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AP Calculus AB Free Response Questions

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  • Unit 1: Limits and Continuity (34)
  • Unit 2: Differentiation: Definition and Fundamental Properties (27)
  • Unit 3: Differentiation: Composite, Implicit, and Inverse Functions (31)
  • Unit 4: Contextual Applications of Differentiation (33)
  • Unit 5: Analytical Applications of Differentiation (38)
  • Unit 6: Integration and Accumulation of Change (30)
  • Unit 7: Differential Equations (28)
  • Unit 8: Applications of Integration (29)
Unit 1: Limits and Continuity

Algebraic Manipulation and Limit Evaluation

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

Easy

Analyzing a Piecewise Function for Continuity

Consider the piecewise function $$ f(x)=\begin{cases} 2x+1, & x<2 \\ x^2-1, & x\geq2 \end{cases}$$.

Easy

Analyzing a Piecewise Velocity Function for Continuity and Limits

A particle moves along a line with a piecewise velocity function given by $$v(t)= \begin{cases} 2*t+

Easy

Analyzing Limit of an Oscillatory Velocity Function

A particle moves along a line with velocity given by $$v(t)= t*\cos\left(\frac{\pi}{t}\right)$$ for

Hard

Analyzing Multiple Discontinuities in a Rational Function

Let $$f(x)= \frac{(x^2-9)(x+4)}{(x-3)(x^2-16)}$$.

Extreme

Application of the Squeeze Theorem

Consider the function defined by $$h(x)=\begin{cases} x^2 \sin\left(\frac{1}{x}\right) & \text{if }

Medium

Area and Volume Setup with Bounded Regions

Consider the region R bounded by the curves $$y = x^2$$, $$y = 4$$, and $$x = 0$$. Though integratio

Hard

Asymptotic Analysis of a Radical Rational Function

Consider the function $$f(x)=\sqrt{4x^2+x}-2x$$ for \(x>0\). Answer the following:

Medium

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 Analysis of a Piecewise Function

Consider the function defined by $$ f(x)=\begin{cases}2x+1, & x<1,\\ x^2, & 1\le x\le 3,\\ 7-x, & x

Medium

Continuity of Constant Functions

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

Easy

Determining Horizontal Asymptotes for Rational Functions

Given the rational function $$R(x)= \frac{2*x^3+ x^2 - x}{x^3 - 4}$$, answer the following:

Medium

Determining Parameters for a Continuous Log-Exponential Function

Suppose a function is defined by $$ v(x)=\begin{cases} \frac{\ln(e^{p*x}+x)-q*x}{x} & \text{if } x \

Hard

Determining Parameters for Continuity in a Piecewise Function

Let the function be defined as $$ f(x)=\begin{cases}ax+3, & x<2,\\ x^2+bx+1, & x\ge2.\end{cases} $$

Medium

End Behavior of Rational Functions

Examine the rational function $$f(x)=\frac{3*x^3-2*x+1}{6*x^3+4*x^2-5}$$. Determine its behavior as

Easy

Evaluating Limits Involving Square Roots

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

Medium

Implicit Differentiation and Tangent Slopes

Consider the circle defined by $$x^2 + y^2 = 25$$. Answer the following:

Medium

Implicit Differentiation in an Exponential Equation

Consider the function defined implicitly by $$e^{x*y} + x - y = 0$$. Answer the following:

Extreme

Intermediate Value Theorem Application

Suppose a continuous function $$f(x)$$ is defined on the interval $$[1,5]$$, with $$f(1)=-3$$ and $$

Easy

Limit Involving a Square Root and Removable Discontinuity

Consider the function $$h(x)=\frac{\sqrt{x+4}-2}{x}$$ for $$x\neq0$$ and $$h(0)=1$$. Answer the foll

Easy

Limits at Infinity and Horizontal Asymptotes

Consider the rational function $$R(x) = \frac{2x^2 - 3x + 4}{x^2 + 5}$$. Analyze its behavior as x a

Easy

Limits at Infinity for Non-Rational Functions

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

Medium

Limits Involving a Removable Discontinuity

Consider the function $$g(x)= \frac{(x+3)(x-2)}{x-2}$$ defined for $$x \neq 2$$. Answer the followin

Easy

Limits Near Vertical Asymptotes

Consider the function $$f(x) = \frac{1}{x - 4}$$. (a) Determine $$\lim_{x \to 4^-} f(x)$$. (b) Dete

Easy

Limits of Absolute Value Functions

Consider the function $$f(x)= \frac{|x-2|}{x-2}$$. Answer the following:

Easy

Modeling Bacterial Growth with a Geometric Sequence

A particular bacterial colony doubles in size every hour. The population at time $$n$$ hours is give

Easy

One-Sided Limits and an Absolute Value Function

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

Easy

Optimization and Continuity in a Manufacturing Process

A company designs a cylindrical can (without a top) for which the cost function in dollars is given

Hard

Oscillatory Behavior and Non-Existence of Limit

Let \(f(x)=\sin(1/x)\) for \(x\neq0\). Answer the following:

Hard

Particle Motion with Squeeze Theorem Application

A particle moves along a line with velocity given by $$v(t)= t^2 \sin(1/t)$$ for $$t>0$$ and is defi

Medium

Return on Investment and Asymptotic Behavior

An investor’s portfolio is modeled by the function $$P(t)= \frac{0.02t^2 + 3t + 100}{t + 5}$$, where

Medium

Squeeze Theorem Application

Consider the function $$f(x)=x^2\sin(\frac{1}{x})$$ for $$x\neq0$$ and $$f(0)=0$$. Answer the follow

Easy

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 Analysis

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

Easy
Unit 2: Differentiation: Definition and Fundamental Properties

Analysis of Motion in the Plane

A particle moves in the plane with its position given by $$\mathbf{s}(t)=\langle t^2 - 4*t,\, 3*t +

Medium

Analyzing the Derivative of a Trigonometric Function

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

Easy

Application of the Quotient Rule: Velocity on a Curve

A car's velocity is modeled by $$v(t)= \frac{2*t+3}{t+1}$$, where $$t$$ is measured in seconds. Anal

Hard

Average vs. Instantaneous Rate of Change

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

Medium

Derivative from First Principles

Let $$f(x)=\sqrt{x}$$. Using the limit definition of the derivative, answer the following:

Medium

Differentiating a Product of Linear Functions

Let $$f(x) = (2*x^2 + 3*x)\,(x - 4)$$. Use the product rule to find $$f'(x)$$.

Easy

Finding the Derivative using the Limit Definition

Let $$h(x)= 5*x^2 + 3*x - 7$$. Use the limit definition of the derivative to determine $$h'(x)$$.

Easy

Graphical Estimation of a Derivative

Consider the graph provided which plots the position $$s(t)$$ (in meters) of an object versus time $

Medium

Higher-Order Derivatives in Motion

A particle moves along a line with its position given by $$s(t)= t^3 - 6*t^2 + 9*t + 5$$, where $$t$

Hard

Instantaneous Acceleration from a Velocity Function

A runner's velocity is given by $$v(t)= 3*t^2 - 12*t + 9$$, where $$t$$ is in seconds. Analyze the r

Easy

Inverse Function Analysis: Cubic Transformation

Consider the function $$f(x)=(x-1)^3$$ defined for all real numbers.

Easy

Inverse Function Analysis: Rational Function 2

Consider the function $$f(x)=\frac{x+4}{x+2}$$ defined for $$x\neq -2$$, with the additional restric

Medium

Inverse Function Analysis: Trigonometric Function with Linear Term

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

Medium

Marginal Cost Function in Economics

A company’s cost function is given by $$C(x)=200+8*x+0.05*x^2$$, where $$C(x)$$ is in dollars and $$

Easy

Medication Infusion with Clearance

A patient receives medication via an IV at a rate of $$f(t)=5*e^{-0.1*t}$$ mg/min, while the body cl

Medium

Optimizing Car Speed: Rate of Change Analysis

A car’s speed in km/h is modeled by the function $$s(t)=50+2*t^2-0.1*t^3$$ for $$0 \leq t \leq 10$$

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

Product Rule Application in Economics

A company's cost function for producing $$x$$ units is given by $$C(x)= (3*x+2)*(x^2+5)$$ (cost in d

Medium

Projectile Motion Analysis

A projectile is launched with its height (in meters) modeled by the function $$f(t)= -5*t^2 + 20*t +

Medium

RC Circuit Voltage Decay

An RC circuit's capacitor voltage is modeled by $$V(t)= V_{0}*e^{-t/(R*C)}$$, where $$V_{0}$$ is the

Medium

Relating Average and Instantaneous Velocity in a Particle's Motion

A particle’s position is modeled by $$s(t)=\frac{4}{t+1}$$, where $$s(t)$$ is in meters and $$t$$ is

Medium

River Crossover: Inflow vs. Damming

A river receives water from two tributaries at rates $$f_1(t)=7+0.5*t$$ and $$f_2(t)=9-0.2*t$$ (lite

Medium

Secant Slope from Tabulated Data

A table below gives values of a function $$f(x)$$ representing the concentration of a solution at di

Medium

Tangent Line Equation for an Exponential Function

Consider the function $$f(x)= e^{x}$$ and its graph.

Easy

Using Derivative Rules on a Trigonometric Function

Consider the function $$f(x)=3*\sin(x)+\cos(2*x)$$. Answer the following questions:

Hard

Using the Difference Quotient with a Polynomial Function

Let $$g(x)=2*x^2 - 5*x + 3$$. Answer the following questions:

Easy

Using the Limit Definition of the Derivative

Consider the function $$g(x)=3*x^3-2*x+5$$, which models the cost (in dollars) of manufacturing $$x$

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

Analyzing Motion in the Plane using Implicit Differentiation

A particle moves in the xy-plane along a path defined implicitly by $$x^2+x*y+y^2=7$$. Determine the

Medium

Chain and Product Rules in a Rate of Reaction Process

In a chemical reaction, the rate is modeled by the function $$R(t)= \sqrt{t+3}*\cos(2*t)$$, where $$

Medium

Chain Rule in an Economic Model

In an economic model, the cost function for producing a good is given by $$C(x)=(3*x+1)^5$$, where $

Easy

Chain Rule in Population Modeling

A biologist models the population of a species with the function $$P(t)= f(g(t))$$, where $$g(t)=25*

Medium

Chain Rule in Temperature Model

A scientist models the temperature in a laboratory experiment by the function $$T(t)=\sqrt{3*t^2+2}$

Easy

Chain Rule with Trigonometric and Exponential Functions

Let $$y = \sin(e^{3*x})$$. Answer the following:

Medium

Chain Rule with Trigonometric Function

Consider the function $$f(x)=\sin(5*x^2)$$. Answer the following:

Easy

Composite Function in Biomedical Model

The concentration C(t) (in mg/L) of a drug in the bloodstream is modeled by $$C(t) = \sin(3*t^2)$$,

Medium

Differentiation of an Inverse Trigonometric Composite Function

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

Medium

Differentiation of Inverse Function with Polynomial Functions

Let \(f(x)= x^3+2*x+1\) be a one-to-one function. Its inverse is denoted by \(f^{-1}\).

Medium

Estimating Derivatives Using a Table

An experiment measures a one-to-one function $$f$$ and its inverse $$g$$, yielding the following dat

Easy

Implicit Differentiation for an Ellipse

Consider the ellipse defined by the equation $$\frac{x^2}{9} + \frac{y^2}{4} = 1$$. This equation re

Medium

Implicit Differentiation in a Cubic Relationship

Examine the curve defined by $$x^3 + y^3 = 6*x*y$$, which represents a complex relationship between

Hard

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 an Economic Model

In an economic model, the relationship between the quantity supplied $$x$$ and the market price $$y$

Hard

Implicit Differentiation in an Ellipse

Consider the ellipse defined by $$4*x^2+9*y^2=36$$.

Medium

Implicit Differentiation in Circular Motion

Given the circle defined by $$x^2 + y^2 = 16$$, analyze its differential properties.

Medium

Implicit Differentiation Involving Logarithms

Consider the equation $$\ln(x) + x*y = \ln(y) + x$$ which relates $$x$$ and $$y$$. Use implicit diff

Medium

Implicit Differentiation of a Logarithmic-Exponential Equation

Consider the equation $$\ln(x+y) + e^{x*y} = 7$$, which implicitly defines $$y$$ as a function of $$

Extreme

Implicit Differentiation of an Exponential-Product Equation

Consider the curve defined implicitly by $$e^(x*y) + x - y = 0.$$ Solve the following:

Medium

Implicit Differentiation on a Circle

Consider the circle defined by $$x^2+y^2=25$$.

Easy

Implicit Differentiation with Product Rule

Consider the implicit equation $$x*y+y^2=6$$ which defines $$y$$ as a function of $$x$$. Use implici

Medium

Implicit Differentiation with Product Rule

Consider the equation $$x*y+e^{y}=x^2$$. 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

Implicitly Defined Inverse Relation

Consider the relation $$y + \ln(y)= x.$$ Answer the following:

Easy

Inverse Derivative of a Sum of Exponentials and Linear Terms

Let $$f(x)= e^(x)+ x$$ and let g be its inverse function satisfying $$g(f(x))= x$$. Answer the follo

Easy

Inverse Function Differentiation in a Biological Growth Curve

A biological measurement is modeled by the function $$f(t)= \frac{4*t}{t+2}$$, which is one-to-one o

Medium

Inverse Trigonometric Function Differentiation

Consider the function $$y= \arcsin\left(\frac{2*x}{1+x^2}\right)$$.

Hard

Pendulum Angular Displacement Analysis

A pendulum's angular displacement is modeled by the function $$\theta(t)=\sin(2t^2+3)$$, where t is

Easy

Rate of Change in a Circle's Shadow

The equation of a circle is given by $$x^2 + y^2 = 36$$. A point \((x,y)\) on the circle corresponds

Easy

Related Rates: Shadow Length

A 1.8 m tall person is walking away from a street lamp that is 5 m tall at a speed of 1.2 m/s. Using

Hard
Unit 4: Contextual Applications of Differentiation

Accelerating Car Motion Analysis

A car's velocity is modeled by $$v(t)=4t^2-16t+12$$ in m/s for $$t\ge0$$. Analyze the car's motion.

Medium

Chemical Reaction Rate Analysis

A chemical reaction follows the concentration model $$c(t)=\frac{100}{1+5e^{-0.3t}}$$, where c is in

Medium

Cost Function Optimization

A company’s cost is modeled by the function $$C(x)=0.5x^3-6x^2+20x+100$$, where x (in hundreds of un

Hard

Dynamics of a Car: Stopping Distance and Deceleration

A car traveling at 30 m/s begins to decelerate at a constant rate. Its velocity is modeled by $$v(t)

Medium

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

Expanding Circular Ripple in a Pond

A circular ripple in a pond has its area increasing at a constant rate of 10 square meters per secon

Easy

Filling a Conical Tank: Related Rates

Water is being pumped into an inverted conical tank at a rate of $$\frac{dV}{dt}=3\;m^3/min$$. The t

Medium

FRQ 16: Implicit Differentiation in Orbital Mechanics

A satellite’s orbit is described by the equation $$x^2 + 2*x*y + y^2 = 25$$, where x and y represent

Medium

FRQ 20: Market Demand Analysis

In an economic market, the demand D (in thousands of units) and the price P (in dollars) satisfy the

Hard

Graphing a Function via its Derivative

Consider the function $$f(x) = x^{1/3}$$ defined for all real numbers.

Medium

Growth Rate Estimation in a Biological Experiment

In a biological experiment, the mass $$M(t)$$ (in grams) of a bacteria colony is recorded over time

Medium

Implicit Differentiation and Related Rates in Conic Sections

A point moves along the ellipse defined by $$\frac{x^2}{9} + \frac{y^2}{16} = 1$$. At a certain inst

Extreme

Inflation of a Balloon: Surface Area Rate of Change

A spherical balloon is being inflated so that its volume increases at a rate of $$\frac{dV}{dt}=50$$

Medium

Inflection Points and Concavity in Business Forecasting

A company's profit is modeled by $$P(x)= 0.5*x^3 - 6*x^2 + 15*x - 10$$, where $$x$$ represents a pro

Medium

Interpretation of the Derivative from Graph Data

The graph provided represents the position function $$s(t)$$ of a particle moving along a straight l

Medium

Linearization and Differentials

Given the function $$f(x)=x^4$$, use linear approximation to estimate the value of $$(3.98)^4$$.

Easy

Linearization of a Nonlinear Function

Suppose $$f(x)=\ln(x)$$. Use linearization about $$x=4$$ to approximate $$\ln(4.1)$$. Answer the fol

Easy

Local Linearization Approximation

Let $$f(x)=x^3.$$ We want to approximate $$f(4.02)$$ using linearization near $$x=4$$.

Easy

Minimizing Materials for a Cylindrical Can

A manufacturer aims to design a closed cylindrical can that holds exactly $$500$$ cubic centimeters

Hard

Optimization: Minimizing Surface Area of a Box

An open-top box with a square base is to have a volume of 500 cubic inches. The surface area (materi

Medium

Particle Acceleration and Direction of Motion

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

Medium

Population Change Rate

The population of a town is modeled by $$P(t)= 50*e^{0.3*t}$$, where $$t$$ is in years and $$P(t)$$

Hard

Population Growth Rate Analysis

A town's population is modeled by the exponential function $$P(t) = 500e^{0.03t}$$, where $$t$$ is i

Medium

Reaction Rate and Temperature

The rate of a chemical reaction is modeled by $$r(T)= 0.5*e^{-0.05*T}$$, where $$T$$ is the temperat

Hard

Related Rates in Shadows: A Lamp and a Tree

A lamp post 5 m tall casts a shadow of a tree. At a certain moment, the tree is 3 m from the lamp an

Hard

Related Rates: Expanding Circle

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

Easy

Related Rates: Expanding Circular Ripple

A circular ripple on a calm water surface is expanding such that its area is increasing at a rate of

Easy

Related Rates: Shadow Length

A 1.8-meter tall person is walking away from a 4.5-meter tall streetlight at a constant speed of 1.2

Easy

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

Route Optimization for a Rescue Boat

A rescue boat must travel from a point on the shore to an accident site located 2 km along the shore

Hard

Seasonal Water Reservoir

A reservoir's water volume (in million m³) changes with the seasons according to $$V(t)=5+2\sin\left

Hard

Temperature Cooling in a Cup of Coffee

The temperature of a cup of coffee is modeled by $$T(t)=70+50e^{-0.1*t}$$ (in °F), where $$t$$ is th

Easy

Water Tank Volume Change

A water tank is being filled and its volume is given by $$V(t)= 4*t^3 - 9*t^2 + 5*t + 100$$ (in gall

Medium
Unit 5: Analytical Applications of Differentiation

Analyzing a Piecewise Function and Differentiability

Let $$f(x)$$ be defined piecewise by $$f(x)= x^2$$ for $$x \le 2$$ and $$f(x) = 4*x - 4$$ for $$x >

Hard

Approximating Displacement from Velocity Data

A vehicle's velocity (in $$m/s$$) over time (in seconds) was recorded during a test run. The table b

Medium

Area Bounded by $$\sin(x)$$ and $$\cos(x)$$

Consider the functions $$f(x)= \sin(x)$$ and $$g(x)= \cos(x)$$ on the interval $$[0, \frac{\pi}{2}]$

Easy

Average Value of a Function and Mean Value Theorem for Integrals

Consider the function $$f(x)= e^{-x}$$ on the interval $$[0, 2]$$. Answer the following:

Hard

Chemical Mixing in a Tank

A 200-liter tank initially contains pure water. A salt solution with a concentration of 0.5 kg/L flo

Medium

Chemical Reaction Rate and Exponential Decay

In a chemical reaction, the concentration of a reactant declines according to $$C(t)= C_0* e^{-k*t}$

Medium

Concavity and Inflection Points of a Cubic Function

Consider the cubic function $$f(x)=x^3-6*x^2+9*x+2$$. Answer the following questions regarding its d

Medium

Derivative of the Natural Log Function by Definition

Let $$f(x)= \ln(x)$$. Use the definition of the derivative to prove that $$f'(a)= \frac{1}{a}$$ for

Easy

Drag Force and Rate of Change from Experimental Data

Drag force acting on an object was measured at various velocities. The table below presents the expe

Medium

Exponential Bacterial Growth

A bacterial culture grows according to $$P(t)= P_0 * e^{k*t}$$, where $$t$$ is in hours. The culture

Easy

FRQ 2: Daily Temperature Extremes and the Extreme Value Theorem

A function modeling the ambient temperature (in $$^\circ C$$) during the first 6 hours of a day is g

Medium

FRQ 11: Particle Motion with Non-Constant Acceleration

A particle moves along a straight line with acceleration given by $$a(t)= 12*t - 6$$ (in m/s²). If t

Hard

FRQ 14: Projectile Motion – Determining Maximum Height

The height of a projectile (in meters) is modeled by $$h(t)= -4.9*t^2 + 20*t + 5$$, where $$t$$ is t

Medium

FRQ 16: Finding Relative Extrema for a Logarithmic Function

Consider the function $$f(x)= \ln(x) - x$$ defined for $$x>0$$.

Medium

FRQ 19: Analysis of an Exponential-Polynomial Function

Consider the function $$f(x)= e^{-x}*x^2$$ defined for $$x \ge 0$$.

Hard

Garden Fence Optimization Problem

A rectangular garden is to be built adjacent to a building. Fencing is required on only three sides

Medium

Graphical Analysis and Derivatives

A function \( f(x) \) is represented by the graph provided below. Answer the following based on the

Medium

Implicit Differentiation and Tangent Lines

Consider the curve defined implicitly by the equation $$x^2 + x*y + y^2= 7$$.

Medium

Inverse Analysis of a Linear Function

Consider the function $$f(x)=3*x+2$$. Analyze its inverse function by answering all parts below.

Easy

Inverse Analysis: Transformation Geometry of a Parabolic Function

Consider the function $$f(x)=4-(x-3)^2$$ with the domain $$x\le 3$$. Analyze its inverse function as

Medium

Limit Analysis of a Piecewise Function Involving a Rational Expression

Consider the function $$ f(x) = \begin{cases} \frac{2x^2-8}{x-2}, & x < 2, \\ x+2, & x \ge 2. \end{

Medium

Liquid Cooling System Flow Analysis

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

Hard

Logarithmic Transformation of Data

A scientist models an exponential relationship between variables by the equation $$y= A*e^{k*x}$$. T

Hard

Logistic Population Model Analysis

Consider the logistic model $$P(t)= \frac{500}{1+ 9e^{-0.4t}}$$, where $$t$$ is in years. Answer the

Hard

Optimization of an Open-Top Box

A company is designing an open-top box with a square base. The volume of the box is modeled by the f

Hard

Piecewise Function and the Mean Value Theorem

Consider the piecewise function $$f(x)= \begin{cases} x^2 & \text{if } x \le 1, \\ 2*x - 1 & \text{

Medium

Polynomial Rational Discontinuity Investigation

Consider the function $$ g(x) = \begin{cases} \frac{x^3 - 8}{x - 2}, & x \neq 2, \\ 5, & x = 2. \en

Easy

Population Growth Analysis via the Mean Value Theorem

A country's population data over a period of years is given in the table below. Use the data to anal

Medium

Production Cost Optimization and the Extreme Value Theorem

A company monitored its production cost as a function of units produced. The following table gives e

Medium

Profit Analysis and Inflection Points

A company's profit is modeled by $$P(x)= -x^3 + 9*x^2 - 24*x + 10$$, where $$x$$ represents thousand

Hard

Profit Function Concavity Analysis

A company’s profit is modeled by $$P(x) = -2*x^3 + 18*x^2 - 48*x + 10$$, where $$x$$ is measured in

Hard

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

Relative Extrema of a Rational Function

Examine the function $$f(x)= \frac{x+1}{x^2+1}$$ and determine its relative extrema using derivative

Medium

Sand Pile Dynamics

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

Medium

Traffic Flow Modeling

A highway segment experiences varying traffic flows. Cars enter at a rate $$I(t)=50+10*\sin(\frac{\p

Medium

Urban Water Supply Management

An urban water supply system receives water from two sources. The inflow rates are $$R_1(t)=15+2*t$$

Medium

Volume of Solid with Square Cross-Sections

Consider the region between $$f(x)= \sin(x)$$ and the x-axis on the interval $$[0, \pi]$$. A solid i

Medium
Unit 6: Integration and Accumulation of Change

Accumulated Change Function Evaluation

Let $$F(x)=\int_{1}^{x} (2*t+3)\,dt$$ for $$x \ge 1$$. This function represents the accumulated chan

Easy

Accumulation Function and Its Derivative

Define the function $$F(x)= \int_0^x \Big(e^{t} - 1\Big)\,dt$$. Answer the following parts related t

Easy

Analyzing Bacterial Growth via Riemann Sums

A biologist measures the instantaneous growth rate of a bacterial population (in thousands of cells

Medium

Antiderivatives of Trigonometric Functions

Evaluate the integral $$\int \sin(2*t)\,dt$$, and then use your result to compute the definite integ

Easy

Antiderivatives with Initial Conditions: Temperature

The rate of temperature change in a chemical reaction is given by $$T'(t)=-0.2*t+3$$ (in °C/min), wi

Easy

Area Under a Piecewise Function

Consider the piecewise function $$ f(x)= \begin{cases} x & \text{for } 0\le x<3,\\ 9-x & \text{for

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

Chemical Reactor Conversion Process

In a chemical reactor, the instantaneous reaction rate is given by $$R(t)=4t^2-t+3$$ mol/min, while

Hard

Definite Integral Approximation Using Riemann Sums

Consider the function $$f(x)= x^2 + 3$$ defined on the interval $$[2,6]$$. A table of sample values

Medium

Electric Charge Accumulation

An electrical circuit records the current (in amperes) at various times during a brief experiment. U

Easy

Elevation Profile Analysis on a Hike

A hiker records the elevation (in meters) along a trail at various distances. Use this data to analy

Medium

Evaluating a Trigonometric Integral Using U-Substitution

Evaluate the integral $$\int_{0}^{\frac{\pi}{2}} \sin(2*x)\,dx$$ using u-substitution.

Easy

Evaluating an Integral with a Trigonometric Function

Evaluate the definite integral $$\int_{0}^{\frac{\pi}{2}} \cos(x)*\sin(x)\,dx$$ using an appropriate

Easy

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

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

Fuel Consumption for a Rocket Launch

During a rocket launch, fuel is consumed at a rate $$F_{cons}(t)=50-3t$$ kg/s while additional fuel

Medium

Implicit Differentiation and Integration Verification

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

Extreme

Integration by U-Substitution in Physics

Consider the integral $$I=\int_0^4 \frac{t}{\sqrt{4+t^2}}\,dt.$$ This integral arises in determining

Hard

Net Surplus Calculation

A consumer's satisfaction is given by $$S(x)=100-4*x^2$$ and the marginal cost is given by $$C(x)=30

Hard

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). T

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 Accumulation Analysis

The rainfall intensity at a location is modeled by the function $$i(t) = 0.5*t$$ (inches per hour) f

Easy

Roller Coaster Work Calculation

An amusement park engineer recorded the force applied by a roller coaster engine (in Newtons) at var

Extreme

Total Distance from Velocity Data

A car’s velocity, in meters per second, is recorded over time as given in the table below: | Time (

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 Integral with U-Substitution

Evaluate the definite integral $$\int_{0}^{\frac{\pi}{4}} \sec^2(t)\tan(t)\,dt$$.

Medium

Volume of a Solid by Washer Method

A region is bounded by the curves $$y=\sqrt{x}$$ and $$y=\frac{x}{2}$$. This region, between the cur

Hard

Volume of a Solid with Square Cross Sections

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

Medium

Water Accumulation in a Tank

Water flows into a tank at a rate given by $$R(t)=2*\sqrt{t}$$ (in m³/min) for t in minutes. Answer

Medium

Water Tank: Accumulation and Maximum Level

A water tank is being filled with water at a rate $$r_{in}(t) = 4 + \sin(t)$$ L/min and is simultane

Medium
Unit 7: Differential Equations

Analyzing Slope Fields for $$dy/dx=x\sin(y)$$

Consider the differential equation $$\frac{dy}{dx}=x\sin(y)$$. A corresponding slope field is provid

Medium

Carbon Dating and Radioactive Decay

Carbon dating is based on the radioactive decay model given by $$\frac{dC}{dt}=-kC$$. Let the initia

Medium

Chemical Reaction Rate

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

Medium

Chemical Reaction Rate and Concentration Change

The rate of a chemical reaction is described by the differential equation $$\frac{dC}{dt}=-0.3*C^2$$

Medium

Cooling with Variable Ambient Temperature

An object cools in an environment where the ambient temperature varies with time. Its temperature $$

Extreme

Drug Concentration Model

The concentration $$C(t)$$ (in mg/L) of a drug in a patient's bloodstream is modeled by the differen

Medium

Drug Concentration with Continuous Infusion

A drug is administered intravenously such that its blood concentration $$C(t)$$ (in mg/L) follows th

Hard

Falling Object with Air Resistance

A falling object experiences air resistance proportional to its velocity. Its motion is modeled by t

Medium

Fishery Harvesting Model

The fish population in a lake is modeled by the differential equation $$\frac{dP}{dt} = 0.8P\left(1-

Hard

Implicit Solution of a Differential Equation

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

Medium

Integrating Factor Initial Value Problem

Solve the initial value problem $$\frac{dy}{dx}+\frac{2}{x}y=x^2$$ for $$x>0$$ with $$y(1)=3$$.

Easy

Logistic Growth Model

A population $$P(t)$$ grows according to the logistic model $$\frac{dP}{dt} = rP\left(1-\frac{P}{K}\

Hard

Motion Along a Curve with Implicit Differentiation

A particle moves along the curve defined by $$x^2+ y^2- 2*x*y= 1$$. At a certain instant, its horizo

Medium

Motion Under Gravity with Air Resistance

An object is falling vertically under the influence of gravity and air resistance. Its velocity $$v(

Medium

Particle Motion in the Plane

A particle moving in the plane has a constant x-component velocity of $$v_x(t)=2$$ m/s, and its y-co

Medium

Radioactive Decay

A radioactive substance decays according to $$\frac{dN}{dt} = -\lambda N$$. Initially, there are 500

Easy

Radioactive Decay

A radioactive substance decays according to the differential equation $$\frac{dN}{dt}=-kN$$. If the

Easy

Radioactive Decay

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

Easy

Radioactive Decay and Half-Life

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

Easy

Radioactive Material with Constant Influx

A laboratory receives radioactive waste material at a constant rate of $$3$$ g/day. Simultaneously,

Easy

Separable Differential Equation with Initial Condition

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

Medium

Separable Differential Equation with Trigonometric Component

Solve the differential equation $$\frac{dy}{dx}=\frac{3x^2}{1+\sin(y)}$$ with the initial condition

Hard

Separable Differential Equation: $$dy/dx = x*y$$

Consider the differential equation $$dy/dx = x*y$$ with the initial condition $$y(0)=2$$. Solve the

Medium

Separable Differential Equation: y' = (2*x)/y

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

Easy

Slope Field Analysis for $$dy/dx = x$$

Consider the differential equation $$dy/dx = x$$. A slope field representing this equation is provid

Easy

Temperature Regulation in a Greenhouse

The temperature $$T$$ (in °F) inside a greenhouse is recorded over time (in hours) as shown. The war

Medium

Traffic Flow Dynamics

On a highway, the density of cars, \(D(t)\) (in cars), changes over time due to a constant inflow of

Easy

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

Accumulated Rainfall Calculation

During a 12-hour storm, the rainfall rate is modeled by $$r(t)=3+\sin(t)$$ (in mm/hr) for $$0 \le t

Easy

Area Between Cost Functions in a Business Analysis

A company analyzes its cost structure using two functions: the fixed-plus-variable cost function $$C

Easy

Area Between Curves in an Ecological Study

In an ecological study, the population densities of two species are modeled by the functions $$P_1(x

Hard

Area Between Curves with Variable Limits

Consider two functions, $$f(x)$$ and $$g(x)$$, whose values are tabulated below. The functions inter

Extreme

Average and Instantaneous Rates in a Cooling Process

A cooling process is modeled by the function $$T(t)= 100*e^{-0.05*t}$$ (in degrees Fahrenheit), wher

Medium

Average Density of a Rod

A rod of length $$10$$ cm has a linear density given by $$\rho(x)= 4 + x$$ (in g/cm) for $$0 \le x \

Medium

Average Drug Concentration in the Bloodstream

The concentration of a drug in the bloodstream is modeled by $$C(t)=\frac{20*t}{1+t^2}$$ (in mg/L) f

Easy

Average Speed from a Velocity Function

A car’s velocity is given by $$v(t)=t^2-4*t+5$$ (in m/s) for $$0 \le t \le 5$$. Assume that $$v(t)$$

Medium

Average Temperature Analysis

A research facility recorded the temperature in a greenhouse over a period of 5 hours. The temperatu

Medium

Average Temperature Analysis

A local weather station recorded the temperature throughout a day using the model $$T(t)=-0.5*t+35$$

Easy

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

Average Value and the Mean Value Theorem

For the function $$f(x)=\cos(x)$$ on the interval [0, $$\pi/2$$], compute the average value and find

Medium

Average Value of a Function in a Production Process

A factory machine's temperature (in $$^\circ C$$) during a production run is modeled by $$T(t)= 5*t

Easy

Car Braking Analysis

A car decelerates with acceleration given by $$a(t)=-4e^{-t/2}$$ (in m/s²) and has an initial veloci

Hard

Cooling Process Analysis

A cup of coffee cools in a room, and its temperature (in °C) is modeled by $$T(t)=30*e^{-0.1*t}+5$$

Hard

Cost Analysis with Discontinuous Pricing

A utility company’s billing is modeled by the function $$C(q)=\begin{cases} 3*q & \text{if } 0\le q\

Medium

Cost Optimization for a Cylindrical Container

A manufacturer wishes to design a closed cylindrical container with a fixed volume $$V_0$$. The cost

Extreme

Funnel Design: Volume by Cross Sections

A funnel is designed by rotating the region bounded by the curve $$y=4-x^2$$ for -2 ≤ x ≤ 2 about th

Extreme

Hiking Trail: Position from Velocity

A hiker's velocity is given by $$v(t)=3\cos(t/2)+1$$ (in km/h) for 0 ≤ t ≤ 2π. Assuming the hiker st

Medium

Medication Dosage Increase

A patient receives a daily medication dose that increases by a fixed amount each day. The first day'

Easy

Motion Analysis Using Integration of a Sinusoidal Function

A car has velocity given by $$v(t)=3*\sin(t)+4$$ (in $$m/s$$) for $$t \ge 0$$, and its initial posit

Hard

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 Growth Rate Analysis

Suppose the instantaneous growth rate of a population is given by $$r(t)=0.04 - 0.002*t$$ for $$t \i

Medium

Population Growth with Variable Growth Rate

A city's population changes with time according to a non-constant growth rate given in thousands per

Medium

Position and Velocity Relationship in Car Motion

A car's position along a highway is modeled by $$s(t)=t^3-6*t^2+9*t+2$$ (in kilometers) with time $$

Medium

Shaded Area between $$y=\sqrt{x}$$ and $$y=\frac{x}{2}$$

Consider the curves $$f(x)=\sqrt{x}$$ and $$g(x)=\frac{x}{2}$$. Use integration to determine the are

Easy

Temperature Average Calculation

A scientist records the temperature in a lab using a continuous function $$T(t)=3*t^2 - 4*t + 5$$, w

Medium

Volume with Square Cross-Sections

Consider the region bounded by the curve $$y=x^2$$ and the line $$y=4$$ for $$0 \le x \le 2$$. Squar

Medium

Water Tank Filling with Graduated Inflow

A water tank is filled daily by adding a certain amount of water that increases by a fixed amount ea

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.