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AP Calculus BC Free Response Questions

The best way to get better at FRQs is practice. Browse through dozens of practice AP Calculus BC FRQs to get ready for the big day.

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  • Unit 1: Limits and Continuity (28)
  • Unit 2: Differentiation: Definition and Fundamental Properties (27)
  • Unit 3: Differentiation: Composite, Implicit, and Inverse Functions (25)
  • Unit 4: Contextual Applications of Differentiation (25)
  • Unit 5: Analytical Applications of Differentiation (30)
  • Unit 6: Integration and Accumulation of Change (27)
  • Unit 7: Differential Equations (35)
  • Unit 8: Applications of Integration (30)
  • Unit 9: Parametric Equations, Polar Coordinates, and Vector-Valued Functions (23)
Unit 1: Limits and Continuity

Algebraic Manipulation in Limit Evaluation

Consider the function $$f(x)=\frac{x^3-8}{x-2}$$ defined for $$x \neq 2$$. Answer the following:

Hard

Algebraic Method for Evaluating Limits

Consider the function $$h(x)=\frac{x^2-9}{x-3}.$$ Answer the following parts.

Easy

Algorithm Time Complexity

A recursive algorithm has an execution time that decreases with each iteration: the first iteration

Medium

Analysis of a Jump Discontinuity

Consider the function $$f(x)=\begin{cases} 3*x+1, & x<4 \\ 2*x-3, & x\geq4 \end{cases}$$.

Medium

Analysis of a Piecewise Function with Multiple Definitions

Consider the function $$h(x)=\begin{cases} \frac{x^2-9}{x-3} & \text{if } x<3, \\ 2*x-1 & \text{if

Medium

Application of the Squeeze Theorem

Let $$f(x)=x^2 * \sin(\frac{1}{x})$$ for $$x \neq 0$$. Answer the following:

Hard

Applying the Squeeze Theorem to a Trigonometric Function

Consider the function $$f(x)= x^2*\sin(\frac{1}{x})$$ for $$x \neq 0$$ with $$f(0)=0$$. Use the Sque

Easy

Calculating Tangent Line from Data

The table below gives a function $$f(x)$$ representing the distance (in meters) of a moving object f

Medium

Evaluating a Rational Function Limit Using Algebraic Manipulation

Consider the function $$f(x)=\frac{x^2-9}{x-3}$$. Analyze the limit as $$x \to 3$$.

Easy

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 Function Limit and Continuity

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

Medium

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

Fuel Efficiency and Speed Graph Analysis

A car manufacturer records data on fuel efficiency (in mpg) as a function of speed (in mph). A graph

Medium

Graphical Analysis of Water Tank Volume

The water volume in a tank over time is recorded and displayed in the graph provided. Due to a senso

Medium

Internet Data Packet Transmission and Error Rates

In a data transmission system, an error correction protocol improves the reliability of transmitted

Extreme

Interplay of Polynomial Growth and Exponential Decay

Consider the function $$s(x)= x\cdot e^{-x}$$.

Medium

L'Hôpital's Rule for Indeterminate Forms

Evaluate the limit $$h(x)=\frac{e^{2*x}-1}{\sin(3*x)}$$ as x approaches 0.

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 Absolute Value Functions

Examine the function $$f(x)= \frac{|x-3|}{x-3}$$ defined for $$x \neq 3$$.

Medium

Limits Involving Trigonometric Functions

Consider the function $$q(x)=\frac{1-\cos(2*x)}{x^2}$$.

Medium

Limits of Composite Trigonometric Functions

Let $$p(x)= \frac{\sin(3x)}{\sin(5x)}$$.

Easy

Limits with Composite Logarithmic Functions

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

Medium

Maclaurin Polynomial Approximation and Error Analysis for $$\ln(1+x)$$

Consider the function $$f(x)=\ln(1+x)$$. You are asked to approximate $$f(0.5)$$ using its Maclaurin

Hard

One-Sided Limits in a Piecewise Function

Consider the function $$f(x)=\begin{cases} \sqrt{x+4}, & x < 5, \\ 3*x-7, & x \ge 5. \end{cases}$$ A

Medium

Rational Functions and Limit at Infinity

Consider the rational function $$r(x)= \frac{2x^2+3x-1}{x^2-4}$$.

Easy

Removable Discontinuity in a Cubic Function

Consider the function $$f(x)=\frac{x^3-8}{x-2}$$ defined for $$x\neq2$$. Answer the following:

Extreme

Removing a Removable Discontinuity in a Piecewise Function

Examine the function $$g(x)= \begin{cases} \frac{x^2-9}{x-3}, & x \neq 3 \\ m, & x=3 \end{cases}$$.

Easy

Using Power Series to Estimate a Trigonometric Function

The power series for $$\sin(x)$$ is $$Q(x)=\sum_{n=0}^{\infty} \frac{(-1)^n*x^{2*n+1}}{(2*n+1)!}.$$

Medium
Unit 2: Differentiation: Definition and Fundamental Properties

Average and Instantaneous Growth Rates in a Bacterial Culture

A bacterial population is modeled by the function $$P(t)= e^{0.3*t} + 10$$, where $$t$$ is measured

Easy

Bacteria Culturing in a Bioreactor

In a bioreactor, the bacterial inflow (growth) rate is given by $$B_{in}(t)=\frac{15}{1+e^{-0.3*(t-5

Extreme

Circular Motion Analysis

An object moves along a circular path with angular position given by $$\theta(t)=2*t-0.1*t^2$$ (in r

Easy

Derivative from First Principles: Reciprocal Function

Let $$f(x)= \frac{1}{x}$$.

Medium

Derivative Using Limit Definition

Let $$f(x)=\frac{1}{x+2}$$. Using the definition of the derivative, find $$f'(x)$$.

Hard

Differentiation in Biological Growth Models

In a biological experiment, the rate of resource consumption is modeled by $$R(t)=\frac{\ln(t^2+1)}{

Hard

Differentiation of Implicitly Defined Functions

An ellipse is defined by the equation $$\frac{x^2}{4} + \frac{y^2}{9} = 1$$. Use implicit differenti

Medium

Economic Model Rate Analysis

A company models its cost variations with respect to price $$p$$ using the function $$C(p)=e^{-p}+\l

Easy

Error Bound Analysis for Cos(x) Approximations in Physical Experiments

In a controlled physics experiment, small angle approximations for $$\cos(x)$$ are critical. Analyze

Medium

Implicit Differentiation in Logarithmic Equations

Consider the relation given by $$x*\ln(y)+y*\ln(x)=5$$, where $$x>0$$ and $$y>0$$.

Hard

Instantaneous Velocity from a Displacement Function

A particle moves along a straight line with its position at time $$t$$ (in seconds) given by $$s(t)

Easy

Instantaneous vs. Average Rate of Change

Consider the trigonometric function $$f(x)= \sin(x)$$.

Medium

Limit Definition of Derivative for a Rational Function

For the function $$f(x)=\frac{1}{x+1}$$, use the limit definition of the derivative to answer the fo

Extreme

Limit Definition of the Derivative for a Quadratic Function

Let $$f(x)= 5*x^2 - 4$$. Use the limit definition of the derivative to compute $$f'(x)$$.

Easy

Maclaurin Series for arctan(x) and Error Estimate

An engineer in signal processing needs the Maclaurin series for $$g(x)=\arctan(x)$$ and an understan

Hard

Motion Along a Line

An object moves along a line with its position given by $$s(t)=4*t^3 - 12*t^2 + 9*t$$, where $$s$$ i

Medium

Pharmacokinetics: Drug Concentration Analysis

The concentration of a drug in the bloodstream is modeled by the function $$C(t)=10*\ln(t+2)*e^{-0.3

Hard

Population Dynamics: Derivative and Series Analysis

A town's population is modeled by the continuous function $$P(t)= 1000e^{0.04t}$$, where t is in yea

Hard

Product and Quotient Rule Application

Consider the function $$f(x)=\frac{x*\ln(x)}{e^{x}+2}$$, defined for $$x>0$$. Analyze its behavior u

Hard

Product Rule in Differentiation

Suppose the cost function is given by $$Q(x)=(3*x^2 - x)*e^{x}$$, which represents a cost (in dollar

Medium

Radioactive Decay with Logarithmic Correction

A radioactive substance decays following the model $$A(t)=A_0*e^{-k*t}+\ln(t+1)$$, where $$t$$ is th

Hard

Revenue Change Analysis via the Product Rule

A company’s revenue (in thousands of dollars) is modeled by $$R(x) = (2*x + 3)*(x^2 - x + 4)$$, wher

Medium

Secant and Tangent Approximations from a Graph

A function f(t) has been graphed from t = 0 to 10 seconds. Use the graph to estimate rates of change

Medium

Tangent Line Estimation in Transportation Modeling

A vehicle's displacement along a highway is modeled by $$s(t)=\ln(3*t+1)*e^{t}$$, where $$t$$ denote

Medium

Traffic Flow Analysis

A highway on-ramp has vehicles entering at a rate of $$V_{in}(t)=30+2*t$$ vehicles per minute and ve

Medium

Urban Population Flow

A city’s population changes due to migration. The inflow of people is modeled by $$M_{in}(t)=8-0.5*t

Medium

Water Reservoir Depth Analysis

The depth of water (in meters) in a reservoir is modeled by $$d(t)=10+3*t-0.5*t^2$$, where $$t$$ is

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

Analysis of a Piecewise Function with Discontinuities

Consider the piecewise function $$ f(x) = \begin{cases} 2*x+1, & x < 1, \\ 3, & 1 \le x \le 2, \\ \s

Easy

Analyzing the Rate of Change in an Economic Model

Suppose the profit function is given by $$P(x)=e^{x}-4*\ln(x+2)$$, where x represents the number of

Easy

Biological Growth Model Differentiation

In a biological model, the concentration of a chemical is modeled by $$C(t)=e^{-0.5*t}+\ln(2*t+3)$$.

Medium

Chain, Product, and Implicit: A Motion Problem

A particle moves along a curve defined by the parametric equations $$x(t)=e^{-t}\cos(t)$$ and $$y(t)

Medium

Composite and Inverse Differentiation in an Electrical Circuit

In an electrical circuit, the current is modeled by $$ I(t)= \sqrt{20*t+5} $$ and the voltage is giv

Hard

Composite Function: Polynomial Exponent

Consider the function $$ f(x)= (2*x^2+3*x-5)^3 $$. Analyze the function's derivative and behavior.

Easy

Differentiation Involving an Inverse Function and Logarithms

Let $$f(x)= x^2+ 3*x+ 2$$, and let y be defined by $$y= \ln(f^{-1}(e^{x}))$$, where $$f^{-1}$$ is th

Extreme

Engine Air-Fuel Mixture

In an engine, the fuel injection rate is modeled by the composite function $$F(t)=w(z(t))$$, where $

Medium

Enzyme Kinetics in a Biochemical Reaction

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

Extreme

Implicit Differentiation in a Conic Section

Consider the curve defined by $$x^2 + x*y + y^2 = 9$$.

Medium

Implicit Differentiation in a Cost-Production Model

In an economic model, the relationship between the production level $$x$$ (in units) and the average

Easy

Implicit Differentiation in a Hyperbola-like Equation

Consider the equation $$ x*y = 3*x - 4*y + 12 $$.

Medium

Implicit Differentiation in a Pressure-Temperature Experiment

In a chemistry experiment, the pressure $$P$$ (in atm) and temperature $$T$$ (in °C) of a system sat

Medium

Implicit Differentiation in an Elliptical Orbit

An orbit of a satellite is modeled by the ellipse $$4*x^2 + 9*y^2 = 36$$. At the point $$\left(1, \f

Medium

Implicit Differentiation in Circular Motion

Consider the circle described by $$x^2+y^2=49$$, representing a particle's path. Answer the followin

Medium

Implicit Differentiation in Geometric Optics

A parabolic mirror used in a geometric optics experiment is described by the implicit equation $$x^2

Hard

Implicit Differentiation with Exponential and Trigonometric Mix

Consider the equation $$e^{x*y} + \sin(x) - y = 0$$. Differentiate implicitly with respect to $$x$$

Extreme

Inverse Analysis of a Log-Polynomial Function

Consider the function $$f(x)=\ln(x^2+1)$$. Analyze its one-to-one property on the interval $$[0,\inf

Easy

Inverse Function Derivative with Logarithms

Let $$f(x)= \ln(x+2) + x$$ with inverse function $$g(x)$$. Find the derivative $$g'(y)$$ in terms of

Medium

Inverse Function Differentiation Basics

Let $$f$$ be a one-to-one differentiable function with $$f(3)=5$$ and $$f'(3)=2$$, and let $$g$$ be

Easy

Inverse Function Differentiation for Cubic Functions

Let $$f(x)= x^3 + 2*x$$, and let $$g(x)$$ be its inverse function. Answer the following:

Medium

Inverse Function Differentiation in Navigation

A vehicle’s distance traveled is modeled by $$f(t)= t^3 + t$$, where $$t$$ represents time in hours.

Medium

Logarithmic and Composite Differentiation

Let $$g(x)= \ln(\sqrt{x^2+1})$$.

Medium

Nested Composite Function Differentiation

Consider the function $$ h(x)= \sqrt{\cos(3*x^2+1)} $$.

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
Unit 4: Contextual Applications of Differentiation

Airplane Passing a Tower: A Related Rates Problem

An airplane is flying at a constant altitude of $$1000\; m$$ horizontally away from an observer on t

Hard

Analyzing Runner's Motion

A runner's displacement is modeled by the function $$s(t)=-t^3+9t^2+1$$, where s(t) is in meters and

Hard

Bacterial Growth and Linearization

A bacterial population is modeled by $$P(t)=100e^{0.3*t}$$, where $$t$$ is in hours. Answer the foll

Medium

Concavity and Acceleration in Motion

A car’s position is modeled by $$s(t)= t^3 - 6*t^2 + 9*t+5$$ with time $$t$$ in seconds. Analyze the

Medium

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

Conical Tank Water Flow

Water is pumped into a conical tank at a rate of $$\frac{dV}{dt}=9\text{ ft}^3/\text{min}$$. The tan

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

Economic Rates: Marginal Profit Analysis

A manufacturer’s profit (in dollars) from producing $$x$$ items is modeled by $$P(x)=500*x-2*x^2$$.

Medium

Estimating the Rate of Change from Reservoir Data

A reservoir's water level h (in meters) was recorded at different times, as shown in the table below

Medium

Exponential Cooling Rate Analysis

The temperature of a cup of coffee is modeled by $$T(t)=70+50e^{-0.05t}$$ (in °F), where t is in min

Medium

Fuel Consumption Rate Analysis

The fuel consumption of a car (in gallons per 100 miles) is modeled by $$C(v)=0.05*v^2+1$$, where $$

Medium

Horizontal Tangents on Cubic Curve

Consider the curve defined by $$x^3 + y^3 - 6*x*y = 0$$.

Extreme

Inflating Balloon: Radius and Surface Area

A spherical balloon is being inflated such that its volume increases at a constant rate of 12 cm³/s.

Medium

Inflating Balloon: Related Rates

A spherical balloon is being inflated such that its volume increases at a constant rate of 10 in³/s.

Medium

Inflating Spherical Balloon: A Related Rates Problem

A spherical balloon is being inflated so that its volume increases at a constant rate of $$12\; in^3

Easy

Linearization and Differentials: Approximating Function Values

Consider the function $$f(x)= x^4$$. Use linearization to estimate the value of the function for a s

Medium

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

Linearization of a Power Function

Let $$f(x)=x^4$$. Use linearization at $$x=4$$ with $$\Delta x=-0.02$$ to approximate $$(3.98)^4$$.

Easy

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

Mixing a Saline Solution: Related Rates

A tank contains a saline solution with a constant volume of 50 liters. Salt is added at a rate of 2

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

Population Growth Rate

The population of a bacteria culture is given by $$P(t)= 500e^{0.03*t}$$, where $$t$$ is in hours. A

Easy

Related Rates in a Circular Pool

A circular pool is being filled such that the surface area increases at a constant rate of $$10$$ ft

Easy

Series Approximation of a Temperature Function

The temperature in a chemical reaction is modeled by $$T(t)= 100 + \sum_{n=1}^{\infty} \frac{(-1)^n

Easy

Water Tank Filling: Related Rates

A cylindrical tank has a fixed radius of 2 m. The volume of water in the tank is given by $$V=\pi*r^

Easy
Unit 5: Analytical Applications of Differentiation

Absolute Extrema and the Candidate’s Test

Let $$f(x)=x^3-3x^2-9x+5$$ be defined on the closed interval $$[-2,5]$$. Answer the following parts:

Medium

Analysis of an Exponential-Linear Function

Consider the function $$p(x)=e^x-4*x$$. Answer the following parts:

Hard

Analysis of Total Distance Traveled

A particle moves along a line with a velocity function given by $$v(t)=t^2-4*t+3$$ for $$t\in[0,5]$$

Medium

Analyzing Convergence of a Modified Alternating Series

Consider the series $$S(x)=\sum_{n=1}^\infty (-1)^n * \frac{(x+2)^n}{n}$$. Answer the following.

Hard

Application of the Mean Value Theorem

Consider the function $$f(t)=t^3-3*t^2+2*t+5$$ representing the position (in meters) of a car along

Medium

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

Area Between a Curve and Its Tangent

Consider the curve $$f(x)=x^2$$ and its tangent line at \(x=1\). Investigate the region bounded by t

Hard

Asymptotic Behavior and Limits of a Logarithmic Model

Examine the function $$f(x)= \ln(1+e^{-x})$$ and its long-term behavior.

Easy

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

Concavity and Inflection Points

Examine the function $$p(x)= x^3-3*x^2-9*x+30$$ to determine its concavity and any inflection points

Medium

Curve Sketching with Second Derivative

Consider the function $$f(x) = x^4 - 4*x^3 + 6*x^2 - 4*x + 1$$.

Medium

Determining Absolute Extrema for a Trigonometric-Polynomial Function

Consider the function $$f(x)= x+\cos(x)$$ defined on the closed interval $$[0, 2\pi]$$. Determine th

Medium

Drug Dosage Infusion

A patient receives an intravenous drug infusion at a rate given by $$D(t)=4*\exp(-0.2*t)$$ mg/min. A

Medium

Dynamic Analysis Under Time-Varying Acceleration in Two Dimensions

A particle moves in the plane with acceleration given by $$\vec{a}(t)=\langle3\cos(t),-2\sin(t)\rang

Extreme

Epidemic Infection Model

In a community experiencing an epidemic, the rate of new infections is modeled by $$I(t)=\frac{200}{

Hard

Graph Analysis of a Logarithmic Function

Consider the function $$g(x)= \ln(x) - \frac{1}{x}$$ defined for $$x>0$$. Analyze its behavior and g

Medium

Interpreting a Velocity-Time Graph

A particle’s velocity over the interval $$[0,6]$$ seconds is depicted in the graph provided.

Medium

Linear Approximation of a Radical Function

For the function $$f(x)= \sqrt{x+1}+x$$, find its linear approximation at $$x=3$$ and use it to appr

Easy

Mean Value Theorem Application

Let $$f(x)=\ln(x)$$ be defined on the interval $$[1, e^2]$$. Answer the following parts using the Me

Medium

Mean Value Theorem in Motion

A car travels along a straight highway with its position given by $$s(t)=4*t^3 - 12*t^2 + 9*t + 5$$

Medium

Mean Value Theorem in River Flow

A river cross‐section’s depth (in meters) is modeled by the function $$f(x) = x^3 - 4*x^2 + 3*x + 5$

Medium

Optimization in a Geometric Setting: Garden Design

A farmer is designing a rectangular garden adjacent to a river. No fence is needed along the river s

Easy

Pharmaceutical Dosage and Metabolism

A patient is administered a medication with an initial dose of 50 mg. Due to metabolism, the amount

Medium

Population Growth Modeling

A region's population (in thousands) is recorded over a span of years. Use the data provided to anal

Medium

Rolle's Theorem: Modeling a Car's Journey

An object moves along a straight line and its position is given by $$s(t)= t^3-6*t^2+9*t$$ for $$t$$

Easy

Series Convergence and Integration in a Physical Model

A physical process is modeled by the power series $$g(x)=\sum_{n=1}^\infty \frac{2^n}{n!} * (x-3)^n$

Medium

Sign Chart Construction from the Derivative

Consider the function $$ f(x)=x^4-4x^3+6x^2.$$ Answer the following parts:

Medium

Square Root Function Inverse Analysis

Consider the function $$f(x)= \sqrt{3*x+4}$$ defined for $$x \ge -\frac{4}{3}$$. Answer the followin

Medium

Water Tank Volume Analysis

Water is being added to a tank at a varying rate given by $$r(t) = 3*t^2 - 12*t + 15$$ (in liters/mi

Medium

Wireless Signal Attenuation

A wireless signal, originally at an intensity of 80 units, passes through a series of walls. Each wa

Medium
Unit 6: Integration and Accumulation of Change

Antiderivative with Initial Condition

Find the general antiderivative of the function $$f(x)=5*x^3-2*x+6$$ and determine the particular an

Easy

Applying the Fundamental Theorem of Calculus

Consider the function $$f(x)=2*x$$. Use the Fundamental Theorem of Calculus to evaluate the definite

Easy

Approximating Energy Consumption Using Riemann Sums

A household’s power consumption (in kW) is recorded over an 8‐hour period. The following table shows

Easy

Approximating Water Volume Using Riemann Sums

A storm causes a varying inflow rate f(t) (in m³/h) into a reservoir. The inflow rate was recorded a

Easy

Arc Length Calculation

Find the arc length of the curve $$y=\frac{1}{3}x^{3/2}$$ from $$x=0$$ to $$x=9$$.

Medium

Area Between Curves

Consider the curves $$f(x)=x^2$$ and $$g(x)=4*x - 3$$. These curves intersect and enclose a region.

Hard

Area Under the Curve for a Quadratic Function

Consider the quadratic function $$h(x)= x^2 + 2*x$$. Find the area between the curve and the $$x$$-a

Hard

Bacterial Growth with Logarithmic Integration

A bacterial culture grows at a rate given by $$P'(t)=100/(t+2)$$ (in bacteria per hour). Given that

Medium

Definite Integration of a Polynomial Function

For the function $$f(x)=5*x^{3}$$ defined on the interval $$[1,2]$$, determine the antiderivative an

Easy

Drug Concentration in a Bloodstream

A patient receives an intravenous drug infusion at a rate $$R_{in}(t)=3e^{-0.1*t}$$ mg/min for $$0 \

Hard

Estimating Integral from Tabular Data

Given the following table of values for $$F(t)$$ over time, estimate the integral $$\int F(t)\,dt$$

Easy

Evaluating a Complex Integral

Evaluate the integral $$\int_{0}^{2} 2*x*(x+1)^3\,dx$$ using an appropriate substitution method.

Hard

Integration of a Trigonometric Product via U-Substitution

Evaluate the indefinite integral $$\int \sin(2*x)\cos(2*x)\,dx$$.

Medium

Limit of a Riemann Sum as a Definite Integral

Consider the limit of the Riemann sum given by $$\lim_{n \to \infty} \sum_{i=1}^{n} \left( \frac{6}{

Medium

Logarithmic Functions in Ecosystem Models

Let \(f(t)= \ln(t+2)\) for \(t \ge 0\) model an ecosystem measurement. Answer the following question

Medium

Modeling Water Inflow Using Integration

Water flows into a tank at a rate given by $$R(t)=4-0.5*t$$ (in liters per minute) for $$t\in[0,8]$$

Easy

Particle Motion with Changing Velocity Signs

A particle is moving along a line with its velocity given by $$v(t)= 6 - 4*t$$ (in m/s) for t betwee

Hard

Particle Motion with Variable Acceleration and Displacement Analysis

A particle moves along a straight line with acceleration given by $$a(t)=4-2*t$$ (in m/s²). At time

Medium

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

Rainfall Accumulation and Runoff

During a storm, rainfall intensity is modeled by $$R(t)=3*t$$ inches per hour for $$0 \le t \le 4$$

Medium

Region Bounded by a Parabola and a Line: Area and Volume

Consider the region bounded by the curves $$y=x^{2}$$ and $$y=2*x+3$$. Answer the following:

Medium

Sandpile Accumulation

At an industrial site, sand is continuously added to and removed from a pile. The addition rate is g

Medium

Taylor/Maclaurin Series Approximation and Error Analysis

Consider the function $$f(x)=\ln(1+x)$$. This function is infinitely differentiable at x = 0 and has

Hard

Trapezoidal Approximation for a Curved Function

Consider the function $$f(x)=x^2+2$$ on the interval [1, 5]. Answer the following:

Easy

Vehicle Distance Estimation from Velocity Data

A vehicle's velocity over time is recorded in the table provided. Use Riemann sums to estimate the v

Medium

Volume of a Solid with Known Cross-sectional Area

A solid extends from $$x=0$$ to $$x=5$$, and its cross-sectional area perpendicular to the x-axis is

Hard

Volume of a Solid with Square Cross-Sections

Consider the region bounded by the curve $$y=x^{2}$$ and the line $$y=4$$. Cross-sections taken perp

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 an Inverse Function from a Differential Equation Solution

Suppose a differential equation is solved to give an increasing function $$f(x)=\ln(2*x+3)$$ defined

Medium

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

Bacterial Growth with Time-Dependent Growth Rate

A bacterial culture grows according to the differential equation $$\frac{dP}{dt}=(r_0+r_1*t)P$$, whe

Hard

Capacitor Discharge in an RC Circuit

In an RC circuit, the voltage $$V(t)$$ across a discharging capacitor obeys the differential equatio

Easy

Car Engine Temperature Dynamics

The temperature $$T(t)$$ (in °C) of a car engine is modeled by the differential equation $$\frac{dT}

Hard

Chemical Reactor Mixing

In a chemical reactor, the concentration $$C(t)$$ (in M) of a chemical is governed by the equation $

Hard

Constructing and Interpreting a Slope Field

Consider the differential equation $$\frac{dy}{dx} = \sin(x) - y$$. Answer the following:

Hard

Cooling of a Smartphone Battery

A smartphone battery cools according to Newton’s law: $$\frac{dT}{dt} = -k*(T-T_{room})$$. Initially

Easy

Differential Equation Involving Logarithms

Consider the differential equation $$\frac{dy}{dx} = (y-1)*\ln|y-1|$$ with the initial condition $$y

Hard

Economic Growth Model

An economy's output \(Y(t)\) is modeled by the differential equation $$\frac{dY}{dt}= a\,Y - b\,Y^2$

Extreme

Exact Differential Equation

Consider the differential equation $$ (2*x*y + y^2)\,dx + (x^2+2*x*y)\,dy = 0 $$. Answer the followi

Hard

Exponential Growth via Slope Field Analysis

Consider the differential equation $$\frac{dy}{dx} = x * y$$ with the initial condition $$y(0)=2$$.

Easy

Exponential Growth with Variable Rate

A bacterial culture grows according to the differential equation $$\frac{dP}{dt}=k(t)P$$, where the

Medium

Free-Fall with Air Resistance

An object falling under gravity experiences air resistance proportional to the square of its velocit

Extreme

Gas Pressure Dynamics

A container is being filled with gas such that the pressure $$P(t)$$ (in psi) increases at a constan

Easy

Implicit Differential Equations and Slope Fields

Consider the implicit differential equation $$x\frac{dy}{dx}+ y = e^x$$. Answer the following parts.

Medium

Implicit Differentiation from an Implicitly Defined Relation

Consider the implicit equation $$x^3 + y^3 - 3*x*y = 0$$ which defines $$y$$ as a function of $$x$$

Hard

Inverse Function Analysis Derived from a Differential Equation Solution

Consider the function $$f(x)=x^3+2$$. Although this function is provided outside of a differential e

Medium

Logistic Differential Equation Analysis

A population $$P(t)$$ grows according to the logistic differential equation $$\frac{dP}{dt} = r\,P\,

Hard

Logistic Growth Population Model

A population is modeled by the logistic differential equation $$\frac{dP}{dt}=k*P\left(1-\frac{P}{M}

Hard

Mixing Problem with Differential Equations

A tank initially contains $$S(0)=S_0$$ grams of salt dissolved in a volume $$V$$ liters of water. Br

Medium

Mixing Tank with Variable Inflow

A tank initially contains 50 L of water with 5 kg of salt dissolved in it. A brine solution with a s

Medium

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

Newton's Law of Cooling is given by the differential equation $$\frac{dT}{dt} = -k*(T-T_a)$$, where

Medium

Picard Iteration for Approximate Solutions

Consider the initial value problem $$\frac{dy}{dt}=y+t, \quad y(0)=1$$. Use one iteration of the Pic

Medium

Population Growth with Harvesting

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

Medium

Predator-Prey Model with Harvesting

Consider a simplified model for the prey population in a predator-prey system that includes constant

Extreme

Radio Signal Strength Decay

A radio signal's strength $$S$$ decays with distance r according to the differential equation $$\fra

Easy

Second-Order Differential Equation: Oscillations

Consider the second-order differential equation $$\frac{d^2y}{dx^2}= -9*y$$ with initial conditions

Medium

Separable Differential Equation with Parameter Identification

A chemical reaction is modeled by the differential equation $$\frac{dC}{dt} = -a*C$$, where $$C(t)$$

Medium

Slope Field Analysis for $$\frac{dy}{dx}=x$$

Consider the differential equation $$\frac{dy}{dx}= x$$. A slope field for this differential equatio

Easy

Slope Field and Sketching a Solution Curve

The differential equation $$\frac{dy}{dx}=x-y$$ has been represented by a slope field. Answer the fo

Medium

Solution and Analysis of a Linear Differential Equation with Equilibrium

Consider the differential equation $$\frac{dy}{dx} = 3*y - 2$$, with the initial condition $$y(0)=1$

Medium

Solution Curve from Slope Field

A differential equation is given by $$\frac{dy}{dx} = -y + \cos(x)$$. A slope field for this equatio

Hard
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

Analyzing Convergence of a Taylor Series

Consider the function $$g(x)= e^{-x^2}$$. Analyze the Maclaurin series for this function.

Hard

Arc Length and Average Speed for a Parametric Curve

A particle moves along a path defined by the parametric equations $$x(t)=t^2$$ and $$y(t)=2*t$$ for

Medium

Arc Length of the Logarithmic Curve

For the function $$f(x)=\ln(x)$$ defined on the interval $$[1,e]$$, determine the arc length of the

Medium

Area Between Curves from Experimental Data

In an experiment, researchers recorded measurements for two functions, $$f(t)$$ and $$g(t)$$, repres

Medium

Area Between Curves in a Physical Context

The heights of two particles moving along parallel tracks are given by $$h_1(t)=t^2$$ and $$h_2(t)=4

Easy

Area Between Curves: Enclosed Region

The curves $$f(x)=x^2$$ and $$g(x)=x+2$$ enclose a region. Answer the following:

Medium

Area Between Curves: Parabolic and Linear Functions

Consider the functions $$f(x)=5*x-x^2$$ and $$g(x)=x$$. Determine the area enclosed between these cu

Medium

Area Between Exponential Curves

Consider the functions $$f(x)=e^{-x}$$ and $$g(x)=e^{-2*x}$$ for $$x\ge0$$. Answer the following:

Hard

Area of One Petal of a Polar Curve

The polar curve defined by $$r = \cos(2\theta)$$ forms a rose with four petals. Find the area of one

Hard

Average and Instantaneous Analysis in Periodic Motion

A particle moves along a line with its displacement given by $$s(t)= 4*\cos(2*t)$$ (in meters) for $

Hard

Average Chemical Concentration Analysis

In a chemical reaction, the concentration of a reactant (in M) is recorded over time as given in the

Easy

Average Reaction Concentration in a Chemical Reaction

In a chemical reaction, the concentration of a reactant is modeled by $$C(t)=20*\exp(-0.5*t)$$ (in m

Hard

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 Value of a Temperature Function

A region’s temperature throughout a day is modeled by the function $$T(t)=10+5*\sin(\frac{\pi}{12}*t

Easy

Bonus Payout: Geometric Series vs. Integral Approximation

A company issues monthly bonuses that decrease by 20% each month. The bonus in the first month is $5

Hard

Center of Mass of a Rod

A thin rod of length 10 m has a linear density given by $$\rho(x)=3+0.4*x$$ (in kg/m) where $$x$$ is

Medium

Cost Analysis: Area Between Production Cost Curves

Suppose two cost functions for producing goods are given by $$f(x)=20+2*x$$ and $$g(x)=5*x-\frac{1}{

Medium

Electric Current and Charge

An electric current in a circuit is defined by $$I(t)=4*\cos\left(\frac{\pi}{10}*t\right)$$ amperes,

Medium

Electrical Charge Distribution

A wire of length 3 meters carries a charge distribution given by $$\rho(x)=\frac{5}{1+x^2}$$ (in cou

Medium

Net Change and Direction of Motion

A particle’s velocity is given by $$v(t)=\sin(t)-\frac{1}{2}*t$$ for $$0 \le t \le 6$$.

Medium

Particle Motion Analysis with Variable Acceleration

A particle moving along a straight line has an acceleration given by $$a(t)=4*e^{-t}-\sin(t)$$ (in m

Medium

River Crossing: Average Depth and Flow Calculation

The depth of a river along a 100-meter cross-section is modeled by $$d(x)=2+\cos\left(\frac{\pi}{50}

Medium

Solid of Revolution using Washer Method

The region bounded by the curves $$y = x^2$$ and $$y = 2 * x$$ is rotated about the x-axis. Answer t

Medium

Volume by the Shell Method: Rotating a Region

Consider the region bounded by the curve $$y=\sqrt{x}$$, the line $$y=0$$, and the vertical line $$x

Medium

Volume of a Solid of Revolution Between Curves

Consider the region bounded by the curves $$y=\sqrt{x}$$ and $$y=\frac{x}{2}$$ for $$x \in [0,4]$$.

Medium

Work Done in Pumping Water from a Parabolic Tank

A water tank has a parabolic cross-section described by $$y=x^2$$ (with y in meters, x in meters). T

Hard

Work Done in Pumping Water from a Tank

A cylindrical tank has a radius of $$3$$ meters and a height of $$10$$ meters. The tank is completel

Hard

Work Done Pumping Water

A water tank is shaped like an inverted circular cone with a height of $$10$$ m and a top radius of

Hard

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

Arc Length of a Parametrically Defined Curve

A curve is defined by the parametric equations $$x(t)=t^2$$ and $$y(t)=\frac{t^3}{3}$$ for $$0 \leq

Medium

Arc Length of a Quarter-Circle

Consider the circle defined parametrically by $$x(t)=\cos(t)$$ and $$y(t)=\sin(t)$$ for $$0 \le t \l

Easy

Arc Length of a Vector-Valued Function

Let $$ r(t)=\langle 3*t,\; 4*\sin(t) \rangle $$ for $$ 0 \le t \le \pi $$. Determine properties rela

Medium

Concavity and Inflection in Parametric Curves

A curve is defined by the parametric functions $$x(t)=t^3-3*t$$ and $$y(t)=t^2$$ for \(-2\le t\le2\)

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

Curvature and Oscillation in Vector-Valued Functions

Given the vector function $$\vec{r}(t)=\langle \ln(t+1), \cos(t) \rangle$$ for $$t \ge 0$$, answer t

Hard

Curve Analysis and Optimization in a Bus Route

A bus follows a route described by the parametric equations $$x(t)=t^3-3*t$$ and $$y(t)=2*t^2-t$$, w

Extreme

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

Dynamics in Polar Coordinates

A particle moves such that its polar coordinates are given by $$ r(\theta)=1+\theta $$, where $$ \th

Medium

Inflow and Outflow in a Water Tank

A water tank has water entering at a rate given by $$I(t)=5+\sin(t)$$ (liters per minute) and water

Medium

Intersection and Area Between Polar Curves

Two polar curves are given by $$r_1(\theta)=2\sin(\theta)$$ and $$r_2(\theta)=1+\cos(\theta)$$.

Extreme

Intersection of Parametric Curves

Consider two particles moving along different paths: Particle A: $$x_A(t)= t^2, \quad y_A(t)= 2t +

Medium

Intersections in Polar Coordinates

Two polar curves are given by $$r = 3 - 2*\sin(\theta)$$ and $$r = 1 + \cos(\theta)$$.

Extreme

Motion Along a Helix

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

Medium

Multi-Step Problem Involving Polar Integration and Conversion

Consider the polar curves $$r_1(\theta)= 2\cos(\theta)$$ and $$r_2(\theta)=1$$.

Extreme

Parametric Curve with a Loop and Tangent Analysis

Consider the parametric curve defined by $$x(t)=t^3-3*t$$ and $$y(t)=t^2$$, where $$t\in[-2,2]$$. An

Extreme

Parametric Equations from Real-World Data

A moving vehicle’s position is modeled by the parametric equations $$ x(t)=3*t+1 $$ and $$ y(t)=t^2-

Medium

Particle Motion in the Plane

A particle moves in the plane with parametric equations $$x(t)= 3\cos(t)$$ and $$y(t)= 3\sin(t)$$ fo

Easy

Particle Motion in the Plane

Consider a particle whose motion in the plane is defined by the parametric equations $$x(t) = t^2 -

Medium

Periodic Motion: A Vector-Valued Function

A point moves on a circle with position given by $$\vec{r}(t)= \langle \cos(2t), \sin(2t) \rangle$$

Easy

Polar Boundary Conversion and Area

A region in the polar coordinate plane is defined by $$1 \le r \le 3$$ and $$0 \le \theta \le \frac{

Hard

Polar Differentiation and Tangent Lines

Consider the polar curve $$r(\theta)= 1+\cos(\theta)$$.

Hard

Vector-Valued Function with Constant Acceleration

A particle moves in the plane with its position given by $$\vec{r}(t)=\langle 5*t, 3*t+2*t^2 \rangle

Medium

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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.