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AP Calculus AB Free Response Questions

The best way to get better at FRQs is practice. Browse through dozens of practice AP Calculus AB 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 (40)
  • Unit 3: Differentiation: Composite, Implicit, and Inverse Functions (31)
  • Unit 4: Contextual Applications of Differentiation (37)
  • Unit 5: Analytical Applications of Differentiation (25)
  • Unit 6: Integration and Accumulation of Change (29)
  • Unit 7: Differential Equations (33)
  • Unit 8: Applications of Integration (27)
Unit 1: Limits and Continuity

Advanced Analysis of a Piecewise Function

Consider the function $$f(x)=\begin{cases} x^2*\sin\left(\frac{1}{x}\right) & x\neq 0 \\ 0 & x=0 \en

Extreme

Advanced Analysis of an Oscillatory Function

Consider the function $$ f(x)= \begin{cases} \sin(1/x), & x\ne0 \\ 0, & x=0 \end{cases} $$.

Extreme

Analyzing a Removable Discontinuity

Consider the function $$f(x) = \frac{x^2 - 4}{x - 2}$$ for $$x \neq 2$$. Notice that f is not define

Easy

Arithmetic Sequence in Temperature Data and Continuity Correction

A temperature sensor records the temperature every minute and the readings follow an arithmetic sequ

Easy

Composite Function and Continuity Analysis

Define \(f(x)=\sqrt{1-\ln(x)}\) for \(x>0\) and consider the composite function \(g(x)=f(e^x)\). Ans

Hard

Continuity and Limit Comparison for Two Particle Paths

Two particles, A and B, travel along the same line. Their position functions are given by $$s_A(t)=

Medium

Continuity at Zero for a Trigonometric Function

Consider the function $$f(x)= x*\sin\left(\frac{1}{x}\right)$$ for x $$\neq 0$$ and $$f(0)=0$$. Answ

Medium

Continuity of Constant Functions

Consider the constant function $$f(x)=7$$ for all x. Answer the following parts.

Easy

Discontinuity Analysis in Piecewise Functions

Consider the piecewise function $$f(x)= \begin{cases} \frac{x^2-4}{x-2} & x\neq2 \\ 5 & x=2 \end{cas

Medium

Factorization and Limit Evaluation

Consider the function $$f(x) = \frac{x^3 - 8}{x - 2}$$. (a) Factor the numerator and simplify the e

Easy

Graph-Based Analysis of Discontinuities

Examine the graph of a function $$ f(x) $$ depicted in the stimulus. The graph shows the function fo

Medium

Graph-Based Analysis of Discontinuity

Examine the graph of a function that appears to be defined by $$f(x)= 3x - 2$$ for all $$x \neq 2$$,

Easy

Graphical Estimation of a Limit

The following graph shows the function $$f(x)$$. Use the graph to answer the subsequent questions re

Medium

Horizontal Asymptote of a Rational Function

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

Medium

Identifying Discontinuities in a Rational Function

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

Medium

Intermediate Value Theorem in Context

Let $$f(x) = x^3 - 6x^2 + 9x + 2$$, which is continuous on the interval [0, 4]. Answer the following

Medium

Limits at Infinity for Non-Rational Functions

Consider the function $$ h(x)=\frac{2*x+3}{\sqrt{4*x^2+7}} $$.

Medium

Mixed Function with Jump Discontinuity at Zero

Consider the function $$f(x)=\begin{cases} 1+x & x<0\\ 2 & x=0\\ \frac{\sin(x)}{x}+1 & x>0 \end{case

Medium

Oscillatory Behavior and Discontinuity

Consider the function $$f(x)=\begin{cases} x\cos(\frac{1}{x}) & x\neq0 \\ 2 & x=0 \end{cases}$$. Ans

Medium

Parameters for Continuity

Consider the function $$f(x)=\begin{cases} a*x^2+3, & x \le 2 \\ b*x+5, & x > 2 \end{cases}$$ Dete

Easy

Piecewise Function with Different Expressions

Consider the function $$f(x)=\begin{cases} 3*x-1 & x<2\\ \frac{x^2-4}{x-2} & x>2\\ 5 & x=2 \end{case

Medium

Rational Functions with Removable Discontinuities

Examine the function $$f(x)= \frac{x^2 - 5x + 6}{x - 2}$$. (a) Factor the numerator and simplify th

Easy

Removable Discontinuity and Redefinition

Consider the function $$f(x)=\frac{x^2-4}{x-2}$$ for $$x\neq2$$. Note that f is undefined at $$x=2$$

Medium

Removal of Discontinuity by Redefinition

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

Easy

Squeeze Theorem Application

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

Medium

Squeeze Theorem for an Oscillatory Function

Define the function $$f(x)= x \cos(\frac{1}{x})$$ for x ≠ 0, and let f(0)= 0.

Hard

Vertical Asymptote and End Behavior

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

Easy

Water Tank Inflow-Outflow Analysis

Consider a water tank operating over the time interval $$0 \le t \le 12$$ minutes. The water inflow

Medium
Unit 2: Differentiation: Definition and Fundamental Properties

Approximating Derivatives Using Secant Lines

For the function $$f(x)=\ln(x)$$, we want to approximate the derivative at $$x=3$$ using secant line

Medium

Approximating the Tangent Slope

Consider the function $$f(x)=3*x^2$$. Answer the following:

Easy

Derivative from First Principles

Derive the derivative of the polynomial function $$f(x)=x^3+2*x$$ using the limit definition of the

Medium

Derivative of a Trigonometric Function

Let \(f(x)=\sin(2*x)\). Answer the following parts.

Easy

Derivatives on an Ellipse

The ellipse given by $$\frac{x^2}{9} + \frac{y^2}{4} = 1$$ represents a race track. Answer the follo

Medium

Differentiating an Absolute Value Function

Consider the function $$f(x)= |3*x - 6|$$.

Medium

Differentiation of Exponential Functions

Consider the function $$f(x)=e^{2*x}-3*e^{x}$$.

Easy

DIY Rainwater Harvesting System

A household's rainwater harvesting system collects rain at a rate of $$f(t)=12-0.5*t$$ (liters/min)

Easy

Event Ticket Sales Dynamics

For a popular concert, tickets are sold at a rate of $$f(t)=100-3*t$$ (tickets/hour) while cancellat

Easy

Exploring the Difference Quotient for a Trigonometric Function

Consider the trigonometric function $$f(x)= \sin(x)$$, where $$x$$ is measured in radians. Use the d

Hard

Exponential Decay Analysis

A radioactive substance decays according to the function $$N(t)=N_0 \cdot e^{-0.03t}$$, where t is m

Easy

Finding Derivatives with Product and Quotient Rule

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

Extreme

Finding the Tangent Line Using the Product Rule

For the function $$f(x)=(3*x^2-2)*(x+5)$$, which models a physical quantity's behavior over time (in

Medium

Graph vs. Derivative Graph

A graph of a function $$f(x)$$ and a separate graph of its derivative $$f'(x)$$ are provided in the

Hard

Graphical Analysis of Secant and Tangent Slopes

A function $$f(x)$$ is represented by the red curve in the graph below. Answer the following questio

Medium

Implicit Differentiation in Motion

A particle’s motion is given by the implicit equation $$y^2 + x*y = 10$$, where x represents time (i

Hard

Instantaneous and Average Velocity

A particle's position is given by $$s(t)=t^3-6*t^2+9*t+2$$, where $$s(t)$$ is in meters and $$t$$ is

Easy

Instantaneous Rate of Change in Motion

A particle’s position along a straight line is given by $$s(t)= 4*t^3 - 12*t^2 + 9*t + 5$$, where $$

Medium

Interpreting Derivative Graphs and Tangent Lines

A graph of the function $$f(x)=x^2 - 2*x + 1$$ along with its tangent line at $$x=2$$ is provided. A

Medium

Inverse Function Analysis: Hyperbolic-Type Function

Consider the function $$f(x)=\sqrt{x^2+1}$$ defined for $$x\geq0$$.

Easy

Inverse Function Analysis: Logarithmic Transformation

Consider the function $$f(x)=\ln(2*x+5)$$ defined for $$x> -\frac{5}{2}$$.

Easy

Inverse Function Analysis: Sum with Reciprocal

Consider the function $$f(x)=x+\frac{1}{x}$$ defined for $$x\geq1$$.

Hard

Limit Definition for a Quadratic Function

For the function $$h(x)=4*x^2 + 2*x - 7$$, answer the following parts using the limit definition of

Medium

Particle Motion on a Straight Road

A particle moves along a straight road. Its position at time $$t$$ seconds is given by $$s(t) = t^3

Medium

Piecewise Function and Discontinuities

A piecewise function $$f$$ is defined by $$ f(x)=\begin{cases} \frac{x^2-4}{x-2} & \text{if } x\ne

Medium

Population Growth Rate

Suppose the population of a species is modeled by $$P(t)= 1000*e^{0.07*t}$$, where $$t$$ is measured

Easy

Product and Quotient Rule Combination

Given $$u(x)=3*x^2+2$$ and $$v(x)=x-4$$, consider the function $$F(x)=\frac{u(x)*v(x)}{x+1}$$. Answe

Hard

Proof of Scaling in Derivatives

Let $$f(x)$$ be a differentiable function and let $$k$$ be a constant. Consider $$g(x)= k*f(x)$$. Us

Easy

Quotient Rule Application

Let $$f(x)= \frac{e^{x}}{x+1}$$, a function defined for $$x \neq -1$$, which involves both an expone

Hard

Quotient Rule Challenge

For the function $$f(x)= \frac{3*x^2 + 2}{5*x - 7}$$, find the derivative.

Hard

Rate of Chemical Reaction

The concentration of a reactant in a chemical reaction is modeled by \(C(t)=10*e^{-0.3*t}\), where \

Medium

Rates of Change from Experimental Data

A chemical experiment yielded the following measurements of a substance's concentration (in molarity

Easy

Real-World Cooling Process

In an experiment, the temperature (in °C) of a substance as it cools is modeled by $$T(t)= 30*e^{-0.

Hard

Related Rates: Conical Tank Draining

A conical water tank drains so that its volume is given by $$V=\frac{1}{3}\pi r^2h$$. The radius r o

Hard

Secant and Tangent Line Approximation in a Real-World Model

A temperature sensor records data modeled by $$g(t)= 0.5*t^2 - 3*t + 2$$, where $$t$$ is in seconds

Medium

Secant Approximation Convergence and the Derivative

Consider the natural logarithm function $$f(x)= \ln(x)$$. Investigate its rate of change using the d

Extreme

Slope of a Tangent Line from Experimental Data

Experimental data recording the distance traveled by an object over time is provided in the table be

Easy

Tangent Line and Instantaneous Rate at a Point with a Radical Function

Consider the function $$f(x)= (x+4)^{1/2}$$, which represents a physical measurement (with the domai

Medium

Temperature Change Analysis

A weather station models the temperature (in °C) with the function $$T(t)=15+2*t-0.5*t^2$$, where $$

Easy

Using the Limit Definition to Derive the Derivative

Let $$f(x)= 3*x^2 - 2*x$$.

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

Advanced Implicit and Inverse Function Differentiation on Polar Curves

Consider the curve defined implicitly by $$x^2+y^2= \sin(x*y)$$. Although not a typical polar curve,

Extreme

Chain Rule with Exponential and Trigonometric Functions

A particle's position is given by $$s(t)=e^{3*t}\sin(2*t)$$. Use appropriate differentiation techniq

Medium

Chain Rule with Inverse Trigonometric Function

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

Easy

Composite Function from an Implicit Equation

Consider the implicit equation $$x^2 + y^2 + x*y = 7$$, which defines $$y$$ as an implicit function

Hard

Composite Function with Logarithm and Trigonometry

Let $$h(x)=\ln(\sin(2*x))$$.

Medium

Composite Function with Nested Exponential and Trigonometric Terms

Let $$f(x)= e^{\sin(4*x)}$$. This function combines exponential and trigonometric elements.

Easy

Composite Functions in Population Dynamics

The population of a species is modeled by the composite function $$P(t) = f(g(t))$$, where $$g(t) =

Easy

Composite Log-Exponential Function Analysis

A function is defined by $$f(x)=\ln\left(e^(2*x^2) + 5\right)$$. This function is composed of an exp

Medium

Composite Temperature Model

Consider a temperature function given by $$T(t) = \sin(t^3 - 2*t)$$, where t is measured in seconds.

Medium

Concavity Analysis of an Implicit Curve

Consider the relation $$x^2+xy+y^2=7$$.

Hard

Differentiation of a Composite Inverse Trigonometric-Log Function

Let $$f(x)= \ln\left(\arctan(e^(x))\right)$$. Differentiate and evaluate as required:

Hard

Differentiation of an Inverse Trigonometric Composite Function

Consider the function $$y = \arctan(\sqrt{3x})$$.

Medium

Differentiation of Inverse Trigonometric Function via Implicit Differentiation

Let $$y=\arcsin(\frac{x}{\sqrt{2}})$$. Answer the following:

Hard

Exponential Form and Chain Rule Complexity

Define $$Q(x)=(\cos(x))^{\sin(x)}$$. Hint: Express Q(x) as an exponential function.

Extreme

Graph Analysis of a Composite Motion Function

A displacement function representing the motion of an object is given by $$s(t)= \ln(2*t+3)$$. The g

Easy

Implicit Differentiation in a Financial Model

An implicit relationship between revenue $$R$$ (in thousands of dollars) and price $$p$$ (in dollars

Medium

Implicit Differentiation in a Population Growth Model

Consider the model $$e^{x*y} + x - y = 5$$ that relates time \(x\) to a population scale value \(y\)

Hard

Implicit Differentiation with Exponential-Trigonometric Functions

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

Hard

Implicit Differentiation with Mixed Trigonometric and Algebraic Components

Consider the equation $$x^2+\sin(y)=y^2$$. Answer the following:

Medium

Implicit Differentiation with Trigonometric Components

Consider the equation $$\sin(x) + \cos(y) = x*y$$, which implicitly defines $$y$$ as a function of $

Extreme

Intersection of Curves via Implicit Differentiation

Two curves are defined by the equations $$y^2= 4*x$$ and $$x^2+ y^2= 10$$. Consider their intersecti

Hard

Inverse Function Derivative in Thermodynamics

A thermodynamic process is modeled by the function $$P(V)= 3*V^2 + 2*V + 5$$, where $$V$$ is the vol

Medium

Inverse Function Derivative with Composite Functions

Consider the function $$f(x)=x^3+2*x+1$$, which is one-to-one on its domain. Given that $$f(1)=4$$,

Extreme

Inverse Function Differentiation for a Log Function

Let $$f(x)=\ln(3*x+2)$$, and assume that $$f$$ is invertible with inverse function $$g$$. Find the d

Medium

Inverse Function Differentiation in an Exponential Model

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

Medium

Inverse Function Differentiation with an Exponential-Linear Function

Let $$f(x)=e^{2*x}+x$$ and assume it is invertible. Answer the following:

Medium

Inverse Function Differentiation with Exponentials and Trigonometry

Let $$f(x)=\sin(e^{x})+e^{x}$$ and assume that it is invertible. Answer the following:

Extreme

Inverse Function in Currency Conversion

A function converting dollars to euros is given by $$f(d) = 0.9*d + 10\ln(d+1)$$ for $$d > 0$$. Let

Medium

Particle Motion: Logarithmic Position Function

The position of a particle moving along a line is given by $$s(t)= \ln(3*t+2)$$, where s is in meter

Easy

Related Rates of a Shadow

A 2 m tall lamp post casts a shadow from a person who is 1.8 m tall. The person is moving away from

Medium

Second Derivative via Implicit Differentiation

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

Extreme
Unit 4: Contextual Applications of Differentiation

Analysis of a Composite Function involving Logarithm

The revenue function is given by $$R(x)= x\ln(100/x)$$ for x > 0, where x is the number of units sol

Medium

Bacterial Growth Analysis

The number of bacteria in a culture is given by $$P(t)=500e^{0.2*t}$$, where $$t$$ is measured in ho

Easy

Chemical Reaction Rate Analysis

The concentration of a reactant in a chemical reaction is modeled by $$C(t)=\frac{10}{1+e^{0.5t}}$$,

Medium

Complex Piecewise Function Analysis

Consider the function $$f(x)=\begin{cases}\frac{\sin(x)}{x} & x<\pi \\ 2 & x=\pi \\ 1+\cos(x-\pi) &

Medium

Cost Analysis through a Rational Function

A company's average cost function is given by $$C(x)= \frac{2*x^3 + 5*x^2 - 20*x + 40}{x}$$, where $

Medium

Cost Efficiency in Production

A firm's cost function for producing $$x$$ items is given by $$C(x)=0.1*x^2 - 5*x + 200$$. Analyze t

Medium

Defect Rate Analysis in Manufacturing

The defect rate in a manufacturing process is modeled by $$D(t)=100e^{-0.05t}+5$$ defects per day, w

Easy

Designing a Flower Bed: Optimal Shape

A landscape designer wants to create a rectangular flower bed with a fixed area of 200 square meters

Easy

Economic Cost Analysis Using Derivatives

A company’s cost function for producing $$x$$ units is given by $$C(x)=0.05*x^3 - 2*x^2 + 40*x + 100

Medium

Economic Cost Function Linearization

A company's production cost is modeled by $$C(x)= 0.02*x^3 - 1.5*x^2 + 40*x + 200$$ dollars, where $

Hard

Elasticity of Demand Analysis

A product’s demand function is given by $$Q(p) = 150 - 10p + p^2$$, where $$p$$ is the price, and $$

Medium

Estimating Function Change Using Differentials

Let $$f(x)=x^{1/3}$$. Use differentials to approximate the change in $$f(x)$$ when $$x$$ increases f

Medium

Estimating Small Changes using Differentials

In an electrical circuit, the resistance is given by $$R(x) = \frac{1}{1+x^2}$$, where x is a parame

Easy

Evaluating Indeterminate Limits via L'Hospital's Rule

Scientists are studying the limit of the function $$L(x)=\frac{5x^3-4x^2+1}{7x^3+2x-6}$$ as $$x \to

Medium

Exponential Decay in Radioactive Material

A radioactive substance decays according to $$M(t)=M_0e^{-0.07t}$$, where $$M(t)$$ is the mass remai

Easy

Falling Object Analysis

An object is dropped from a height and its position is modeled by $$s(t)=100-4.9t^2$$ (in meters), w

Medium

FRQ 2: Balloon Inflation Analysis

A spherical balloon is being inflated. Its volume is given by $$V = \frac{4}{3}\pi r^3$$, and the ra

Medium

Kinematics on a Straight Line

A particle moves along a straight line with a position function $$s(t)=t^3 - 6*t^2 + 9*t + 2$$, wher

Medium

L'Hôpital's Rule in Analysis of Limits

Consider the limit $$L = \lim_{x\to \infty} \frac{5*x^3 - 4*x^2 + 1}{7*x^3 + 2*x - 6}$$. Use L'Hôpit

Medium

Linear Approximation of ln(1.05)

Let $$f(x)=\ln(x)$$. Use linearization at x = 1 to approximate the value of $$\ln(1.05)$$.

Easy

Logarithmic Differentiation in Exponential Functions

Let $$y = (2x^2 + 3)^{4x}$$. Use logarithmic differentiation to find $$y'$$.

Hard

Logarithmic Profit Optimization

A company’s profit is modeled by $$P(x) = 50x \ln(x) - 100x$$, where $$x$$ (in thousands) is the num

Hard

Marginal Analysis in Economics

A company’s cost function is given by $$C(x)=0.5*x^3 - 3*x^2 + 5*x + 8$$, where $$x$$ represents the

Medium

Motion Analysis from Velocity Function

A particle moves along a straight line with a velocity given by $$v(t) = t^2 - 4t + 3$$ (in m/s). Th

Hard

Optimization in a Manufacturing Process

A company designs an open-top container whose volume is given by $$V = x^2 y$$, where x is the side

Medium

Optimizing Water Flow in a Tank

A water tank is being filled with water. The volume $$V(t)$$ (in cubic meters) at time $$t$$ (in min

Easy

Projectile Motion Analysis

A projectile is launched vertically, and its height (in meters) as a function of time is given by $$

Medium

Projectile Motion and Maximum Height

A basketball is thrown such that its height (in meters) is modeled by $$h(t)= -4.9*t^2 + 14*t + 1.5$

Medium

Rate of Change in a Population Model

A population model is given by $$P(t)=30e^{0.02t}$$, where $$P(t)$$ is the population in thousands a

Medium

Rate of Change of Temperature

The temperature of a cooling liquid is modeled by $$T(t)= 100*e^{-0.5*t} + 20$$, where $$t$$ is in m

Easy

Reaction Rates in Chemistry

In a chemical reaction, the concentration of a reactant is modeled by $$C(t)=50e^{-0.3*t}+10$$, wher

Easy

Related Rates in a Spherical Balloon

A spherical balloon is being inflated, and its volume $$V$$ (in cubic inches) is related to its radi

Medium

Revenue and Cost Analysis

A company’s revenue is modeled by $$R(t)=200e^{0.05t}$$ and its cost by $$C(t)=10t^3-30t^2+50t+200$$

Hard

Revenue Sensitivity to Advertising

A firm's revenue (in thousands of dollars) is given by $$R(t)=50\sqrt{t+1}$$, where $$t$$ represents

Easy

Runner’s Speed Analysis

During a sprint, a runner's distance from the starting line is modeled by $$d(t)=-2t^2+12t$$, where

Easy

Savings Account Growth Modeled by a Geometric Sequence

A savings account has an initial balance of $$B_0=1000$$ dollars. The account earns compound interes

Easy

Using L'Hospital's Rule to Evaluate a Limit

Consider the limit $$L=\lim_{x\to\infty}\frac{5x^3-4x^2+1}{7x^3+2x-6}$$. Answer the following:

Medium
Unit 5: Analytical Applications of Differentiation

Application of Rolle's Theorem to a Trigonometric Function

Consider the function $$f(x) = \sin(x)$$ defined on the interval $$[0, \pi]$$. Answer the following:

Easy

Average Rate of Change from Experimental Data

A scientist recorded the temperature of a chemical reactor over time. The table below shows temperat

Medium

Behavior Analysis of a Logarithmic Function

Consider the function $$f(x)= \frac{\ln(x)}{x}$$ for $$x>0$$. Analyze the critical points and concav

Medium

Biological Growth and the Mean Value Theorem

In a bacterial culture, the population is modeled by $$P(t)= 4*t^2 + 3*t + 7$$ for $$t$$ in hours on

Easy

Comprehensive Analysis of a Rational Function

Given the rational function $$f(x)= \frac{x^2-4}{x^2+1}$$, perform a comprehensive analysis includin

Extreme

Concavity Analysis of a Trigonometric Function

For the function $$f(x)= \sin(x) - \frac{1}{2}\cos(x)$$ defined on the interval $$[0,2\pi]$$, analyz

Medium

Concavity and Inflection Points in a Quartic Function

Analyze the concavity and determine any points of inflection for the function $$f(x)= x^4 - 4*x^3$$.

Medium

Discontinuity and Derivative in a Hybrid Piecewise Function

Consider the function $$ f(x) = \begin{cases} x^2, & x \le 1, \\ 3x - 2, & x > 1. \end{cases} $$ A

Medium

FRQ 3: Relative Extrema for a Cubic Function

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

Medium

Inverse Analysis of a Function with Square Root and Linear Term

Consider the function $$f(x)=\sqrt{3*x+1}+x$$. Answer the following questions regarding its inverse.

Hard

Investigating Limits and Discontinuities in a Rational Function with Complex Denominator

Consider the function $$ f(x) = \begin{cases} \frac{x^2-9}{x-3}, & x < 3, \\ \frac{x^2-9}{x-3} + 4,

Hard

Investigating the Behavior of a Composite Function

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

Hard

Liquid Cooling System Flow Analysis

A specialized liquid cooling system operates with non-linear flow rates. The inflow rate is given by

Hard

Motion Analysis with Acceleration Function

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

Medium

Optimizing a Cylindrical Water Tank

A cylindrical water tank without a top is to be built with a fixed surface area of 100 m². Let $$r$$

Extreme

Projectile Motion and Derivatives

A projectile is launched so that its height is given by $$h(t) = -4.9*t^2 + 20*t + 1$$, where $$t$$

Easy

Radioactive Substance Decay

A radioactive substance decays according to the model $$A(t)= A_0 * e^{-\lambda*t}$$, where $$t$$ is

Medium

Rational Function Behavior and Extreme Values

Consider the function $$f(x)= \frac{2*x^2 - 3*x + 1}{x - 2}$$ defined for $$x \neq 2$$ on the interv

Hard

Sand Pile Dynamics

A sand pile is being formed on a surface where sand is both added and selectively removed. The inflo

Medium

Tangent Line and MVT for ln(x)

Consider the function $$f(x)= \ln(x)$$ on the interval $$[1,e]$$.

Hard

Temperature Change and the Mean Value Theorem

A temperature model for a day is given by $$T(t)= 2*t^2 - 3*t + 5$$, where $$t$$ is measured in hour

Medium

Transcendental Function Analysis

Consider the function $$f(x)= \frac{e^x}{x+1}$$ defined for $$x > -1$$ and specifically on the inter

Hard

Using Derivatives to Solve a Rate-of-Change Problem

A particle’s displacement is given by $$s(t) = t^3 - 9*t^2 + 24*t$$ (in meters), where \( t \) is in

Hard

Volume of Solid by Cylindrical Shells

Consider the region bounded by $$f(x)= e^x$$ and $$g(x)= e^{2 - x}$$. Analyze the region and set up

Hard

Water Reservoir Net Change

A water reservoir receives water from a river at a rate $$R_{in}(t)=5+0.5*t$$ and discharges water a

Easy
Unit 6: Integration and Accumulation of Change

Area Between Curves

An engineering design problem requires finding the area of the region enclosed by the curves $$y = x

Hard

Area Between Curves: $$y=x^2$$ and $$y=4*x$$

Consider the curves defined by $$f(x)= x^2$$ and $$g(x)= 4*x$$. Answer the following questions to de

Medium

Area Under a Curve Using Riemann Sums

A function $$f(x)$$ is defined over the interval $$[1,7]$$ and its values are provided in the table

Medium

Average Value of a Function

The temperature over the first 8 hours of a day is modeled by $$T(t)=-0.5*t^2 + 4*t + 10$$, where t

Easy

Average Value of a Log Function

Let $$f(x)=\ln(1+x)$$ for $$x \ge 0$$. Find the average value of $$f(x)$$ on the interval [0,3].

Hard

Composite Functions and Accumulation

Let the accumulation function be defined by $$F(x)=\int_{2}^{x} \sqrt{t+1}\,dt.$$ Answer the followi

Medium

Estimating River Flow Volume

A river's flow rate (in cubic meters per second) has been measured at various times during an 8-hour

Hard

Evaluating an Integral with U-substitution

Evaluate the integral $$\int_{1}^{3} 2*(x-1)^5\,dx$$ using u-substitution. Answer the following ques

Easy

Exact Area Under a Parabolic Curve

Find the exact area under the curve $$f(x)=3*x^2 - x + 4$$ between $$x=1$$ and $$x=4$$.

Easy

Experimental Data Analysis using Trapezoidal Sums

A chemical reaction is monitored over time, and the reaction rate $$f(t)$$ (in moles per minute) is

Hard

Exploring the Fundamental Theorem of Calculus

Let the function $$F(x) = \int_{1}^{x} \frac{1}{t^2+1}\,dt$$ represent an accumulation function. Ans

Medium

Finding the Area of a Parabolic Arch

An architect designs an arch described by the parabola $$y = 10 - \frac{x^{2}}{5}$$. The arch spans

Hard

FRQ4: Inverse Analysis of a Trigonometric Accumulation Function

Let $$ H(x)=\int_{0}^{x} (\sin(t)+2)\,dt $$ for $$ x \in [0,\pi] $$, representing a displacement fun

Medium

FRQ8: Inverse Analysis of a Piecewise-Defined Accumulation Function

Let $$ R(x)=\begin{cases} \int_{1}^{x} t\,dt, & 1 \le x \le 3 \\ \int_{1}^{x} (2*t-1)\,dt, & x > 3 \

Hard

FRQ11: Inverse Analysis of a Parameterized Function

For a positive constant a, consider the function $$ F(x)=\int_{a}^{x} \frac{1}{t+a}\,dt $$ for x > a

Medium

Fuel Consumption: Approximating Total Fuel Use

A car's fuel consumption rate (in liters per hour) is modeled by $$f(t)=0.05*t^2 - 0.3*t + 2$$, wher

Medium

Fundamental Theorem and Net Change

A function $$f$$ has a derivative given by $$f'(x)=3\sqrt{x}$$ for $$0\le x\le 9$$, and it is known

Medium

Integration by Parts: Evaluating $$\int_1^e \ln(x)\,dx$$

Evaluate the integral $$\int_1^e \ln(x)\,dx$$ using integration by parts.

Hard

Modeling Savings with a Geometric Sequence

A person makes annual deposits into a savings account such that the first deposit is $100 and each s

Medium

Motion Under Variable Acceleration

A particle moves along the x-axis with acceleration $$a(t) = 6 - 4*t$$ (in m/s²) for $$0 \le t \le 3

Medium

Population Growth: Accumulation through Integration

A certain population grows at a rate modeled by $$R(t)= 0.5*t^2 - 3*t + 10$$ (individuals per year),

Medium

Rainfall and Evaporation in a Greenhouse

In a greenhouse, rainfall is modeled by $$R(t)= 8\cos(t)+10$$ mm/hr, while evaporation occurs at a c

Easy

Rainwater Collection in a Reservoir

Rainwater falls into a reservoir at a rate given by $$R(t)= 12e^{-0.5t}$$ L/min while evaporation re

Medium

Riemann Sum Approximation from a Table

The table below gives values of a function $$f(x)$$ at selected points: | x | 0 | 2 | 4 | 6 | 8 | |

Medium

Ski Lift Passengers: Boarding and Alighting Rates

On a ski lift, passengers board at a rate $$B(t)= 3e^{-t/2}$$ persons/min and alight at a constant r

Medium

Total Cost Function from Marginal Cost

The marginal cost of production for a company is given by $$MC(q)=6+0.5*q$$ dollars per unit for pro

Easy

Trapezoidal Rule in Estimating Accumulated Change

A rising balloon has its height measured at various times. A portion of the recorded data is given i

Medium

Trigonometric Integration via U-Substitution

Evaluate the integral $$I=\int_{0}^{\frac{\pi}{4}} \tan(x)*\sec^2(x)\,dx.$$ Answer the following par

Medium

Volume of a Solid: Exponential Rotation

Consider the region bounded by the curve $$y=e^{-x}$$, the x-axis, and the vertical lines $$x=0$$ an

Medium
Unit 7: Differential Equations

Analysis of an Autonomous Differential Equation

Consider the autonomous differential equation $$\frac{dy}{dx}=y(4-y)$$ with the initial condition $$

Medium

Bacterial Nutrient Depletion

A nutrient in a bacterial culture is depleting over time according to the differential equation $$\f

Easy

Boat Crossing a River with Current

A boat is attempting to cross a river flowing at a constant speed of $$2$$ m/s. The boat is directed

Easy

Charging of a Capacitor

The voltage $$V$$ (in volts) across a capacitor being charged in an RC circuit is recorded over time

Medium

Chemical Reaction in a Vessel

A 50 L reaction vessel initially contains a solution of reactant A at a concentration of 3 mol/L (i.

Easy

Chemical Reaction Rate

The concentration $$y$$ (in moles per liter) of a reactant in a chemical reaction is modeled by the

Medium

Cooling of a Cup of Coffee

Newton's Law of Cooling states that the rate of change of temperature of an object is proportional t

Medium

Economic Growth with Investment Outflow

A company’s investment fund grows continuously at an annual rate of $$5\%$$, but expenses lead to a

Medium

Heating and Cooling in an Electrical Component

An electronic component experiences heating and cooling according to the differential equation $$\fr

Medium

Implicit Differentiation and Slope Analysis

Consider the function defined implicitly by $$y^2+ x*y = 8$$. Answer the following:

Easy

Implicit Differentiation with Trigonometric Functions

Consider the equation $$\sin(x*y)= x+ y$$. Answer the following:

Hard

Implicit IVP with Substitution

Solve the initial value problem $$\frac{dy}{dx}=\frac{y}{x}+\frac{x}{y}$$ with $$y(1)=2$$. (Hint: Us

Hard

Implicit Solution of a Differential Equation

The differential equation $$\frac{dy}{dx} = \frac{2x}{1+y^2}$$ requires an implicit solution.

Medium

Linear Differential Equation and Integrating Factor

Consider the differential equation $$\frac{dy}{dx} = y - x$$. Use the method of integrating factor t

Medium

Mixing Problem with Time-Dependent Inflow Concentration

A tank initially contains 100 liters of water with 8 kg of dissolved salt. Brine enters the tank at

Medium

Newton's Law of Cooling

An object is heated to $$100^\circ C$$ and left in a room at $$20^\circ C$$. According to Newton's l

Medium

Nonlinear Differential Equation with Powers

Consider the differential equation $$\frac{dy}{dx} = 4*y^{3/2}$$ with the initial condition $$y(1)=1

Hard

Oil Spill Cleanup Dynamics

To mitigate an oil spill, a cleanup system is employed that reduces the volume of oil in contaminate

Medium

Population Model with Harvesting

A fish population in a lake is modeled by the differential equation $$\frac{dP}{dt}=0.3*P\left(1-\fr

Hard

Radioactive Material with Continuous Input

A radioactive substance decays at a rate proportional to its amount while being produced continuousl

Easy

Reaction Kinetics in a Tank

In a chemical reactor, the concentration $$C(t)$$ of a reactant decreases according to the different

Medium

Reversible Chemical Reaction

In a reversible chemical reaction, the concentration $$C(t)$$ of a product is governed by the differ

Medium

Salt Mixing Problem

A tank initially contains $$100$$ kg of salt dissolved in $$1000$$ L of water. A salt solution with

Medium

Separable Differential Equation with Trigonometric Factor

Consider the differential equation $$\frac{dy}{dx}=(2y+3)\cos(x)$$. Answer each part using separatio

Medium

Slope Field Analysis and Asymptotics

Consider the differential equation $$\frac{dy}{dx}=\frac{x}{1+y^2}$$. Solve the equation and analyze

Hard

Slope Field Analysis for a Linear Differential Equation

Consider the linear differential equation $$\frac{dy}{dx}=\frac{1}{2}*x-y$$ with the initial conditi

Medium

Slope Field Interpretation for $$\frac{dy}{dx} = y-x$$

A slope field for the differential equation $$\frac{dy}{dx}=y-x$$ is provided in the stimulus. Use t

Easy

Solving a Differential Equation Using the SIPPY Method

Solve the differential equation $$\frac{dy}{dx}=4*x^3*e^{-y}$$ with the initial condition $$y(1)=0$$

Medium

Tank Draining Differential Equation

Water drains from a tank at a rate that depends on the square root of the volume, according to $$\fr

Medium

Tank Mixing with Salt

In a mixing problem, a tank contains salt that is modeled by the differential equation $$\frac{dS}{d

Easy

Vehicle Deceleration

A vehicle undergoing braking has its speed $$v$$ (in m/s) recorded over time (in seconds) as shown.

Easy

Volume by Revolution of a Differential Equation Derived Region

The function $$y(x) = e^{-x} + x$$, which is a solution to a differential equation, and the line $$y

Hard

Water Temperature Regulation in a Reservoir

A reservoir’s water temperature adjusts according to Newton’s Law of Cooling. Let $$T(t)$$ (in \(^{\

Easy
Unit 8: Applications of Integration

Area Between Curves: Revenue and Cost Analysis

A company’s revenue and cost are modeled by the functions $$f(x)=10-x^2$$ and $$g(x)=2*x$$, where $$

Medium

Average Concentration Calculation

In a continuous stirred-tank reactor (CSTR), the concentration of a chemical is given by $$c(t)=5+3*

Easy

Average Concentration in a Reaction

A chemical reaction has its concentration modeled by the function $$C(t)=50e^{-0.2*t}+5$$ (in mg/L)

Medium

Average Concentration in Medical Dosage

A patient’s bloodstream concentration of a medication is modeled by the function $$C(t)=\frac{100}{1

Hard

Average Temperature of a Cooling Liquid

The temperature of a cooling liquid is modeled by $$T(t)=50*e^{-0.1*t}+20$$ (in $$^\circ C$$) for $$

Medium

Charity Donations Over Time

A charity receives monthly donations that form an arithmetic sequence. The first donation is $$50$$

Easy

Comparing Sales Projections

A company’s projected sales (in thousands of dollars) are modeled by the function $$f(x)=5*x-x^2$$ w

Medium

Implicit Differentiation in Thermodynamics

In a thermodynamics experiment, the pressure $$P$$ and volume $$V$$ of a gas are related by the equa

Hard

Integrated Motion Analysis

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

Medium

Manufacturing Output Increase

A factory produces goods with weekly output that increases by a constant number of units each week.

Easy

Modeling Bacterial Growth

A bacterial culture grows at a rate modeled by $$g(t)=a*e^{0.3*t}$$, where $$t$$ is time in hours an

Medium

Particle Motion Along a Straight Line

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

Medium

Particle Motion on a Line

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

Medium

Particle Motion with Exponential Acceleration

A particle moves along a straight line with acceleration given by $$a(t)=2*e^{-t} - 1$$ (in m/s²) fo

Hard

Population Dynamics in a Wildlife Reserve

A wildlife reserve monitors the change in the number of a particular species. The rate of change of

Easy

Probability from a Density Function

Let a continuous random variable $$X$$ be defined on $$[0,20]$$ with the probability density functio

Medium

Projectile Motion: Time of Maximum Height

A projectile is launched vertically upward with an initial velocity of $$50\,m/s$$ and an accelerati

Medium

Revenue Optimization via Integration

A company’s revenue is modeled by $$R(t)=1000-50*t+2*t^2$$ (in dollars per hour), where $$t$$ (in ho

Medium

Stress Analysis in a Structural Beam

A beam in a building experiences a stress distribution along its length given by $$\sigma(x)=100-15*

Medium

Tank Draining with Variable Flow Rates

A water tank is undergoing simultaneous inflow and outflow. The inflow rate is given by $$I(t)=10+2\

Medium

Traveling Particle with Piecewise Motion

A particle moves along a line with a piecewise velocity function defined as follows: For $$t \in [0

Easy

Volume by the Washer Method: Solid of Revolution

Consider the region bounded by the curves $$y=x$$ and $$y=x^2$$ for $$0 \le x \le 1$$. This region i

Medium

Volume of a Solid of Revolution: Disc Method

Consider the region R bounded by $$y= \sqrt{x}$$, the x-axis, and the vertical line $$x=4$$. When R

Medium

Volume of a Solid Using the Washer Method

Consider the region bounded by $$y=x$$ and $$y=x^2$$ between $$x=0$$ and $$x=1$$. This region is rev

Medium

Volume of a Solid with Square Cross Sections

A solid has a base in the xy-plane given by the region bounded by the curves $$y=x$$ and $$y=\sqrt{x

Medium

Water Flow in a River: Average Velocity and Flow Rate

A river has a width of 10 meters. The velocity of the water at a distance $$x$$ (in meters) from one

Medium

Water Reservoir Inflow‐Outflow Analysis

A water reservoir receives water through an inflow pipe and loses water through an outflow valve. Th

Medium

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FAQWe thought you might have some questions...
Where can I find practice free response questions for the AP Calculus AB 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 ABFree 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 AB 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 AB 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 AB 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 AB free-response questions?
Answering AP Calculus AB free response questions the right way is all about practice! As you go through the AP AP Calculus AB 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.