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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 (28)
  • Unit 3: Differentiation: Composite, Implicit, and Inverse Functions (25)
  • Unit 4: Contextual Applications of Differentiation (29)
  • Unit 5: Analytical Applications of Differentiation (20)
  • Unit 6: Integration and Accumulation of Change (27)
  • Unit 7: Differential Equations (34)
  • Unit 8: Applications of Integration (28)
  • Unit 9: Parametric Equations, Polar Coordinates, and Vector-Valued Functions (31)
Unit 1: Limits and Continuity

Absolute Value Function Limit Analysis

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

Medium

Absolute Value Function Limits

Examine the function $$f(x)=\frac{|x-2|}{x-2}$$.

Hard

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 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

Applying the Squeeze Theorem

Let $$f(x)=x^2\sin(1/x)$$ for $$x \neq 0$$ and define $$f(0)=0$$. Use the Squeeze Theorem to evaluat

Medium

Compound Interest and Loan Repayment

A simplified model for a loan repayment assumes that a borrower owes $$10,000$$ dollars and the rema

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

Evaluating a Limit Involving a Radical and Trigonometric Component

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

Medium

Finding a Parameter in a Limit Involving Logs and Exponentials

Consider the function $$ f(x)= \frac{\ln(1+kx) - (e^x - 1)}{x^2}, $$ for $$x \neq 0$$. Assume that $

Easy

Graphical Analysis of Volume with a Jump Discontinuity

A graph of the water volume \(V(t)\) in a tank shows a jump discontinuity at \(t=6\) minutes. Answer

Medium

Infinite Limits and Vertical Asymptotes

Let $$g(x)=\frac{1}{(x-2)^2}$$. Answer the following:

Medium

Intermediate Value Theorem Application

Suppose $$f(x)$$ is a continuous function on the interval $$[1, 5]$$ with $$f(1) = -2$$ and $$f(5) =

Easy

Investigating a Function with a Removable Discontinuity

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

Easy

Jump Discontinuity Analysis using Table Data

A function f is defined by experimental measurements near $$x=2$$. Use the table provided to answer

Medium

Limit Definition of the Derivative for a Polynomial Function

Let $$f(x)=3*x^2-2*x+1$$. Use the limit definition of the derivative to find $$f'(2)$$.

Easy

Limits Involving Absolute Value Functions

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

Easy

Limits Involving Exponential Functions

Consider the function $$f(x)= \frac{e^{2*x}-1}{x}$$ defined for $$x\neq0$$.

Medium

Limits Involving Trigonometric Functions

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

Medium

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

Limits via Improper Integration Representation

Consider the function defined by the integral $$f(x)= \int_{1}^{x} \frac{1}{t^2} dt$$ for x > 1. Add

Hard

One-Sided Limits for a Piecewise Inflow

In a pipeline system, the inflow rate is modeled by the piecewise function $$R_{in}(t)= \begin{case

Easy

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

Oscillatory Behavior and Limits

Consider the function $$f(x)=x\sin(1/x)$$ for x \neq 0, with f(0) defined to be 0. Use the following

Medium

Piecewise Function Continuity and Differentiability

Consider the piecewise function $$ f(x)= \begin{cases} \frac{\sin(x)}{x} & \text{if } x \neq 0, \\

Hard

Removable Discontinuity and Limit Evaluation

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

Easy

Telecommunications Signal Strength

A telecommunications tower emits a signal whose strength decreases by $$20\%$$ for every additional

Medium

Temperature Change Analysis

The function $$T(t)$$ represents the temperature (in $$^\circ C$$) in a chemical reactor as a functi

Easy
Unit 2: Differentiation: Definition and Fundamental Properties

Analysis of a Quadratic Function

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

Easy

Analyzing Car Speed from a Distance-Time Table

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

Easy

Car Motion: Velocity and Acceleration

A car’s position along a straight road is given by $$s(t)=t^3-9*t$$, where $$t$$ is in seconds and $

Hard

Continuous Compound Interest Analysis

For an investment, the amount at time $$t$$ (in years) is modeled by $$A(t)=P*e^{r*t}$$, where $$P$$

Easy

Cost Minimization in Packaging

A company's packaging box has a cost function given by $$C(x)=0.05*x^2 - 4*x + 200$$, where $$x$$ is

Hard

Derivative from First Principles

Let $$f(x)=\sqrt{x}$$. Use the limit definition of the derivative to find $$f'(x)$$.

Medium

Difference Quotient and the Derivative Definition

Let $$g(x)=2*x^2-3*x+5$$. Use the difference quotient method to explore the rate of change of this f

Easy

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

Estimating Temperature Change

A scientist recorded the temperature of a liquid at different times (in minutes) as it was heated. U

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 in Circular Motion

A particle moves along the circle defined by $$x^2 + y^2 = 25$$. Answer the following parts.

Medium

Implicit Differentiation with Exponential and Trigonometric Functions

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

Hard

Linearization and Tangent Approximations

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

Easy

Market Price Rate Analysis

The market price of a product (in dollars) has been recorded over several days. Use the table below

Medium

Parametric Analysis of a Curve

A particle moves along a path given by the parametric equations $$x(t)=t^2+1$$ and $$y(t)=t^3-3*t$$,

Medium

Population Growth Rate

A population is modeled by $$P(t)=\frac{3*t^2 + 2}{t+1}$$, where $$t$$ is measured in years. Analyze

Medium

Population Model Rate Analysis

A city's population is modeled by $$P(x)=2000+500\ln(x)$$, where $$x$$ represents years since a base

Easy

Position Recovery from a Velocity Function

A particle moving along a straight line has a velocity function given by $$v(t)=6-3*t$$ (in m/s) for

Medium

Product Rule Application in Kinematics

A particle’s distance along a path is given by $$s(t)= t*e^(2*t)$$, where $$t$$ is in seconds. Answe

Hard

Related Rates: Changing Shadow Length

A 1.8 m tall man is walking away from a 5 m tall lamp at a constant speed of 1.2 m/s. The lamp casts

Medium

Related Rates: Expanding Balloon

A spherical balloon is being inflated so that its volume $$V$$ (in m³) and radius $$r$$ (in m) satis

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

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

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

Trigonometric Function Differentiation

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

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

Velocity and Acceleration Analysis

A particle moving along a straight line has a velocity function given by $$v(t)=2*t^2 - 8*t + 3$$ (i

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

Chain Rule and Implicit Differentiation in a Pendulum Oscillation Experiment

In a pendulum experiment, the angle \(\theta(t)\) in radians satisfies the relation $$\cos(\theta(t)

Hard

Chemical Mixing: Implicit Relationships and Composite Rates

In a chemical mix tank, the solute amount (in grams) and the concentration (in mg/L) are related by

Hard

Design Optimization for a Cylindrical Can

A manufacturer wants to design a cylindrical can that holds a fixed volume of $$V = 1000$$ cm³. The

Medium

Differentiation in a Logistic Population Model

The population of a species is modeled by the logistic function $$P(t)= \frac{1000}{1+e^{-0.3*(t-5)}

Medium

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

Differentiation of Composite Exponential and Trigonometric Functions

Let $$f(x) = e^{\sin(x^2)}$$ be a composite function. Differentiate $$f(x)$$ and simplify your answe

Extreme

Financial Flow Analysis: Investment Rates

An investment fund experiences deposits at a rate modeled by the composite function $$D(t)=g(h(t))$$

Easy

Higher Order Implicit Differentiation in a Nonlinear Model

Assume that \(x\) and \(y\) are related by the nonlinear equation $$e^{x*y} + x - \ln(y) = 5$$ with

Extreme

Implicit Differentiation in a Hyperbola-like Equation

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

Medium

Implicit Differentiation in a Radical Equation

The relationship between $$x$$ and $$y$$ is given by $$\sqrt{x} + \sqrt{y} = 6$$.

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 Involving Exponential Functions

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

Hard

Implicit Differentiation of a Circle

The equation of a circle is given by $$x^2+y^2=25$$. Answer the following parts:

Easy

Implicit Differentiation of a Product and Composite Function

Consider the equation $$x^2*\sin(y)+e^{y}=x$$, which defines y implicitly as a function of x. Answer

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: Circle and Tangent Line

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

Medium

Inverse Function Derivative for the Natural Logarithm

Consider the function $$f(x) = \ln(x+1)$$ for $$x > -1$$ and let $$g$$ be its inverse function. Anal

Easy

Inverse Function Derivative in a Cubic Function

Let $$f(x)= x^3+ 2*x - 1$$, a one-to-one differentiable function. Its inverse function is denoted as

Medium

Inverse Function Derivatives in a Sensor Model

An instrument outputs a reading defined by $$f(x)= x^3 + 2$$, where $$x$$ represents the voltage inp

Easy

Inverse Function Differentiation for a Trigonometric-Polynomial Function

Let $$f(x)= \sin(x) + x^2$$ be defined on the interval $$[0, \pi/2]$$ so that it is invertible, with

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

Polar and Composite Differentiation: Arc Slope for a Polar Curve

Consider the polar curve $$r(\theta)=2+\cos(\theta)$$. Answer the following parts:

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

Second Derivative via Chain Rule

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

Hard

Tangent Line to an Ellipse

Consider the ellipse given by $$\frac{x^2}{16} + \frac{y^2}{9} = 1$$. Determine the slope of the tan

Easy
Unit 4: Contextual Applications of Differentiation

Analysis of a Piecewise Function with Discontinuities

Consider the function $$ f(x)= \begin{cases} \frac{x^2-4}{x-2} &\text{if } x \neq 2 \\ 3 &\text{if }

Medium

Analyzing a Motion Graph

A car's velocity over time is modeled by the piecewise function given in the graph. For $$0 \le t <

Medium

Arc Length Calculation

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

Hard

Business Profit Optimization

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

Medium

Compound Interest Rate Change

An investment grows according to $$A(t)=5000e^{0.07t}$$, where t is measured in years. Answer the fo

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

Cubic Curve Linearization

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

Hard

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

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

Easy

Draining Hemispherical Tank

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

Hard

Ellipse Tangent Line Analysis

Consider the ellipse given by the implicit equation $$9*x^2 + 4*y^2 = 36$$. Answer the following par

Medium

Exponential Relation

Consider the equation $$e^{x*y} = x + y$$.

Hard

Financial Model Inversion

Consider the function $$f(x)=\ln(x+2)+x$$ which models a certain financial indicator. Although an ex

Hard

Graphical Analysis of an Inverse Function

Consider a continuously differentiable and strictly increasing function $$f(x)$$ as depicted in the

Hard

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

Infrared Sensor Distance Analysis

An infrared sensor measures the distance to a moving target using the function $$d(t)=50*e^{-0.2*t}+

Medium

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

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

Linear Account Growth in Finance

The amount in a savings account during a promotional period is given by the linear function $$A(t)=1

Easy

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

Optimizing Area of a Rectangular Field

A farmer has 100 meters of fencing to enclose three sides of a rectangular field (the fourth side be

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 Analysis

A certain bacterial population in a lab grows according to the model $$P(t)=100\cdot e^{0.03*t}$$, w

Medium

Population Growth Differential

Consider an implicit relationship between a population $$N$$ and time $$t$$ given by $$\ln(N) + t =

Hard

Related Rates: Pool Water Level

Water is being drained from a circular pool. The surface area of the pool is given by $$A=\pi*r^2$$.

Medium

Security Camera Angle Change

A security camera is mounted on a 15 m tall tower. Let $$x$$ denote the horizontal distance from the

Medium

Series Integration for Work Calculation

A force along a displacement is given by $$F(x)= \sum_{n=0}^{\infty} \frac{(-1)^n x^n}{n+2}$$ (in Ne

Medium

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 Filtration Plant Analysis

A water filtration plant processes water entering at a rate of $$I(t)=60-2t$$ (liters per minute) an

Hard
Unit 5: Analytical Applications of Differentiation

Analysis of a Function with Oscillatory Behavior

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

Medium

Analysis of a Quartic Function as a Perfect Power

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

Hard

Analysis of an Exponential Function

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

Extreme

Analysis of Critical Points for Increasing/Decreasing Intervals

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

Medium

Application of Rolle's Theorem

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

Medium

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

Let $$f(x)=x^3-6x^2+9x+2.$$ Answer the following parts:

Easy

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

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

Expanding Oil Spill - Related Rates

A circular oil spill is expanding such that its area is given by $$A(t) = \pi*[r(t)]^2$$. The radius

Easy

Exponential Growth and Logarithmic Transformation

A bacteria population is modeled by $$P(t)= A*e^{k*t}$$, where $$t$$ is measured in hours, A is the

Medium

Extreme Value Theorem for a Piecewise Function

Let $$h(x)$$ be defined on $$[-2,4]$$ as $$ h(x)= \begin{cases} -x^2+4 & \text{if } x \le 1, \\ 2x-

Hard

Implicit Differentiation in Economic Context

Consider the curve defined implicitly by $$x*y + y^2 = 12$$, representing an economic relationship b

Easy

Logarithmic Function Derivative Analysis

Consider the function $$f(x)= \ln(x^2+1)$$. Answer the following questions about its behavior.

Easy

Logarithmic-Exponential Function Analysis

Consider the function $$f(x)= e^(x) + x$$ defined for all real numbers. Answer the following questio

Hard

MVT Application: Rate of Temperature Change

The temperature in a room is modeled by $$T(t)= -2*t^2+12*t+5$$, where $$t$$ is in hours. Analyze th

Easy

Particle Motion on a Curve

A particle moves along a straight-line path with its position given by \( s(t)=t^3 - 6*t^2 + 9*t + 1

Easy

Related Rates: Changing Shadow Length

A 2-meter tall lamppost casts a shadow of a 1.6-meter tall person who is walking away from the lampp

Medium

Rolle's Theorem on a Cubic Function

Consider the cubic function $$f(x)= x^3-3*x^2+2*x$$ defined on the interval $$[0,2]$$. Verify that t

Medium

Ski Resort Snow Accumulation and Melting

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

Medium
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

Accumulated Displacement from a Piecewise Velocity Function

A particle moves along a line with velocity given by $$ v(t)= \begin{cases} t^2-1, & 0 \le t < 2, \\

Medium

Accumulated Displacement from Acceleration

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

Hard

Accumulation Function from a Rate Function

The rate at which water flows into a tank is given by $$r(t)=3\sqrt{t}$$ (in liters per minute) for

Easy

Approximating an Exponential Integral via Riemann Sums

Consider the function $$h(x)=e^{-x}$$ on the interval $$[0,2]$$. A table of values is provided below

Easy

Area Between the Curves f(x)=x² and g(x)=2x+3

Given the two functions $$f(x)= x^2$$ and $$g(x)= 2*x+3$$ on the interval where they intersect, dete

Medium

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

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

Medium

Average Value of an Exponential Function

For the function $$f(x)= x*e^{-x}$$, determine the average value on the interval $$[0,2]$$. Answer t

Easy

Bacteria Population Accumulation

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

Medium

Bacterial Growth Accumulation

The instantaneous growth rate of a bacterial culture is modeled by $$r(t)= 0.3*t$$ million cells per

Extreme

Chemical Reactor Concentration

In a chemical reactor, a reactant enters at a rate of $$C_{in}(t)=5+t$$ grams per minute and is simu

Medium

Comparing Riemann Sum Approximations for an Increasing Function

A function f(x) is given in the table below: | x | 0 | 2 | 4 | 6 | |---|---|---|---|---| | f(x) | 3

Medium

Cost Accumulation from Marginal Cost Function

A company’s marginal cost function $$MC(q)$$ (in dollars per unit) for producing $$q$$ units is give

Medium

Cross-Sectional Area of a River Using Trapezoidal Rule

The depth $$h(x)$$ (in meters) of a river’s cross-section is measured at various points along a hori

Hard

Estimating Area Under a Curve via Riemann Sums

The following table shows values of a function f(t): | t | 0 | 2 | 4 | 6 | 8 | |---|---|---|---|---

Medium

Evaluating a Trigonometric Integral

Evaluate the integral $$\int_{0}^{\pi/2} \cos(3*x)\,dx$$.

Easy

Heat Energy Accumulation

The rate of heat transfer into a container is given by $$H(t)= 5\sin(t)$$ kJ/min for $$t \in [0,\pi]

Medium

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 by Parts: Logarithmic Function

Evaluate the definite integral $$\int_{1}^{3} x*\ln(x) dx$$ using integration by parts. Answer the f

Medium

Numerical Approximation: Trapezoidal vs. Simpson’s Rule

The function $$f(x)=\frac{1}{1+x^2}$$ is to be integrated over the interval [-1, 1]. A table of valu

Extreme

Rainfall Accumulation Over Time

A storm produces rainfall at a rate modeled by the function $$r(t)=6 * t^(1/2)$$, where $$0 \le t \l

Easy

Riemann Sum from a Table: Plant Growth Data

A function $$f(t)$$ describes the height (in cm) of a plant over time (in days). The measurements ar

Medium

Series Representation and Term Operations

Consider the power series for $$\arctan(x)$$ given by $$\sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{2n+

Medium

Signal Energy through Trigonometric Integration

A signal is described by $$f(t)=3*\sin(2*t)+\cos(2*t)$$. The energy of the signal over one period

Extreme

Vehicle Motion and Inverse Time Function

A vehicle’s displacement (in meters) is modeled by the function $$f(t)= t^2 + 4$$ for $$t \ge 0$$ se

Easy

Water Accumulation Using Trapezoidal Sum

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

Medium

Work on a Nonlinear Spring

A nonlinear spring exerts a force given by $$F(x)=8 * e^(0.3 * x)$$ (in Newtons) as a function of di

Medium
Unit 7: Differential Equations

Capacitor Discharge in an RC Circuit

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

Easy

City Population with Migration

The population $$P(t)$$ of a city changes as individuals migrate in at a constant rate of $$500$$ pe

Easy

Coffee Cooling: Differential Equation Application

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

Medium

Combined Differential Equations and Function Analysis

Consider the function $$y(x)$$ defined by the differential equation $$\frac{dy}{dx} = \frac{2*x}{1+y

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

Compound Interest with Continuous Payment

An investment account grows with a continuous compound interest rate $$r$$ and also receives continu

Easy

Cooling of an Object Using Newton's Law of Cooling

An object cools in a room with constant ambient temperature. The cooling process is modeled by Newto

Medium

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

Free-Fall with Air Resistance

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

Extreme

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 6: Radioactive Decay

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

Easy

FRQ 8: RC Circuit Analysis

In an RC circuit, the voltage across the capacitor, $$V(t)$$, satisfies the differential equation $$

Medium

FRQ 11: Linear Differential Equation via Integrating Factor

Solve the differential equation $$\frac{dy}{dx}+\frac{1}{x}y=\sin(x)$$ with the initial condition $$

Hard

FRQ 18: Enzyme Reaction Rates

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

Easy

Implicit Differential Equations and Slope Fields

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

Medium

Logistic Growth Model

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

Medium

Logistic Population Model

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

Medium

Mixing in a Chemical Reaction

A solution in a tank is undergoing a chemical reaction described by the differential equation $$\fra

Medium

Mixing Problem with Constant Flow Rate

A tank holds 500 L of water and initially contains 10 kg of dissolved salt. Brine with a salt concen

Easy

Modeling Disease Spread with Differential Equations

In a simple model for disease spread, the number of infected individuals, $$I(t)$$, evolves accordin

Hard

Modeling Exponential Growth

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

Easy

Newton's Law of Cooling

An object cools according to Newton's Law of Cooling: $$\frac{dT}{dt} = -k*(T-20)$$, where the ambie

Easy

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

Population Dynamics in Ecology

Consider the differential equation that models the growth of a fish population in a lake: $$\frac{dP

Medium

Population Dynamics with Harvesting

A fish population in a lake is modeled by the logistic equation with harvesting: $$\frac{dP}{dt}=r\,

Medium

Radio Signal Strength Decay

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

Easy

RL Circuit Analysis

An RL circuit is described by the differential equation $$L\frac{di}{dt} + R*i = V$$, where $$L=0.5\

Medium

Saltwater Mixing Problem

A tank initially contains 1000 L of a salt solution with a concentration of 0.2 kg/L (thus 200 kg of

Medium

Second-Order Differential Equation: Oscillations

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

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

Tank Mixing Problem

A tank contains 1000 L of a well‐mixed salt solution. Brine containing 0.5 kg/L of salt flows into t

Hard

Verification of Integral Representation of Solutions

Let $$y(x)=\int_0^x e^{-(x-t)} f(t)\,dt$$, where $$f(t)$$ is a continuous function. Answer the follo

Extreme

Viral Spread on Social Media

Let $$V(t)$$ denote the number of viral posts on a social media platform. Posts go viral at a consta

Easy

Water Tank Inflow-Outflow Model

A water tank is subject to an inflow and outflow. The inflow rate is given by $$I(t)=3*t+2$$ liters

Medium
Unit 8: Applications of Integration

Accumulated Rainfall

The rainfall intensity in a region is given by $$R(t)=0.2*t^2+1$$ (in cm/hour), where $$t$$ is measu

Medium

Analyzing a Reservoir's Volume Over Time

Water flows into a reservoir at a variable rate given by $$R(t)=50e^{-0.1*t}$$ m³/hour and simultane

Medium

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

Approximating Functions using Taylor Series

Consider the function $$f(x)= \ln(1+2*x)$$. Use Taylor series methods to approximate and analyze thi

Hard

Area Between a Rational Function and Its Asymptote

Consider the function $$f(x)=\frac{2*x+3}{x+1}$$ and its horizontal asymptote $$y=2$$ over the inter

Hard

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 Calculation: Region Under a Parabolic Curve

Let $$f(x)=4-x^2$$. Consider the region bounded by the curve $$f(x)$$ and the x-axis.

Easy

Average Temperature Calculation

A city's temperature during a day is modeled by $$T(t)=10+5*\sin\left(\frac{\pi*t}{12}\right)$$ for

Easy

Average Value of a Polynomial Function

Consider the function $$f(x)=2*x^2 - 4*x + 3$$ defined on the closed interval $$[0,4]$$. Answer the

Easy

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

Fluid Flow Rate and Total Volume

A river has a flow rate given by $$Q(t)=50+10*\cos(t)$$ (in cubic meters per second) for $$t\in[0,\p

Easy

Inflow vs Outflow: Water Reservoir Capacity

A reservoir receives water with an inflow rate given by $$I(t)=20+5\sin(t)$$ (liters/min) and discha

Hard

Kinematics: Motion with Variable Acceleration

A particle moves along a straight line with acceleration $$a(t)=4-2*t$$ (in m/s²). The particle has

Medium

Particle Position and Distance Traveled

A particle moves along a line with velocity $$v(t)=t^3-6*t^2+9*t$$ (m/s) for $$t\in[0,5]$$. Given th

Hard

Pollution Concentration in a Lake

A lake has a pollution concentration modeled by $$C(x) = 16 - x^2$$ (in mg/L), where $$x$$ (in meter

Easy

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

Power Series Representation for ln(x) about x=4

The function $$f(x)=\ln(x)$$ is to be expanded as a power series centered at $$x=4$$. Find this seri

Extreme

Projectile Motion Analysis

A projectile is launched vertically upward with an initial velocity of $$20$$ m/s. The only accelera

Medium

Projectile Motion under Gravity

An object is projected vertically upward with an initial velocity of $$20$$ m/s and from an initial

Easy

Savings Account with Decreasing Deposits

An individual opens a savings account with an initial deposit of $1000 in the first month. Every sub

Easy

Volume by the Washer Method: Between Curves

Consider the region between the curves $$y=\sqrt{x}$$ and $$y=\frac{x}{2}$$ for $$x$$ between their

Medium

Volume of a Solid Obtained by Rotation

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

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

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 with Square Cross Sections

The region in the $$xy$$-plane is bounded by $$y=\sqrt{x}$$, $$y=0$$, and $$x=4$$. A solid is formed

Medium

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

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

Analyzing a Cycloid

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

Medium

Analyzing a Looping Parametric Curve

The curve is defined by the equations $$x(t)=t^3-3t$$ and $$y(t)=t^2$$ for \(-2\le t\le 2\). Due to

Hard

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

Area between Two Polar Curves

Given the polar curves $$R(\theta)=3$$ and $$r(\theta)=2$$ for $$0 \le \theta \le 2\pi$$, find the a

Hard

Area Between Two Polar Curves

Consider the two polar curves $$r_1(θ)= 3+\cos(θ)$$ and $$r_2(θ)= 1+\cos(θ)$$. Answer the following:

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

Conversion of Polar to Cartesian Coordinates

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

Easy

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

Exponential Decay in Vector-Valued Functions

A particle moves in the plane with its position given by the vector-valued function $$\vec{r}(t)=\la

Hard

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

Implicit Differentiation with Implicitly Defined Function

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

Medium

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

Intersection of Parametric Curves

Two curves are given by the parametric equations $$x_1(t)=t^2,\; y_1(t)=t^3$$ and $$x_2(s)=1-s^2,\;

Extreme

Kinematics in Polar Coordinates

A particle’s position in polar coordinates is given by $$r(t)= \frac{5*t}{1+t}$$ and $$\theta(t)= \f

Hard

Kinematics on a Circular Path

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

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

Optimization on a Parametric Curve

A curve is described by the parametric equations $$x(t)= e^{t}$$ and $$y(t)= t - e^{t}$$.

Hard

Parameter Elimination in Logarithmic and Quadratic Relationships

Given the parametric equations $$x(t)= \ln(t)$$ and $$y(t)= t^2 - 4*t + 3$$ for $$t > 0$$, eliminate

Easy

Parametric Curves and Concavity

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

Hard

Parametric Equations and Intersection Points

Consider the curves defined parametrically by $$x_1(t)=t^2, \; y_1(t)=2t$$ and $$x_2(s)=s+1, \; y_2(

Extreme

Parametric Intersection and Tangency

Two curves are given in parametric form by: Curve 1: $$x_1(t)=t^2,\, y_1(t)=2t$$; Curve 2: $$x_2(s

Medium

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

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 Trajectory in Parametric Motion

A particle moves along a curve with parametric equations $$x(t)= t^2 - 4*t$$ and $$y(t)= t^3 - 3*t$$

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

Projectile Motion via Vector-Valued Functions

A projectile is launched from the origin with an initial velocity given by \(\mathbf{v}(0)=\langle 5

Medium

Symmetry and Area in Polar Coordinates

Consider the polar curve given by $$r=2\cos(\theta)$$. Answer the following:

Easy

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

Velocity and Acceleration of a Particle

A particle’s position in three-dimensional space is given by the vector-valued function $$\mathbf{r}

Easy

Wind Vector Analysis in Navigation

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

Hard

Work Done by a Force along a Path

A force acting on an object is given by the vector function $$\vec{F}(t)= \langle 3t,\; 2,\; t^2 \ra

Medium

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Need to review before working on AP Calculus BC FRQs?

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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.