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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 (26)
  • Unit 2: Differentiation: Definition and Fundamental Properties (19)
  • Unit 3: Differentiation: Composite, Implicit, and Inverse Functions (25)
  • Unit 4: Contextual Applications of Differentiation (33)
  • Unit 5: Analytical Applications of Differentiation (30)
  • Unit 6: Integration and Accumulation of Change (29)
  • Unit 7: Differential Equations (36)
  • Unit 8: Applications of Integration (28)
  • Unit 9: Parametric Equations, Polar Coordinates, and Vector-Valued Functions (24)
Unit 1: Limits and Continuity

Analyzing a Piecewise Defined Function Near a Boundary

Consider the function $$g(x)=\begin{cases} \frac{x^2-4}{x-2} & \text{if } x<2, \\ 2*x+1 & \text{if

Medium

Application of the Squeeze Theorem with Trigonometric Functions

Let $$f(x)= x^2 \sin\left(\frac{1}{x}\right)$$ for $$x\neq0$$, and $$f(0)=0$$. Analyze the behavior

Medium

Asymptotic Behavior and Horizontal Limits

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

Medium

Asymptotic Behavior of a Water Flow Function

In a reservoir, the net water flow rate is modeled by the rational function $$R(t)=\frac{6\,t^2+5\,t

Hard

Calculating Tangent Line from Data

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

Medium

Continuity Across Piecewise‐Defined Functions with Mixed Components

Let $$ f(x)= \begin{cases} e^{\sin(x)} - \cos(x), & x < 0, \\ \ln(1+x) + x^2, & 0 \le x < 2, \\

Extreme

Continuity Analysis of an Integral Function

Let $$F(x)=\int_{0}^{x} f(t)\,dt,$$ where $$ f(t)= \begin{cases} t+1, & t < 2 \\ 3, & t \ge 2 \end{

Medium

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 for a Logarithmic Transform Function

Consider the function $$f(x)= \ln\Bigl(\frac{e^{3x}-1}{x}\Bigr)$$ for $$x \neq 0$$ and define $$f(0)

Medium

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

Electricity Consumption Rate Analysis

A table provides the instantaneous electricity consumption, $$E(t)$$ (in kW), at various times durin

Medium

Epsilon-Delta Proof for a Polynomial Function

Let $$f(x)=x^2+3*x+2$$. Answer the following:

Hard

Experimental Data Limit Estimation from a Table

Using the table below, estimate the behavior of a function f(x) as x approaches 1.

Easy

Exploring the Squeeze Theorem

Define the function $$ f(x)= \begin{cases} x^2*\cos\left(\frac{1}{x}\right), & x\neq0 \\ 0, & x=0

Medium

Exponential Function Limits at Infinity

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

Easy

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

Indeterminate Forms in Log‐Exponential Context

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

Medium

Intermediate Value Theorem Application

Let $$g(x)=x^3+2*x-1$$ be defined on the interval [0, 1].

Medium

Limits Involving Infinity and Vertical Asymptotes

Consider the function $$f(x)=\frac{1}{x-3}$$. Answer the following:

Easy

Limits of Composite Trigonometric Functions

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

Easy

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

One-Sided Limits and Discontinuities

Consider the function $$p(x)=\begin{cases} x^2+1, & x<2, \\ 4*x-3, & x\ge2. \end{cases}$$ Answer t

Easy

One-Sided Limits and Jump Discontinuities

Consider the piecewise function $$j(x)=\begin{cases}x+2 & \text{if } x< 3,\\ 5-x & \text{if } x\ge 3

Medium

Piecewise Function Critical Analysis

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

Easy

Squeeze Theorem with a Log Function

Let $$f(x)= x\,\ln\Bigl(1+\frac{1}{x}\Bigr)$$ for $$x > 0$$. Use the Squeeze Theorem to determine $$

Easy

Squeeze Theorem with an Oscillatory Factor

Consider the function $$f(x)= x*\cos(\frac{1}{x})$$ for $$x \neq 0$$, with f(0) defined as 0. Use th

Easy
Unit 2: Differentiation: Definition and Fundamental Properties

Calculating Velocity and Acceleration from a Position Function

A car’s position along a straight road is given by the function $$s(t)= 0.5*t^3 - 3*t^2 + 4*t + 2$$

Easy

Derivative Using Limit Definition

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

Hard

Graphical Estimation of Tangent Slopes

Using the provided graph of a function g(t), analyze its rate of change at various points.

Hard

Growth Rate of a Bacterial Colony

The radius of a bacterial colony is modeled by $$r(t)= \sqrt{4*t+1}$$, where t (in hours) represents

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

Implicit Differentiation: Conic with Mixed Terms

Consider the curve defined by $$x*y + y^2 = 6$$.

Medium

Inflection Points and Concavity Analysis

Consider the function $$f(x)=x^3 - 6*x^2+9*x+1$$. Answer the following:

Medium

Instantaneous Rate of Change of a Trigonometric Function

Consider the function $$h(t)=\sin(2*t) + \cos(t)$$ which models the displacement (in centimeters) of

Medium

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

Linearization and Tangent Approximations

Let $$f(x)=e^{-x}$$ represent a cost decay function over time. Use linear approximation near $$x=0$$

Easy

Optimization in a Chemical Reaction

The rate of a chemical reaction is modeled by the function $$R(x)=x*e^{-x}+\ln(x+2)$$, where $$x$$ r

Hard

Rate of Change in a Logarithmic Function

Consider the function $$f(x)=\frac{\ln(x)}{x}$$ defined for \(x>0\). Answer the following:

Medium

Related Rates: Constant Area Rectangle

A rectangle maintains a constant area of $$A = l*w = 50$$ m², where the length l and width w vary wi

Medium

Related Rates: Sweeping Spotlight

A spotlight located at the origin rotates at a constant rate of $$2 \text{ rad/s}$$. A wall is posit

Easy

Tangent Line Approximation vs. Taylor Series for ln(1+x)

An engineer studying the function $$f(x)=\ln(1+x)$$ is comparing the tangent line approximation with

Medium

Taylor Series of ln(x) Centered at x = 1

A researcher studies the natural logarithm function $$f(x)=\ln(x)$$ by constructing its Taylor serie

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

Traffic Flow and Instantaneous Rate

The number of cars passing through an intersection per minute is modeled by $$F(t)= 3t^2 + 2t + 10$$

Medium

Water Tank: Inflow-Outflow Dynamics

A water tank initially contains $$1000$$ liters of water. Water enters the tank at a rate of $$R_{in

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

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 and Higher-Order Derivatives

Given the function $$f(x)= \ln(\sqrt{1 + e^{3*x}})$$, answer the following parts:

Hard

Coffee Cooling Dynamics using Inverse Function Differentiation

A cup of coffee cools according to the model $$T=100-a\,\ln(t+1)$$, where $$T$$ is the temperature i

Hard

Combined Differentiation: Composite, Implicit, and Inverse Analysis

A complex system is modeled by the equation $$\sqrt{1+xy}+\ln(x+y)=3,$$ which relates the variables

Extreme

Composite, Implicit, and Inverse: A Multi-Method Analysis

Let $$F(x)=\sqrt{\ln(5*x+9)}$$ for all x such that $$5*x+9>0$$, and let y = F(x) with g as the inver

Hard

Differentiation in an Economic Cost Function

The cost of producing $$q$$ units is modeled by $$C(q)= (5*q)^{3/2} + 200*\ln(1+q)$$. Differentiate

Medium

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

Differentiation of an Inverse Trigonometric Composite Function

Let $$f(x)= \arctan(e^{2*x})$$. Answer the following parts:

Medium

Differentiation of an Inverse Trigonometric Composite Function

Let $$y = \arcsin(\sqrt{x})$$. Answer the following:

Easy

Differentiation of Inverse Trigonometric Functions

Consider the function $$f(x)= \sin(x)$$ for $$x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$$ and

Easy

Differentiation of the Inverse Function in a Mechanics Experiment

An object's displacement is described by a one-to-one differentiable function \(s(t)\). It is given

Easy

Engine Air-Fuel Mixture

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

Medium

Fuel Tank Dynamics

A fuel storage tank is being filled by a pump at a rate given by the composite function $$P(t)=(4*t+

Hard

Implicit Differentiation and Concavity of a Logarithmic Curve

The curve defined implicitly by $$y^3 + x*y - \ln(x+y) = 5$$ is given. Use implicit differentiation

Hard

Implicit Differentiation for an Elliptical Path

An ellipse is given by the equation $$4*x^2 + y^2 = 16$$. Answer the following parts.

Medium

Implicit Differentiation of a Composite Equation

Given the implicit relation $$x^2*y + \sin(y) = x$$, answer the following:

Medium

Implicit Differentiation with Logarithmic Equation

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

Hard

Implicit Differentiation with Trigonometric Components

Consider the equation $$x*\sqrt{y} + \cos(y) = x^2$$, where $$y$$ is a function of $$x$$. Differenti

Hard

Implicit Differentiation: Second Derivatives of a Circle

Given the circle $$x^2+y^2=10$$, answer the following parts:

Medium

Implicit Equation with Logarithmic and Exponential Terms

The relation $$\ln(x+y)+e^{x-y}=3$$ defines y implicitly as a function of x. Answer the following pa

Hard

Inverse Function Differentiation in Economics

In an economic model, the price function $$f(x)$$ is differentiable and one-to-one, mapping the quan

Easy

Inverse Trigonometric Differentiation

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

Medium

Population Dynamics in a Fishery

A lake is being stocked with fish as part of a conservation program. The number of fish added per da

Medium

Reservoir Levels and Evaporation Rates

A reservoir is being filled with water from an inflow while losing water through controlled release

Medium

Second Derivative of an Implicit Function

The curve is defined implicitly by the equation $$x^2*y+y^3=4$$. Answer the following parts:

Hard
Unit 4: Contextual Applications of Differentiation

Air Pressure Change in a Sealed Container

The air pressure in a sealed container is modeled by $$P(t)=\frac{100}{1+e^{-0.5*(t-10)}}$$, where $

Hard

Analyzing Concavity through the Second Derivative

A particle’s position is given by $$x(t)=\ln(t^2+1)$$, where $$t$$ is in seconds.

Medium

Area and Volume of Bounded Polynomials

Consider the region in the first quadrant bounded by the curves $$y = x^2$$ and $$y = 4 - x$$. Use t

Medium

Balloon Inflation and Related Rates

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

Easy

Biological Growth Rate

A bacterial culture grows according to the model $$P(t)= 500*e^{0.8*t}$$, where \(P(t)\) is the popu

Medium

Boat Crossing a River: Relative Motion

A boat must cross a 100 m wide river. The boat's speed relative to the water is 5 m/s (directly acro

Medium

Comparison of Series Convergence and Error Analysis

Consider the series $$S(x)= \sum_{n=0}^{\infty} \frac{(-1)^n x^n}{2^n}$$ and $$T(x)= \sum_{n=0}^{\in

Hard

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

Expanding Pool Rates

The area $$A$$ of a circular swimming pool is given by $$A=\pi*r^2$$. The pool is being filled so th

Easy

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

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

Inversion of an Absolute Value Function

Consider the function $$f(x)=|x-3|+2$$ with the domain restricted to $$x\ge3$$. Analyze its inverse.

Medium

L'Hôpital's Analysis

Evaluate the limit $$\lim_{x\to\infty}\frac{5*x^3-4*x^2+1}{7*x^3+2*x-6}$$. Answer the following part

Medium

Linearization Approximation Problem

Given the function $$f(x)=\sqrt{x+4}$$, use linearization to approximate the value of $$f(5.1)$$.

Easy

Linearization for Approximating Function Values

Let $$f(x)= \sqrt{x}$$. Use linearization at $$x=10$$ to approximate $$\sqrt{10.1}$$. Answer the fol

Easy

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

Maximizing a Rectangular Enclosure Area

A farmer has 100 m of fencing to enclose a rectangular area. Answer the following:

Easy

Maximizing Efficiency: Derivative Analysis in a Production Process

The efficiency of a production process is modeled by $$E(x)=50+10*\ln(x)-0.5*x$$, where $$x$$ repres

Medium

Maximizing the Area of an Inscribed Rectangle

A rectangle is inscribed in a semicircle of radius $$R$$, with its base along the diameter. The rect

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

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

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

Particle Motion Along a Line with Polynomial Velocity

A particle moves along the x-axis with velocity $$v(t)=4*t^3-9*t^2+6*t-1$$ (m/s). Given that $$s(0)=

Medium

Pool Water Volume Change

The volume of water in a pool is described by the function $$V(t)=8*t^2-32*t+4$$, where $$V$$ is mea

Easy

Projectile Motion Analysis

A projectile is launched such that its horizontal and vertical positions are modeled by the parametr

Hard

Rational Function Particle Motion Analysis

A particle moves along a straight line with its position given by $$s(t)=\frac{t^2+1}{t-1}$$, where

Hard

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

A spherical balloon is being inflated so that its volume, given by $$V= \frac{4}{3}\pi*r^3$$, increa

Medium

Series Approximation in an Exponential Population Model

A population is modeled by $$P(t)= 1000 \times \sum_{n=0}^{\infty} \frac{(0.05t)^n}{n!}$$, which is

Hard

Solids of Revolution: Washer vs Shell Methods

Consider the region enclosed by $$y = \sin(x)$$ and $$y = \cos(x)$$ for $$0 \le x \le \frac{\pi}{4}$

Hard

Temperature Change of Coffee: Exponential Cooling

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

Easy

Vehicle Motion on a Curved Path

A vehicle moving along a straight road has its position given by $$s(t)= 4*t^3 - 24*t^2 + 36*t + 5$$

Medium

Vertical Projectile Motion

An object is thrown vertically upward with an initial velocity of 20 m/s and experiences a constant

Medium
Unit 5: Analytical Applications of Differentiation

Amusement Park Ride Braking Distance

An amusement park ride uses a sequence of friction pads to stop a roller coaster. The first pad diss

Easy

Analysis of a Cubic Function

Consider the function $$f(x)=x^3-6*x^2+9*x+2$$. Using this function, answer the following parts.

Medium

Analysis of an Absolute Value Function

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

Hard

Application of Rolle's Theorem

Consider the function $$f(x) = x^2 - 4*x + 4$$ on the interval $$[0,4]$$.

Easy

Bacterial Culture with Periodic Removal

A laboratory experiment involves a bacterial culture that, at the beginning of an hour, has 200 bact

Medium

Concavity and Inflection Points in a Trigonometric Function

Consider the function $$f(x)=\sin(x)-\frac{1}{2}*x$$ on the interval [0, 2π]. Answer the following p

Medium

Cumulative Angular Displacement Analysis

A rotating wheel has an angular acceleration given by $$\alpha(t)=4-0.6*t$$ (in rad/s²), with an ini

Medium

Derivative Sign Chart and Function Behavior

Given the function $$ f(x)=\frac{x}{x^2+1},$$ answer the following parts:

Medium

Differentiability of a Piecewise Function

Consider the piecewise function $$r(x)=\begin{cases} x^2, & x \le 2 \\ 4*x-4, & x > 2 \end{cases}$$.

Hard

Echoes in an Auditorium

In an auditorium, an audio signal produces echoes. The first echo has an intensity that is 70% of th

Medium

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 Experimental Data

A set of experimental measurements was recorded over time. Analyze the following data regarding the

Easy

Graph Interpretation of a Function's Second Derivative

Using the provided graph of the second derivative, analyze the concavity of the original function $$

Medium

Increasing/Decreasing Intervals for a Rational Function

Consider the function $$f(x) = \frac{x^2}{x+2}$$, defined for $$x > -2$$ (with $$x \neq -2$$).

Hard

Integration of a Series Representing an Economic Model

An economist models the marginal cost by the power series $$MC(q)=\sum_{n=0}^\infty (-1)^n * \frac{q

Easy

Inverse Analysis for a Function with Multiple Transformations

Consider the function $$f(x)= 2*(x-1)^3 + 3$$ for all real numbers. Answer the following parts.

Easy

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

Linear Particle Motion Analysis

A particle moves along a straight line with acceleration given by $$a(t)=6-4*t$$ (in m/s²) for $$t\g

Medium

Mean Value Theorem Application

Let \( f(x) = x^3 - 3*x^2 + 2 \) be defined on the closed interval \([0,3]\). Answer the following p

Easy

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 Motion

A car travels along a straight road and its position is modeled by $$s(x) = x^2$$ (in kilometers), w

Medium

Modeling Real World with the Mean Value Theorem

A car travels along a straight road with its position at time $$t$$ (in seconds) given by $$ s(t)=0.

Hard

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

Rate of Change in a Chemical Reaction

The concentration of a reactant in a chemical reaction is modeled by $$C(t)=10*e^{-0.5*t} + 2$$ (in

Hard

Rational Function Discontinuities

Consider the rational function $$ R(x)=\frac{(x-3)(x+2)}{(x-3)(x-1)}.$$ Answer the following parts:

Medium

Relative Extrema Using the First Derivative Test

Consider the function $$ f(x)=e^{-x^2}.$$ Answer the following parts:

Easy

Taylor Polynomial for $$\cos(x)$$ Centered at $$x=\pi/4$$

Consider the function $$f(x)=\cos(x)$$. You will generate the second degree Taylor polynomial for f(

Hard

Taylor Series for an Integral Function: $$F(x)=\int_0^x \sin(t^2)\,dt$$

Because the antiderivative of $$\sin(t^2)$$ cannot be expressed in closed form, use its power series

Hard

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
Unit 6: Integration and Accumulation of Change

Accumulated Change Prediction

A population grows continuously at a rate proportional to its size. Specifically, the growth rate is

Hard

Advanced U-Substitution with a Quadratic Expression

Evaluate the indefinite integral $$\int \frac{2*x}{\sqrt{x^{2}+1}}\,dx$$ using u-substitution.

Hard

Antiderivative Application in Crop Growth

A crop field grows at a rate modeled by the function $$G'(t)=4*t-3$$ (in square meters per week). Th

Medium

Antiderivatives and the Fundamental Theorem

Suppose a continuous function $$h(x)$$ is defined on [2, 8] and its graph (provided) shows that it i

Easy

Arc Length of $$y=x^{3/2}$$ on $$[0,4]$$

The curve defined by $$y=x^{3/2}$$ is given for $$x\in[0,4]$$. The arc length of a curve is determin

Hard

Arc Length of a Power Function

Find the arc length of the curve $$y=\frac{1}{3}*x^{3/2}$$ on the interval $$[0,9]$$.

Hard

Area and Volume of a Region Bounded by Trigonometric Functions

Consider the curves $$y=\sin(x)$$ and $$y=\cos(x)$$ for $$0 \le x \le \frac{\pi}{4}$$. Answer the fo

Medium

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

Car Acceleration, Velocity, and Distance

In a physics experiment, the acceleration of a car is modeled by the function $$a(t)=4*t-1$$ (in m/s

Hard

Composite Functions and Inverses

Consider \(f(x)= x^2+1\) for \(x \ge 0\). Answer the following questions regarding \(f\) and its inv

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

Determining the Average Value via Integration

Find the average value of the function $$f(x)=3*x^2-2*x+1$$ on the interval $$[1,4]$$.

Easy

Distance Traveled by a Particle

A particle has a velocity function given by $$v(t)=t^2-4*t+3$$ (in m/s) for $$t\in [0,5]$$ seconds.

Medium

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 Rainfall Accumulation

Rainfall intensity measurements (in mm/hr) at various times are recorded in the table. Use Riemann s

Medium

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 by U-Substitution and Evaluation of a Definite Integral

Evaluate the definite integral $$\int_{0}^{1} \frac{2*t}{(t^2+1)^2}\, dt$$ by applying U-substitut

Medium

Integration via U-Substitution for a Composite Function

Evaluate the integral of a composite function and its definite form. In particular, consider the fun

Medium

Logistic Growth and Population Integration

A population grows according to the logistic differential equation $$\frac{dP}{dt}=k*P\left(1-\frac

Easy

Motion and Accumulation: Particle Displacement

A particle moving along a straight line has a velocity function given by $$v(t)=3*t^{2} - 12*t + 5$$

Medium

Rate of Production in a Factory

A factory has a production rate given by $$R(t)=100+20*\cos\left(\frac{\pi*t}{4}\right)$$ units per

Hard

Riemann Sum Approximations: Midpoint vs. Trapezoidal

Consider the function $$f(x)=e^(-x)$$ on the interval [0, 3]. Compare the approximations for the def

Easy

Total Cost from a Marginal Cost Function

A company’s marginal cost function is given by $$MC(x)= 4*x+7$$ (in dollars per unit), where x repre

Easy

Trapezoidal Approximation for a Curved Function

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

Easy

Trapezoidal Sum Approximation for $$f(x)=\sqrt{x}$$

Consider the function $$f(x)=\sqrt{x}$$ on the interval $$[0,4]$$. Use a trapezoidal sum with 4 equa

Easy

U-Substitution in Accumulation Functions

In a chemical reactor, the accumulation rate of a substance is given by $$R(x)= 3*(x-2)^4$$ units pe

Medium

Volume by Cross-Section: Squares on a Parabolic Base

A solid has a base in the xy-plane bounded by the curves $$y=x^2$$ and $$y=4$$. Cross-sections perpe

Hard

Water Volume Accumulation with a Faulty Sensor Reading

Water flows into a container at a rate given by $$ r(t)= \begin{cases} 4t, & 0 \le t < 3, \\ 10, & t

Extreme

Work Done by an Exponential Force

A variable force acting along the x-axis is given by $$F(x)=5 * e^(0.5 * x)$$ (in Newtons) for 0 \(\

Medium
Unit 7: Differential Equations

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

Chain Reaction in a Nuclear Reactor

A simplified model for a nuclear chain reaction is given by the logistic differential equation $$\fr

Extreme

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

Cooling Cup of Coffee

A cup of coffee at an initial temperature of $$95^\circ C$$ is placed in a room. For the first 5 min

Medium

Differential Equation in a Gravitational Context

Consider the differential equation $$\frac{dv}{dt}= -G\,\frac{M}{(R+t)^2}$$, which models a simplifi

Extreme

Euler's Method Approximation

Let the differential equation be $$\frac{dy}{dx} = x+y$$ with the initial condition $$y(0)=1$$. Usin

Hard

Exact Differential Equations

Consider the differential equation $$ (2*x + y) + (x + 3*y)\,\frac{dy}{dx} = 0$$.

Hard

Exponential Growth and Decay

A bacterial population grows according to the differential equation $$\frac{dy}{dt}=k\,y$$ with an i

Easy

Exponential Growth with Shifted Dependent Variable

The differential equation $$\frac{dy}{dx} = e^{x}*(y+2)$$ is used to model a growth process where th

Medium

FRQ 2: Separable Differential Equation with Initial Condition

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

Medium

FRQ 13: Cooling of a Planetary Atmosphere

A planetary atmosphere cools according to Newton's Law of Cooling: $$\frac{dT}{dt}=-k(T-T_{eq})$$, w

Medium

FRQ 18: Enzyme Reaction Rates

A chemical concentration $$C(t)$$ in a reaction decreases according to the differential equation $$\

Easy

Implicit Differentiation in a Differential Equation Context

Suppose the function $$y(x)$$ satisfies the implicit equation $$x\,e^{y}+y^2=7$$. Differentiate both

Medium

Logistic Growth in Populations

A population of organisms is modeled by the logistic differential equation $$\frac{dP}{dt} = rP \lef

Medium

Logistic Population Growth Model

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

Medium

Maclaurin Series Solution for a Differential Equation

Given the differential equation $$\frac{dy}{dx} = y * \cos(x)$$ with initial condition $$y(0)=1$$, f

Hard

Modeling Exponential Growth

A population follows the differential equation $$\frac{dP}{dt} = k*P$$. Given that the population do

Easy

Modeling Temperature in a Biological Specimen

A biological specimen initially at $$37^\circ C$$ is cooling in an environment where the ideal ambie

Medium

Newton's Law of Cooling: Temperature Change

A hot object is cooling in a room with an ambient temperature of 20°C. Measurements of the object's

Medium

Nonlinear Differential Equation with Implicit Solution

Consider the nonlinear differential equation $$\frac{dy}{dx} = \frac{x^2+y^2}{2xy}$$. An implicit so

Extreme

Parameter Identification in a Cooling Process

The temperature of an object cooling in an environment at $$20^\circ C$$ is modeled by Newton's Law

Medium

Parametric Equations and Differential Equations

A particle moves in the plane along a curve defined by the parametric equations $$x(t)=\ln(t)$$ and

Hard

Pollutant Concentration in a Lake

A lake receives a pollutant at a constant rate of $$5$$ kg/day and the pollutant is removed at a rat

Easy

Population Dynamics with Harvesting

A species population is modeled by the differential equation $$\frac{dP}{dt} = 0.2*P\left(1-\frac{P}

Hard

Population Dynamics with Harvesting

Consider a population model that includes constant harvesting, given by the differential equation $$

Hard

Radio Signal Strength Decay

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

Easy

Radioactive Decay Data Analysis

A radioactive substance is decaying over time. The following table shows the measured mass (in grams

Medium

Second-Order Differential Equation in a Mass-Spring System

A mass-spring system without damping is modeled by the differential equation $$m\frac{d^2x}{dt^2}+kx

Medium

Separable Differential Equation with Initial Condition

Solve the differential equation $$\frac{dy}{dx} = \frac{x}{y}$$ subject to the initial condition $$y

Easy

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 and Solution Curve Sketching for $$\frac{dy}{dx}= x - y$$

Consider the differential equation $$\frac{dy}{dx} = x - y$$ with initial condition $$y(0)=1$$. You

Medium

Slope Field and Equilibrium Analysis for $$\frac{dy}{dx}= y*(1-y)$$

Consider the autonomous differential equation $$\frac{dy}{dx}= y*(1-y)$$. Answer the following:

Medium

Solution Curve Sketching Using Slope Fields

Given the differential equation $$\frac{dy}{dx} = x - y$$, a slope field is provided. Use the field

Medium

Solving a Linear Differential Equation using an Integrating Factor

Consider the linear differential equation $$\frac{dy}{dx} + \frac{2}{x} * y = \frac{\sin(x)}{x}$$ wi

Hard

Spring-Mass System with Damping

A spring-mass system with damping is modeled by the differential equation $$m\frac{d^2y}{dt^2}+ c\fr

Hard

Water Pollution with Seasonal Variation

A river receives a pollutant with a time-varying influx modeled by $$I(t)=20+5\cos(0.5*t)$$ kg/day,

Medium
Unit 8: Applications of Integration

Area Between a Function and Its Tangent Line

Let $$f(x)=x^3-x$$. At the point $$x=1$$, find the tangent line to the curve and determine the area

Hard

Area Between a Parabola and a Line

Let $$f(x)= x^2$$ and $$g(x)= 2*x + 3$$. Determine the area of the region bounded by these two curve

Hard

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 Under a Parametric Curve

Consider the parametric equations $$x= t^2$$ and $$y= t^3 + t$$ for $$t \in [0,2]$$. Find the area u

Extreme

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 function modeling the temperature (in °F) throughout a day is given by $$T(t)= 3*\sin\left(\frac{\

Easy

Balloon Inflation Related Rates

A spherical balloon is being inflated such that its radius $$r(t)$$ (in centimeters) increases at a

Easy

Car Braking and Stopping Distance

A car decelerates with an acceleration given by $$a(t)=-2*t$$ (in m/s²) and has an initial velocity

Medium

Complex Integral Evaluation with Exponential Function

Evaluate the integral $$I=\int_1^e \frac{2*\ln(t)}{t}dt$$ by applying a suitable substitution.

Medium

Consumer Surplus Analysis

The demand function for a product is given by $$D(p)=120-2*p$$, where \(p\) is the price in dollars.

Medium

Cost Analysis of a Water Channel

A water channel has a cross-sectional shape defined by the region bounded by $$y=\sqrt{x}$$ and $$y=

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

Distance Traveled from a Velocity Function

A car has a velocity function given by $$v(t)=t^2-4*t+3$$ (in m/s) for $$t$$ in seconds from 0 to 5.

Medium

Distance Traveled versus Displacement

A particle moves along a line with velocity given by $$v(t)=t^3 - 6*t^2 + 9*t$$ (in m/s) for $$t\in[

Medium

Fluid Flow in a River

The rate of water flow in a river is given by $$Q(t)=50+10*\sin\left(\frac{\pi}{6}*t\right)$$ cubic

Easy

Fluid Force on a Submerged Plate

A vertical plate submerged in water experiences a force due to fluid pressure given by $$F(y)=\rho*g

Hard

Moment of Inertia of a Thin Plate

A thin plate occupies the region bounded by the curves $$y= x$$ and $$y= x^2$$ for $$0 \le x \le 1$$

Medium

Optimization and Integration: Maximum Volume

A company designs open-top cylindrical containers to hold $$500$$ liters of liquid. (Recall that $$1

Extreme

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

Population Growth: Cumulative Increase

A bacterial culture grows at a rate given by $$r(t)=3*e^{0.2*t}$$ (in thousands per hour), where $$t

Medium

Series Convergence and Approximation

Consider the function defined by the infinite series $$f(x)=\sum_{n=0}^{\infty} \frac{(-1)^n*x^{2*n}

Extreme

Surface Area of a Solid of Revolution

Consider the curve $$y= \sqrt{x}$$ over the interval $$0 \le x \le 4$$. When this curve is rotated a

Hard

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 by the Washer Method

The region bounded by $$y=x^2$$ and $$y=4$$ is rotated about the x-axis, forming a solid with a hole

Hard

Volume of a Solid Rotated about y = -1

The region bounded by $$y=\sqrt{x}$$ and $$y=0$$ for $$x\in[0,9]$$ is rotated about the line $$y=-1$

Hard

Volume of a Solid with Equilateral Triangle Cross Sections

Consider the region bounded by $$y= \sqrt{x}$$ and $$y=0$$ for $$x \in [0,1]$$. A solid is formed by

Hard

Volume Using Washer Method

Consider the region bounded by the curves $$y=\sqrt{x}$$ and $$y=\frac{x}{2}$$. This region is rotat

Hard

Work Done by a Variable Force

A force acting on an object moving along a straight line is given by $$F(x)= 6 - x$$ (in Newtons) as

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

Acceleration in Polar Coordinates

An object moves in the plane with its position expressed in polar coordinates by $$r(t)= 4+\sin(t)$$

Medium

Analysis of a Cycloid

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

Extreme

Arc Length of a Cycloid

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

Medium

Arc Length of a Parabolic Curve

The parametric curve defined by $$x(t)=t$$ and $$y(t)=t^2$$ models a portion of a parabolic path for

Easy

Arc Length of a Parametric Curve

Consider the parametric equations $$x(t) = t^2$$ and $$y(t) = t^3$$ for $$0 \le t \le 2$$.

Medium

Arc Length of a Parametric Curve

Consider the parametric curve defined by $$ x(t)=t^2 $$ and $$ y(t)=t^3 $$ for $$ 0 \le t \le 2 $$.

Medium

Arc Length of a Polar Curve

Consider the polar curve given by $$r(θ)= 1+\sin(θ)$$ for $$0 \le θ \le \pi$$. Answer the following:

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

Conversion Between Polar and Cartesian Coordinates

Given the polar equation $$r=4\cos(\theta)$$, explore its conversion and properties.

Easy

Conversion of Polar to Cartesian Coordinates

Consider the polar curve $$ r=4*\cos(\theta) $$. Analyze its Cartesian equivalent and some of its pr

Easy

Converting Polar to Cartesian and Computing Slope

The polar curve is given by the equation $$r=4\cos(\theta)$$. Answer the following:

Medium

Designing a Parametric Curve for a Cardioid

A cardioid is described by the polar equation $$r(\theta)=1+\cos(\theta)$$.

Medium

Intersection of Polar Curves

Consider the polar curves given by $$r=2\sin(\theta)$$ and $$r=1+\cos(\theta)$$. Answer the followin

Medium

Logarithmic Exponential Transformations in Polar Graphs

Consider the polar equation $$r=2\ln(3+\cos(\theta))$$. Answer the following:

Extreme

Motion Along a Helix

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

Medium

Motion of a Particle in the Plane

A particle moves in the plane with parametric equations $$x(t)=t^2-4*t$$ and $$y(t)=2*t^3-6*t^2$$ fo

Medium

Parametric and Polar Conversion Challenge

Consider the parametric equations $$x(t)= \frac{1-t^2}{1+t^2}$$ and $$y(t)= \frac{2*t}{1+t^2}$$ for

Extreme

Parametric Egg Curve Analysis

An egg-shaped curve is modeled by the parametric equations $$x(t)= \cos(t)+0.5\cos(2t)$$ and $$y(t)=

Hard

Parametric Intersection of Curves

Consider the curves $$C_1: x(t)=\cos(t),\, y(t)=\sin(t)$$ for $$0 \le t \le 2\pi$$ and $$C_2: x(s)=1

Hard

Polar Coordinate Area Calculation

Consider the polar curve $$r=4*\sin(θ)$$ for $$0 \le θ \le \pi$$. This equation represents a circle.

Easy

Projectile Motion using Parametric Equations

A projectile is launched with an initial speed of $$v_0 = 20\,\text{m/s}$$ at an angle of $$30^\circ

Medium

Vector-Valued Functions and 3D Projectile Motion

An object's position in three dimensions is given by $$\mathbf{r}(t)=\langle 3t, 4t, 10t-5t^2 \rangl

Medium

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

Weather Data Analysis from Temperature Table

A meteorologist records the temperature (in $$^\circ C$$) at a weather station at various times (in

Easy

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Tips from Former AP Students

FAQ

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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.