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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 $AP Calculus BC FRQs to get ready for the big day.

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  • Unit 1: Limits and Continuity (30)
  • Unit 2: Differentiation: Definition and Fundamental Properties (34)
  • Unit 3: Differentiation: Composite, Implicit, and Inverse Functions (22)
  • Unit 4: Contextual Applications of Differentiation (29)
  • Unit 5: Analytical Applications of Differentiation (27)
  • Unit 6: Integration and Accumulation of Change (22)
  • Unit 7: Differential Equations (30)
  • Unit 8: Applications of Integration (25)
  • Unit 9: Parametric Equations, Polar Coordinates, and Vector-Valued Functions (31)
Unit 1: Limits and Continuity

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

Analyzing Limits of a Combined Exponential‐Log Function

Consider $$f(x)= e^{-x}\,\ln(1+\sqrt{x})$$ for $$x \ge 0$$. Analyze the limits as $$x \to 0^+$$ and

Medium

Asymptotic Behavior in Rational Functions

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

Medium

Caffeine Metabolism in the Human Body

A person consumes a cup of coffee containing 100 mg of caffeine at the start, and then drinks one cu

Hard

Continuity and the Intermediate Value Theorem in Temperature Modeling

A temperature function $$T(t)$$, continuous for $$t\in[0,24]$$ (in hours), satisfies $$T(6)=10^\circ

Medium

Continuity in Piecewise Defined Functions

Consider the piecewise function $$f(x)= \begin{cases} x^2+1, & \text{if } x \leq 3 \\ 2*x+k, & \text

Easy

Electricity Consumption Rate Analysis

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

Medium

Endpoint Behavior of a Continuous Function

Let $$m(x)=\sqrt{x+4}$$ be defined on the interval $$[-4,5]$$. Answer the following:

Easy

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

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

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

Intermediate Value Theorem Application

Let $$f(x)=x^3-4*x+1$$, which is continuous on the real numbers. Answer the following:

Hard

Intermediate Value Theorem in Temperature Analysis

A city's temperature during a day is modeled by a continuous function $$T(t)$$, where t (in hours) l

Easy

Investigating Limits Involving Nested Rational Expressions

Evaluate the limit $$\lim_{x\to3} \frac{\frac{x^2-9}{x-3}}{x-2}$$. (a) Simplify the expression and e

Easy

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

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

Medium

Limits Involving Absolute Value Functions

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

Easy

Limits Involving Radicals

Consider the function $$f(x)=\frac{\sqrt{x+4}-2}{x}$$ defined for $$x \neq 0$$. Answer the following

Hard

Limits Involving Trigonometric Functions and the Squeeze Theorem

Examine the following trigonometric limits: (a) Evaluate $$\lim_{x\to0} \frac{\sin(4*x)}{x}$$. (b) E

Medium

Modeling Temperature Change with Continuity

A city’s temperature throughout the day is modeled by the continuous function $$T(t)=\frac{1}{2}t^2-

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

Non-Existence of a Limit due to Oscillation

Consider the function $$h(x)= \sin(\frac{1}{x})$$. Answer the following regarding its limit as x app

Medium

One-Sided Limits and Jump Discontinuity Analysis

Consider the piecewise function $$ f(x)= \begin{cases} x+2, & x < 1 \\ 3-x, & x \ge 1 \end{cases} $

Medium

Physical Applications: Temperature Continuity

A temperature sensor records temperature (in °C) over time according to the function $$T(t)=\frac{t^

Medium

Piecewise Functions and Continuity

Consider the function defined by $$ f(x)=\begin{cases} \frac{x^2-1}{x-1}, & x \neq 1 \\ k, & x=1

Easy

Pond Ecosystem Nutrient Levels

In a pond ecosystem, nutrient input occurs from periodic runoff events. Each runoff adds 20 kg of nu

Hard

Reciprocal Function Behavior and Asymptotes

Examine the function $$f(x)= \frac{1}{x-1}$$.

Easy

Related Rates with an Expanding Spherical Balloon

A spherical balloon is being inflated so that its volume increases at a rate of $$\frac{dV}{dt}=100\

Medium

Squeeze Theorem in Oscillatory Functions

Consider the function $$f(x)= x\sin(1/x)$$ for $$x \neq 0$$ and define $$f(0)=0$$.

Hard

Trigonometric Rate Function Analysis

A pump’s output is modified by a trigonometric factor. The outflow rate is recorded as $$R(t)=\frac{

Easy
Unit 2: Differentiation: Definition and Fundamental Properties

Analyzing Motion Through Derivatives

A car’s position as a function of time is given by $$s(t) = 4*t^3 - 12*t^2 + 9*t + 5,$$ where $$s

Medium

Applying Product and Quotient Rules

For the function $$h(x)=\frac{(3*x^2+2)*(x-4)}{x+1}$$, determine its derivative by appropriately app

Hard

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

Biochemical Reaction Rates: Derivative from Experimental Data

The concentration of a reactant in a chemical reaction is modeled by $$C(t)= 8 - 5t + t^2$$ (in M) w

Medium

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

Car Acceleration: Secant and Tangent Slope

A car's position along a straight road is given by $$s(t)= 2t^3 - 9t^2 + 12t$$, where s is in meters

Medium

Cost Optimization in Production: Derivative Application

A company's cost function is given by $$K(x)= 0.1x^3 - 2x^2 + 15x + 100$$, where x represents the nu

Medium

Derivative from First Principles: Quadratic Function

Consider the function $$f(x)= 3*x^2 + 2*x - 5$$. Use the limit definition of the derivative to compu

Easy

Derivative of a Composite Function Using the Limit Definition

Consider the function $$h(x)=(2*x+3)^3$$. Use the limit definition of the derivative to answer the f

Hard

Differentiation from First Principles

Let $$h(x)=3*x^2+2*x-1$$. Use the limit definition of the derivative to analyze this function.

Medium

Differentiation of Functions with Variable Exponents

Consider the function $$Z(x)=x^{\sin(x)}$$ which represents a complex growth model. Differentiate th

Extreme

Differentiation of Inverse Functions

Let $$f(x)=3*x+2$$ and let $$f^{-1}(x)$$ denote its inverse function. Answer the following:

Easy

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

Graphical Estimation of Tangent Slopes

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

Hard

Implicit Differentiation and Tangent Line Slope

Consider the curve defined by $$x^2 + x*y + y^2 = 7$$. Answer the following:

Medium

Implicit Differentiation of a Circle

Given the equation of a circle $$x^2 + y^2 = 25$$,

Easy

Implicit Differentiation: Exponential-Polynomial Equation

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

Hard

Implicit Differentiation: Square Root Equation

Consider the curve defined by $$\sqrt{x} + \sqrt{y} = \sqrt{10}$$, where $$x, y \ge 0$$.

Hard

Inflection Points and Concavity Analysis

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

Medium

Limit Definition of the Derivative for a Trigonometric Function

Consider the function $$f(x)= \cos(x)$$.

Hard

Maclaurin Polynomial for √(1+x)

A scientist approximates the function $$f(x)=\sqrt{1+x}$$ for small values of x using its Maclaurin

Easy

Optimization in Engineering Design

A manufacturer designs a cylindrical can with a fixed volume of $$1000\,cm^3$$. The surface area of

Hard

Population Growth Approximation using Taylor Series

A biologist models population growth with the exponential function $$P(t)=e^{0.05*t}$$. To estimate

Hard

Population Growth Rates

A city’s population (in thousands) was recorded over several years. Use the data provided to analyze

Medium

Rate Function Involving Logarithms

Consider the function $$h(x)=\ln(x+3)$$.

Medium

Rate of Change and Change in Market Trends

A company’s profit (in thousands of dollars) is modeled by the quadratic function $$P(x)=-2*x^2+40*x

Medium

Secant and Tangent Slope Analysis

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

Medium

Tangent Line Approximation

Consider the function $$f(x)=\cos(x)$$. Answer the following:

Easy

Tangent Line Estimation from Experimental Graph Data

A function $$f(x)$$ is represented by the following graph of experimental data approximating $$f(x)=

Medium

Tangent Line to a Curve

Consider the function $$f(x)=\sqrt{x+4}$$ modeling a physical quantity. Analyze the behavior at $$x=

Medium

Tangent Line to a Logarithmic Function

Consider the function $$f(x)= \ln(x+1)$$.

Medium

Temperature Function Analysis

Suppose the temperature over time is modeled by $$T(t)=e^(2*t)*\sin(t)$$, where $$t$$ is measured in

Medium

Velocity Function from a Cubic Position Function

An object’s position along a line is modeled by $$s(t) = t^3 - 6*t^2 + 9*t$$, where s is in meters a

Medium

Vibration Analysis: Rate of Change in Oscillatory Motion

The displacement of a vibrating string is given by $$f(t)= 3\sin(4t) - 2\cos(4t)$$, where t is in se

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

Analyzing a Composite Function with Nested Radicals

Consider the function $$h(x)=\sqrt{1+\sqrt{2+3x}}$$. Answer the following parts:

Medium

Chain Rule and Higher-Order Derivatives

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

Hard

Chain Rule for Inverse Trigonometric Functions in Optics

In an optics experiment, the angle of incidence $$\theta(t)$$ (in radians) is modeled by $$\theta(t)

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

Complex Composite and Implicit Function Analysis

Consider the equation $$e^{x*y}+\ln(x+y)=2$$, where y is defined implicitly as a function of x. Answ

Extreme

Composite Exponential Logarithmic Function Analysis

Consider the function $$f(x)=\ln(2*e^{3*x}+5)$$ which models a logarithmic transformation of an expo

Medium

Composite Function Modeling with Chain Rule

A chemical reaction rate is modeled by the composite function $$R(x)=f(g(x))$$ where $$f(u)=\sin(u)$

Easy

Composite Functions in Biological Growth

Let a model for bacteria growth be represented by $$f(t)=e^{2*t}$$, and let the effect of nutrient c

Medium

Composite Population Growth Function

A population model is given by $$P(t)= e^{3\sqrt{t+1}}$$, where $$t$$ is measured in years. Analyze

Medium

Differentiation of an Inverse Trigonometric Composite Function

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

Medium

Implicit Differentiation in Exponential Equation

Consider the equation $$e^{x*y}+x^2-y^3=0$$ that relates x and y. Answer the following parts:

Medium

Implicit Differentiation with Exponential and Trigonometric Components

Consider the relation $$ (x^2 + y^2) * e^{y} = x $$. Answer the following:

Medium

Implicit Differentiation with Product and Chain Rule in a Thermal Expansion Model

A material's length $$L$$ (in meters) under thermal expansion satisfies the equation $$L - \sin(L *

Extreme

Inverse Analysis of a Composite Exponential-Trigonometric Function

Let $$f(x)=e^x+\cos(x)$$. Analyze the behavior of its inverse function under appropriate domain rest

Extreme

Inverse Function Analysis for Exponential Functions

Let $$f(x)=e^{2*x}+1$$ and let g be the inverse function of f. Answer the following parts.

Medium

Inverse Function Derivative Calculation

Let $$f$$ be a one-to-one differentiable function for which the table below summarizes selected info

Easy

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 Trigonometric Differentiation

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

Medium

Modeling with Composite Functions: Pollution Concentration

The pollutant concentration in a lake is modeled by $$C(t) = \sqrt{100 - 2*e^{-0.1*t}}$$, where $$t$

Medium

Related Rates: Ladder Sliding Down a Wall

A ladder of length $$10\, m$$ leans against a wall such that its position is governed by $$x^2 + y^2

Easy

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

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

Analyzing Motion on a Curved Path

A particle moves along a path defined by $$x(t)=\cos(t)$$ and $$y(t)=\sin(t)$$ for $$t \in [0,2\pi]$

Medium

Arc Length Calculation

Consider the curve $$y = \sqrt{x}$$ for $$x \in [1, 4]$$. Determine the arc length of the curve.

Hard

Area Under a Curve: Definite Integral Setup

Consider the function $$f(x) = x^3 - 4x + 1$$ on the interval $$[0, 3]$$. Explore the area between t

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

Business Profit Optimization

A firm's profit is modeled by $$P(x)= -4*x^2 + 240*x - 1000$$, where $$x$$ (in hundreds) represents

Medium

City Population Migration

A city's population is influenced by immigration at a rate of $$I(t)=100e^{-0.2t}$$ (people per year

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

Cooling Coffee Temperature

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

Medium

Draining Hemispherical Tank

A hemispherical tank of radius $$5$$ m is draining. The volume of water in the tank is given by $$V

Hard

Graphical Analysis of Derivatives

A function $$f(x)$$ is plotted on the graph provided below. Using this graph, answer the following:

Hard

Horizontal Tangents on Cubic Curve

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

Extreme

Integration of Flow Rates Using the Trapezoidal Rule

A tank is being filled with water, and the flow rate Q (in L/min) is recorded at several time interv

Medium

Inverse of a Trigonometric Function

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

Easy

Inversion in a Light Intensity Decay Model

A laboratory experiment records the decay of light intensity over time, modeled by $$f(t)=80*e^{-0.2

Medium

L'Hôpital’s Rule in Chemical Reaction Rates

In a chemical reaction, the ratio of certain concentrations is modeled by $$R(x)=\frac{3*x^2-2*x+1}{

Easy

L'Hospital's Rule for Indeterminate Limits

Evaluate the limit $$\lim_{x \to \infty} \frac{5*x^3 - 4*x^2 + 1}{7*x^3 + 2*x - 6}$$ using L'Hospita

Medium

Linearization Approximation

Let $$f(x)=x^4$$. Linearization can be used to approximate small changes in a function's values. Ans

Easy

Minimizing Travel Time in Mixed Terrain

A hiker travels from point A to point B. On a flat plain the hiker walks at 5 km/h, but on an uphill

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

Polar Curve: Slope of the Tangent Line

Consider the polar curve defined by $$r(\theta)=10e^{-0.1*\theta}$$.

Extreme

Pollution Accumulation in a Lake

A lake is subject to pollution with pollutants entering at a rate of $$I(t)=3e^{0.1t}$$ (kg per day)

Hard

Production Cost Analysis

A company’s production cost $$C$$ (in dollars) and production level $$x$$ (in thousands of units) sa

Medium

Series Approximation for a Displacement Function

A displacement function is modeled by $$s(t)= \sum_{n=1}^{\infty} \frac{(-1)^{n+1} t^n}{n}$$, which

Medium

Sliding Ladder

A 10 m long ladder rests against a vertical wall. Let $$x$$ be the distance from the foot of the lad

Medium

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

Vector Function: Particle Motion in the Plane

A particle moves in the plane with a position vector given by $$\mathbf{r}(t)=\langle t, t^2 \rangle

Medium

Water Tank Flow Analysis

A water tank receives water from an inlet at a rate given by $$I(t)=4+\cos(t)$$ (liters per minute)

Medium
Unit 5: Analytical Applications of Differentiation

Analyzing Inverses in a Rate of Change Scenario

Consider the function $$f(x)= \ln(x+5) + x$$ defined for $$x > -5$$. This function models a system's

Medium

Analyzing The Behavior of a Log-Exponential Function Over a Specified Interval

Consider the function $$h(x)= \ln(x) + e^{-x}$$. A portion of its values is given in the following t

Medium

Area Between Curves and Rates of Change

An irrigation canal has a cross-sectional shape described by \( y=4-x^2 \) for \( |x| \le 2 \). The

Hard

Bacterial Culture with Periodic Removal

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

Medium

Bank Account Growth and Instantaneous Rate

A bank account balance is modeled by the function $$B(t) = 1000*e^{0.05*t}$$, where t (in years) rep

Easy

Chemical Reaction Rate

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

Easy

Combining Series and Integration to Analyze a Population Model

A population's growth rate is approximated by the series $$P'(t)=\sum_{n=0}^\infty \frac{t^n}{(n+1)!

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

Concavity in an Economic Model

Consider the function $$f(x)= x^3 - 3*x^2 + 2$$, which represents a simplified economic trend over t

Medium

Discounted Cash Flow Analysis

A project is expected to return cash flows that decrease by 10% each year from an initial cash flow

Hard

Epidemic Infection Model

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

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

Mean Value Theorem Application for Mixed Log-Exponential Function

Let $$h(x)= \ln(x) + e^{-x}$$ be defined on the interval [1,3]. Analyze the function using the Mean

Medium

Parameter Estimation in a Log-Linear Model

In a scientific experiment, the data is modeled by $$P(t)= A\,\ln(t+1) + B\,e^{-t}$$. Given that $$P

Medium

Projectile Motion Analysis

A projectile is launched vertically with its height given by $$s(t) = -16*t^2 + 64*t + 80$$ (in feet

Medium

Rate of Change and Inverse Functions

Let $$f(x)=x^3 + 3*x + 1$$, which is one-to-one. Investigate the rate of change of \(f(x)\) and its

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

Roller Coaster Height Analysis

A roller coaster's height (in meters) as a function of time (in seconds) is given by $$h(t) = -0.5*t

Hard

Salt Tank Mixing Problem

In a mixing tank, salt is added at a constant rate of $$A(t)=10$$ grams/min while the salt solution

Medium

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

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

Tangent Line to a Parametric Curve

A curve is defined by the parametric equations $$x(t) = \cos(t)$$ and $$y(t) = \sin(t) + \frac{t}{2}

Medium

Travel Distance from Speed Data

A traveler’s speed (in km/h) is recorded at various times during a trip. Use the data to approximate

Easy

Wastewater Treatment Reservoir

At a wastewater treatment reservoir, wastewater enters at a rate of $$W_{in}(t)=12+2*t$$ m³/min and

Extreme

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 a Velocity Function

A car’s velocity is given by the function $$v(t)=4 + t$$ (in m/s) over the interval [0, 8] seconds.

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

Area and Volume for an Exponential Function Region

Consider the curve $$y=e^{-x}$$ for $$x\ge0$$. Answer the following:

Medium

Average Value of a Function on an Interval

Let $$f(x)=\ln(x)$$ be defined on the interval $$[1,e]$$. Determine the average value of $$f(x)$$ on

Medium

Bacteria Growth with Nutrient Supply

A bacterial culture in a laboratory is provided with nutrients at a rate of $$N(t)=6*\ln(t+1)$$ mg/m

Medium

Bacterial Population Growth from Accumulated Rate

A bacteria population grows according to the rate function $$r(t)=k*t*e^{-t}$$ (in cells/hour) for \

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

Cost Accumulation via Integration

A manufacturing process has a marginal cost function given by $$MC(x)= 4 + 3*x$$ dollars per item, w

Easy

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

Differentiation and Integration of a Power Series

Consider the function given by the power series $$f(x)=\sum_{n=0}^\infty \frac{x^n}{2^n}$$.

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

Economics: Accumulated Earnings

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

Medium

Filling a Tank: Antiderivative with Initial Value

Water is entering a tank at a rate given by $$r(t)= \frac{2}{t+1}$$ liters per minute. The initial

Easy

Integrated Growth in Economic Modeling

A company experiences revenue growth at an instantaneous rate given by $$r(t)=0.5*t+2$$ (in millions

Medium

Integration of a Trigonometric Function by Two Methods

Evaluate the definite integral $$\int_0^{\frac{\pi}{2}} \sin(x)*\cos(x)\,dx$$ using two different me

Medium

Particle Motion and the Fundamental Theorem of Calculus

A particle moves along a straight line with its velocity given by $$v(t)=3*t^2-12*t+9$$ (in m/s) for

Medium

Particle Motion with Variable Acceleration

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

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

Scooter Motion with Variable Acceleration

A scooter's acceleration is given by $$a(t)= 2*t - 1$$ (m/s²) for $$t \in [0,5]$$, with an initial v

Hard

Temperature Change Analysis

A series of temperature readings (in °C) are recorded over the day as shown in the table. Analyze th

Medium

Work Done by a Variable Force

A variable force acting along a track is given by $$F(x)=6*\sqrt{x}$$ (in Newtons). Compute the work

Easy
Unit 7: Differential Equations

Autocatalytic Reaction Dynamics

Consider an autocatalytic reaction described by the differential equation $$\frac{dy}{dt} = k*y*\ln|

Extreme

Complex Related Rates Problem Involving a Moving Ladder

A 10-meter ladder leans against a vertical wall. The bottom of the ladder slides away from the wall

Extreme

Cooling of a Metal Rod

A metal rod cools according to the differential equation $$\frac{dT}{dt}=-k\,(T-25)$$ with an initia

Medium

Differential Equations in Compound Interest

An investment account grows with continuously compounded interest following $$\frac{dA}{dt}=rA$$, wh

Hard

Direction Fields and Isoclines

Examine the differential equation $$\frac{dy}{dx} = \frac{x+y}{x-y}$$. A direction field displaying

Extreme

Disease Spread Model

In a simplified epidemiological model, the number of infected individuals \(I(t)\) evolves according

Hard

Economic Investment Growth Model with Regular Deposits

An investment account grows with continuously compounded interest at a rate $$r$$ and receives conti

Medium

Forced Oscillation in a Damped System

Consider the differential equation $$\frac{dx}{dt}=-0.2*x+\sin(t)$$ with initial condition $$x(0)=1$

Medium

FRQ 4: Newton's Law of Cooling

A cup of coffee cools according to Newton's Law of Cooling, where the temperature $$T(t)$$ satisfies

Medium

FRQ 6: Radioactive Decay

A radioactive substance decays according to the differential equation $$\frac{dN}{dt}=-\lambda * N$$

Easy

FRQ 9: Epidemiological Model Differential Equation

An epidemic evolves according to the differential equation $$\frac{dI}{dt}=r*I*(M-I)$$, where $$I(t)

Hard

FRQ 12: Bacterial Growth with Limiting Resources

A bacterial culture grows according to the differential equation $$\frac{dP}{dt}=r*P-c*P^2$$, where

Hard

FRQ 17: Slope Field Analysis and Particular Solution

Consider the differential equation $$\frac{dy}{dx}=x-y$$. Answer the following parts.

Hard

Loan Balance with Continuous Interest and Payments

A loan has a balance $$L(t)$$ (in dollars) that accrues interest continuously at a rate of $$5\%$$ p

Hard

Logistic Growth in Populations

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

Medium

Logistic Model with Harvesting

A fish population is modeled by a modified logistic differential equation that includes harvesting.

Hard

Logistic Population Model

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

Medium

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 100 L of water with 5 kg of dissolved salt. Brine containing 0.1 kg of sal

Hard

Mixing Problem with Constant Rates

A tank contains $$200\,L$$ of a well-mixed saline solution with $$5\,kg$$ of salt initially. Brine w

Medium

Modeling Medication Concentration in the Bloodstream

A patient receives an intravenous drug at a constant rate $$R$$ (mg/min) and the drug is eliminated

Hard

Particle Motion in the Plane with Non-constant Acceleration

A particle moves in the $$xy$$-plane with an acceleration vector given by $$a(t)=\langle 2, e^t \ran

Medium

Particle Motion with Damping

A particle moving along a straight line is subject to damping and its motion is modeled by the secon

Hard

Particle Motion with Variable Acceleration

A particle moves along a straight line with an acceleration given by $$a(t)=6-4*t$$. At time t = 0,

Medium

Particle Motion with Variable Acceleration

A particle moving along a straight line has an acceleration given by $$a(t) = 6 - 4*t$$ (in m/s²). A

Medium

Reservoir Contaminant Dilution

A reservoir has a constant volume of 10,000 L and contains a pollutant with amount $$Q(t)$$ (in kg)

Medium

Separable DE: Basic SIPPY Problem

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

Medium

Series Convergence and Error Analysis

Consider the power series representation $$f(x)=\sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}$

Medium

Slope Field and Solution Curve Sketching

Consider the differential equation $$\frac{dy}{dx} = x - 1$$. A slope field for this differential eq

Easy

Solving a Nonlinear Differential Equation by Separation

Given the differential equation $$\frac{dy}{dx} = \frac{1+y^2}{1+x^2}$$ with the initial condition $

Hard
Unit 8: Applications of Integration

Analysis of Particle Motion in the Plane

A particle’s position is given by the vector function $$\mathbf{r}(t)=\langle e^{-t},\,\sin(t)\rangl

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: 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 Nonlinear Curves

Consider the functions $$f(x)=e^{-x}$$ and $$g(x)=\frac{x}{3}$$. Determine the area between these tw

Hard

Area Between Two Curves in a Water Channel

A channel cross‐section is defined by two curves: the upper boundary is given by $$f(x)=12-0.8*x$$ a

Easy

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 Temperature Over a Day

The temperature in a city over a 24-hour period is modeled by $$T(t)=10+5\sin\left(\frac{\pi}{12}*t\

Easy

Average Value of a Piecewise Function

Consider the function $$g(x)$$ defined piecewise on the interval $$[0,6]$$ by $$g(x)=\begin{cases} x

Hard

Average Value of a Velocity Function

The velocity of a car is modeled by $$v(t)=3*t^2-12*t+9$$ (m/s) for $$t\in[0,5]$$ seconds. Answer th

Medium

Average Value of a Velocity Function

A particle moves along a line with its velocity given by $$v(t)= 2*\cos(t) + \sin(t)$$ for $$t \in [

Easy

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

Cost Function from Marginal Cost

A manufacturing process has a marginal cost function given by $$MC(q)=3*\sqrt{q}$$, where $$q$$ (in

Medium

Displacement vs. Distance: Analysis of Piecewise Velocity

A particle moves along a line with velocity given by $$v(t)=\begin{cases} t^2, & 0 \le t < 2,\\ 8-t^

Hard

Draining a Conical Tank Related Rates

Water is draining from a conical tank that has a height of $$8$$ meters and a top radius of $$3$$ me

Hard

Integral Approximation Using Taylor Series

Approximate the integral $$\int_{0}^{0.2} \sin(2*x)\,dx$$ by using the Taylor series expansion of $$

Medium

Optimizing the Shape of a Parabolic Container

A container is formed by rotating the region under the curve $$y=8 - x^2$$ for $$0 \le x \le \sqrt{8

Extreme

Position from Velocity Function

A particle moves along a horizontal line with a velocity function given by $$v(t)=4*\cos(t) - 1$$ fo

Medium

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

Temperature Modeling: Applying the Average Value Theorem

The temperature of a chemical solution in a tank is modeled by $$T(t)=20+5\cos(0.5*t)$$ (°C) for $$t

Medium

Volume by Cross-Sectional Area (Square Cross-Sections)

A solid has a base in the xy-plane bounded by the curve $$y=\sqrt{x}$$ and the x-axis for $$x\in[0,4

Medium

Volume by Shell Method: Rotated Parabolic Region

Consider the region in the first quadrant bounded by the curve $$y=x^2$$ and the horizontal line $$y

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

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 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 Oscillatory Motion in Parametric Form

The motion of an oscillating particle is given by $$x(t)=e^{-t}\cos(2t)$$ and $$y(t)=e^{-t}\sin(2t)$

Medium

Arc Length of a Cycloid

A cycloid is generated by a circle of radius \(r=1\) rolling along a straight line. The cycloid is g

Hard

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 curve defined by $$x(t)= t^2$$ and $$y(t)= t^3$$ for $$0 \le t \le 1$$. Anal

Medium

Arc Length of a Parametric Curve with Logarithms

Consider the curve defined parametrically by $$x(t)=\ln(t+1)$$ and $$y(t)=\sqrt{t}$$ for $$0 \le t \

Medium

Arc Length of a Vector-Valued Curve

A vector-valued function is given by $$\mathbf{r}(t)=\langle e^t,\, \sin(t),\, \cos(t) \rangle$$ for

Hard

Area Between Polar Curves

Consider the polar curves defined by $$r_1= 4$$ and $$r_2= 2+2\cos(\theta)$$. Find the area of the r

Medium

Area between Two Polar Curves

Given two polar curves: $$r_1 = 1+\cos(\theta)$$ and $$r_2 = 2\cos(\theta)$$, consider the region wh

Hard

Area Between Two Polar Curves

Consider the polar curves $$ r_1=2*\sin(\theta) $$ and $$ r_2=\sin(\theta) $$. Determine the area of

Medium

Comparing Representations: Parametric and Polar

A curve is represented by the parametric equations $$x(t)=3\cos(t)-\sin(t)$$ and $$y(t)=3\sin(t)+\co

Hard

Conversion and Analysis of Polar and Rectangular Forms

Consider the polar equation $$r=3e^{\theta}$$. Answer the following:

Hard

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

Converting Polar to Cartesian and Computing Slope

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

Medium

Finding the Slope of a Tangent to a Parametric Curve

Consider the parametric equations $$x(t)=e^t$$ and $$y(t)=e^{-t}$$, where $$t \in \mathbb{R}$$.

Medium

Integrating a Vector-Valued Function

A particle has a velocity given by $$\vec{v}(t)= \langle e^t, \cos(t) \rangle$$. Its initial positio

Medium

Kinematics in Polar Coordinates

An object moves so that its position in polar coordinates is given by $$r(t)= 4 - t$$ and $$\theta(t

Hard

Motion Along a Parametric Curve

Consider the parametric equations $$x(t)= 2*t - 3$$ and $$y(t)= t^2$$, where $$t$$ represents time i

Medium

Motion Analysis via a Vector-Valued Function

An object's position is described by the vector function $$\mathbf{r}(t)= \langle e^{-t}, \; \ln(1+t

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 Equations and Tangent Slopes

Consider the parametric equations $$x(t)= t^3 - 3*t$$ and $$y(t)= t^2$$, for $$t \in [-2, 2]$$. Anal

Medium

Parametric Intersection and Enclosed Area

Consider the curves C₁ given by $$x=\cos(t)$$, $$y=\sin(t)$$ for $$0 \le t \le 2\pi$$, and C₂ given

Hard

Parametric Motion and Change of Direction

A particle moves along a path defined by the parametric equations $$x(t)=t^3-3t$$ and $$y(t)=2t^2$$

Medium

Particle Motion in Circular Motion

A particle moves along a circular path given by the parametric equations $$x(t)= 5\cos(t)$$ and $$y(

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

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

Polar Equations and Slope Analysis

Given the polar curve $$r = 4\sin(\theta)$$, analyze its Cartesian form and tangent properties.

Medium

Polar Spiral: Area and Arc Length

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

Medium

Sensitivity Analysis and Linear Approximation using Implicit Differentiation

The variables $$x$$ and $$y$$ satisfy the equation $$xy+\ln(y)=5$$.

Hard

Tangent Line to a Parametric Curve

Consider the circle parametrized by $$x(t)=3\sin(t)$$ and $$y(t)=3\cos(t)$$ for $$0\le t\le 2\pi$$.

Medium

Vector-Valued Function Integration

A particle moves along a straight line with constant acceleration given by $$ a(t)=\langle 6,\;-4 \r

Easy

Vector-Valued Functions in Motion

A particle's position is given by the vector-valued function $$\mathbf{r}(t)=\langle \sin(t), \cos(t

Medium

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FAQWe thought you might have some questions...
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.