve.
  • Reddit it also a great place to find AP free response questions that other students may have access to.
How do I practice for AP AP Calculus BC Exam FRQs?
Once you’re done reviewing your study guides, find and bookmark all the free response questions you can find. The question above has some good places to look! while you’re going through them, simulate exam conditions by setting a timer that matches the time allowed on the actual exam. Time management is going to help you answer the FRQs on the real exam concisely when you’re in that time crunch.
What are some tips for AP Calculus BC free response questions?
Before you start writing out your response, take a few minutes to outline the key points you want to make sure to touch on. This may seem like a waste of time, but it’s very helpful in making sure your response effectively addresses all the parts of the question. Once you do your practice free response questions, compare them to scoring guidelines and sample responses to identify areas for improvement. When you do the free response practice on the AP Calculus BC Free Response Room, there’s an option to let it grade your response against the rubric and tell you exactly what you need to study more.
How do I answer AP Calculus BC free-response questions?
Answering AP Calculus BC free response questions the right way is all about practice! As you go through the AP AP Calculus BC Free Response Room, treat it like a real exam and approach it this way so you stay calm during the actual exam. When you first see the question, take some time to process exactly what it’s asking. Make sure to also read through all the sub-parts in the question and re-read the main prompt, making sure to circle and underline any key information. This will help you allocate your time properly and also make sure you are hitting all the parts of the question. Before you answer each question, note down the key points you want to hit and evidence you want to use (where applicable). Once you have the skeleton of your response, writing it out will be quick, plus you won’t make any silly mistake in a rush and forget something important.