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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 (26)
  • Unit 3: Differentiation: Composite, Implicit, and Inverse Functions (28)
  • Unit 4: Contextual Applications of Differentiation (30)
  • Unit 5: Analytical Applications of Differentiation (31)
  • Unit 6: Integration and Accumulation of Change (39)
  • Unit 7: Differential Equations (27)
  • Unit 8: Applications of Integration (35)
Unit 1: Limits and Continuity

Analyzing a Velocity Function with Nested Discontinuities

A particle’s velocity along a line is given by $$v(t)= \frac{(t-1)(t+3)}{(t-1)*\ln(t+2)}$$ for $$t>0

Hard

Analyzing Process Data for Continuity

A manufacturing process produces items whose lengths (in mm) are recorded as a function f(x) of time

Medium

Application of the Intermediate Value Theorem

Let the function $$f(x)= x^3 - 4*x - 1$$ be continuous on the interval $$[0, 3]$$. Answer the follow

Easy

Applying the Squeeze Theorem with Trigonometric Function

Consider the function $$ f(x)= x^2 \sin(1/x) $$ for $$x\ne0$$, with $$f(0)=0$$. Use the Squeeze Theo

Medium

Arithmetic Sequence in Temperature Data and Continuity Correction

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

Easy

Continuity of a Composite Function

Let $$g(x) = \sqrt{x+3}$$ and $$h(x) = x^2 - 4$$. Define the composite function $$f(x) = g(h(x))$$.

Medium

Continuity of Composite Functions

Let $$f(x)=x+2$$ for all x, and let $$g(x)=\begin{cases} \sqrt{x}, & x \geq 0 \\ \text{undefined},

Easy

Ensuring Continuity for a Piecewise-Defined Function

Consider the piecewise function $$p(x)= \begin{cases} ax + 3 & \text{if } x < 2, \\ x^2 + b & \text{

Easy

Evaluating a Compound Limit Involving Rational and Trigonometric Functions

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

Medium

Exponential Limit Parameter Determination

Consider the function $$f(x)=\frac{e^{3*x} - e^{k*x}}{x}$$ for $$x \neq 0$$, and define $$f(0)=L$$,

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

Factorization and Removable Discontinuity

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

Easy

Graph Analysis: Identify Limits and Discontinuities

A graph of a function f(x) is provided in the stimulus. The graph shows a removable discontinuity at

Hard

Graph Reading: Left and Right Limits

A graph of a function f is provided below which shows a discontinuity at x = 2. Use the graph to det

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

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

Implicit Differentiation and Tangent Slopes

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

Medium

Intermediate Value Theorem in Particle Motion

Consider a particle with position function $$s(t)= t^3 - 7*t+3$$. According to the Intermediate Valu

Easy

Limit and Integration in Non-Polynomial Particle Motion

A particle moves along a line with velocity defined by $$v(t)= \frac{e^{2*t}-e^{4}}{t-2}$$ for \(t \

Extreme

Limit Evaluation using Conjugate Multiplication

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

Medium

Limits and the Squeeze Theorem Application

Consider two scenarios: (1) A function f(x) satisfying $$ -|x| \le f(x) \le |x| $$ for all x near 0,

Easy

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 Involving Radical Functions

Evaluate the limit $$\lim_{x \to 4} \frac{\sqrt{x} - 2}{x - 4}$$.

Medium

Limits Involving Radicals and Algebra

Consider the function $$f(x)= \sqrt{x^2 + x} - x$$. Answer the following parts.

Extreme

Limits Near Vertical Asymptotes

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

Easy

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

Rational Function Limits and Removable Discontinuities

Consider the function $$f(x)=\frac{(x+3)(x-2)}{(x+3)(x+5)}$$. Answer the following:

Easy

Real-World Application: Temperature Sensor Calibration

A temperature sensor in a lab records temperatures (in °C) according to the function $$f(t)= \frac{t

Medium

Removable Discontinuity and Limit Evaluation

Consider the function $$f(x) = \frac{(x + 3) * (x - 2)}{x + 3}$$ for $$x \neq -3$$. Answer the follo

Easy

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 with Trigonometric Function

Consider the function \(h(x)=x^2\cos(1/x)\) for \(x\neq0\) with \(h(0)=0\). Answer the following:

Medium

Table Analysis for Estimating a Limit

The table below shows values of the function $$g(x)$$ for x near 4. Use this data to answer the foll

Easy

Trigonometric Limit Evaluation

Examine the function $$ f(x)= \frac{\sin(3*x)}{x} $$ for $$x\ne0$$.

Easy
Unit 2: Differentiation: Definition and Fundamental Properties

Analyzing Differentiability of an Absolute Value Function

Consider the function $$f(x)= |x-2|$$.

Medium

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

Car's Position and Velocity

A car’s position is modeled by \(s(t)=t^3 - 6*t^2 + 9*t\), where \(s\) is in meters and \(t\) is in

Medium

Concavity and the Second Derivative

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

Medium

Derivative Applications in Population Growth

A population of cells is modeled by $$P(t)=100*e^{0.2*t}$$, where $$t$$ is in hours. Answer the foll

Medium

Derivative of the Square Root Function via Limit Definition

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

Medium

Deriving the Derivative from First Principles for a Reciprocal Square Root Function

Let $$f(x)=\frac{1}{\sqrt{x}}$$ for $$x > 0$$. Using the definition of the derivative, show that $$f

Extreme

Differentiability of an Absolute Value Function

Consider the function $$f(x)=|x-3|$$, representing the error margin (in centimeters) in a calibratio

Medium

Economic Marginal Revenue

A company's revenue function is given by \(R(x)=x*(50-0.5*x)\) dollars, where \(x\) represents the n

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

Finding Derivatives of Composite Functions

Let $$f(x)= (3*x+1)^4$$.

Medium

Implicit Differentiation of a Circle

Consider the equation $$x^2 + y^2 = 25$$ representing a circle with radius 5. Answer the following q

Easy

Instantaneous Rate of Change of Temperature

The temperature in a room is modeled by $$T(t)=20+5*\sin\left(\frac{\pi*t}{12}\right)$$, where $$t$$

Medium

Instantaneous Rate of Temperature Change in a Coffee Cup

The temperature of a cup of coffee is recorded at several time intervals as shown in the table below

Easy

Inverse Function Analysis: Cosine and Linear Combination

Consider the function $$f(x)=\cos(x)+x$$ defined on the interval $$[0,\frac{\pi}{2}]$$.

Medium

Inverse Function Analysis: Sum with Reciprocal

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

Hard

Kinematics and Position Function Analysis

A particle’s position is modeled by $$s(t)=4*t^3-12*t^2+5*t+2$$, where $$s(t)$$ is in meters and $$t

Medium

Linking Derivative to Kinematics: the Position Function

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

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

Position Function from a Logarithmic Model

A particle’s position in meters is modeled by $$s(t)= \ln(t+1)$$ for $$t \geq 0$$ seconds.

Easy

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

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: Balloon Surface Area Change

A spherical balloon has volume $$V=\frac{4}{3}\pi r^3$$ and surface area $$S=4\pi r^2$$. If the volu

Hard

Riemann Sums and Derivative Estimation

A car’s position $$s(t)$$ in meters is recorded in the table below at various times $$t$$ in seconds

Medium

Tangent Lines and Local Linearization

Consider the function $$f(x)=\sqrt{x}$$.

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

Advanced Composite Function Differentiation in Biological Growth

A biologist models bacterial growth by the function $$P(t)= e^{\sqrt{t+1}}$$, where $$t$$ is time in

Hard

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

Analyzing a Function and Its Inverse

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

Medium

Chain Rule in Angular Motion

An object rotates such that its angular position is given by $$\theta(t)= \arctan(3*t^2)$$, where $$

Medium

Chain Rule in Temperature Variation

A metal rod's temperature along its length is given by the function $$T(x)= \cos((4*x+2)^2)$$, where

Medium

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 Logarithmic and Radical Functions

Let $$R(x)=\sqrt{\ln(1+x^2)}$$.

Hard

Composite Differentiation of an Inverse Trigonometric Function

Let $$H(x)= \arctan(\sqrt{x+3})$$.

Hard

Composite Function and Tangent Line

Consider the function $$f(x)=\sqrt{3*x^2+2*x+1}$$. (Note: All derivatives are to be computed without

Easy

Composite Function Chain Reaction

A chemist models the concentration of a reacting solution at time $$t$$ (in seconds) with the compos

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: Engineering Stress-Strain Model

In an engineering context, the stress σ as a function of strain ε is given by $$\sigma(\epsilon) = \

Hard

Composite, Implicit, and Inverse Combined Challenge

Consider a dynamic system defined by the equation $$\sin(y)+\sqrt{x+y}=x$$, which implicitly defines

Extreme

Differentiation of Nested Exponential Functions

Let $$F(x)=e^{\sin(x^2)}$$.

Medium

Implicit Curve Analysis: Horizontal Tangents

Consider the curve defined implicitly by $$x^2+ e^(y)= 5$$. Answer the following:

Medium

Implicit Differentiation in a Circle

Consider the circle $$x^2 + y^2 = 25$$. Answer the following parts.

Easy

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 a Trigonometric Composite Function

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

Easy

Implicit Differentiation on a Circle

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

Easy

Implicit Differentiation with Chain and Product Rules

Consider the curve defined implicitly by $$e^{xy} + x^2y = 10$$. Assume that the point $$(1,2)$$ lie

Hard

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

Inverse Function Derivative and Recovery

Let $$f(x)=x^3+x$$, which is one-to-one on a suitable interval. Answer the following parts.

Medium

Inverse Function Differentiation

Let $$f(x)=x^3+x+1$$, a one-to-one function, and let $$g$$ be the inverse of $$f$$. Use inverse func

Easy

Inverse Function Differentiation in an Exponential Context

Let $$f(x)= e^(3*x) - 2$$ and let $$g(x)$$ be the inverse function of f. Answer the following:

Medium

Inverse Trigonometric Function Differentiation

Consider the function $$y=\arctan(2*x)$$. Answer the following:

Medium

Manufacturing Optimization via Implicit Differentiation

A manufacturing cost relationship is given implicitly by $$x^2*y + x*y^2 = 1000$$, where $$x$$ repre

Extreme

Multilayer Composite Function Differentiation

Let $$y=\cos(\sqrt{5*x+3})$$. Answer the following:

Medium

Population Dynamics via Composite Functions

A biological population is modeled by $$P(t)= \ln\left(20*e^(0.1*t^2)+ 5\right)$$, where t is measur

Medium
Unit 4: Contextual Applications of Differentiation

Airplane Altitude Change

An airplane's altitude (in meters) as a function of time is modeled by $$A(t)= 500*t - 4.9*t^2 + 100

Medium

Analysis of a Piecewise Function with Discontinuities

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

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

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

Cooling Coffee: Exponential Decay Model

A cup of coffee cools according to $$T(t) = 70 + 50e^{-0.1t}$$, where $$T(t)$$ (in °F) is the temper

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

Error Estimation in Pendulum Period

The period of a simple pendulum is given by $$T=2\pi\sqrt{\frac{L}{g}}$$, where $$L$$ is the length

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

Expanding Balloon: Related Rates with a Sphere

A spherical balloon is being inflated so that its volume increases at a constant rate of $$dV/dt = 1

Medium

FRQ 1: Vessel Cross‐Section Analysis

A designer is analyzing the cross‐section of a vessel whose shape is given by the ellipse $$\frac{x^

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

FRQ 4: Revenue and Cost Implicit Relationship

A company’s revenue (R) and cost (C) are related by the equation $$R^2 + 3*R*C + C^2 = 1000$$. Treat

Medium

FRQ 12: Airplane Climbing Dynamics

An airplane’s altitude is modeled by the equation $$y = 0.1*x^2$$, where x (in km) is the horizontal

Medium

L'Hôpital's Rule in Chemical Kinetics

In a chemical kinetics experiment, the reaction rate is modeled by the function $$f(x)=\frac{\ln(1+3

Easy

Limit Evaluation Using L'Hôpital's Rule

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

Medium

Linearization and Differential Approximations

Let $$f(x)=x^4$$. Use linearization to approximate $$f(3.98)$$ near $$x=4$$.

Easy

Linearization and Differentials

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

Easy

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

Open-top Box Optimization

A manufacturer wants to design an open‐top rectangular box with a square base that has a fixed volum

Medium

Optimization in Packaging

An open-top box with a square base is to be constructed so that its volume is fixed at $$1000\;cm^3$

Hard

Particle Motion Analysis

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

Medium

Particle Motion with Changing Acceleration

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

Medium

Projectile Motion Analysis

The height of a projectile is modeled by the function $$h(t) = -4.9t^2 + 20t + 2$$, where $$t$$ is i

Easy

Projectile Motion with Velocity Components

A projectile is launched from the ground with a constant horizontal velocity of 15 m/s and a vertica

Medium

Related Rates in a Conical Tank

Water is being poured into a conical tank at a rate of $$\frac{dV}{dt}=10$$ cubic meters per minute.

Hard

Shadow Length Problem

A person 1.80 m tall walks away from a 3.0 m tall lamppost at a rate of 1.2 m/s. Let $$x$$ be the di

Medium

Temperature Change Analysis

The temperature of a chemical solution is recorded over time. Use the table below, where $$T(t)$$ (i

Medium

Vehicle Deceleration Analysis

A car's position function is given by $$s(t)= 3*t^3 - 12*t^2 + 5*t + 7$$, where $$s(t)$$ is measured

Medium

Vehicle Position and Acceleration

A vehicle's position along a straight road is modeled by $$s(t)=4\sqrt{t+1}$$ (in kilometers), where

Easy
Unit 5: Analytical Applications of Differentiation

Analyzing a Supply and Demand Model Using Derivatives

A product's price as a function of the number of units produced is given by $$P(q)= 50 - 3*q + 0.5*q

Hard

Application of Rolle's Theorem for a Quadratic Function

Let $$f(x)= x^2 - 4$$ be defined on the interval $$[-2,2]$$. In this problem, you will verify the co

Easy

Application of the Extreme Value Theorem

Consider the function $$f(x) = \sqrt{x} + (8-x)$$ defined on the interval $$[0, 8]$$. Answer the fol

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

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

Composite Function with Piecewise Exponential and Logarithmic Parts

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

Medium

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

Derivative and Concavity of f(x)= e^(x) - ln(x)

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

Hard

Designing an Enclosure along a River

A farmer wants to build a rectangular enclosure adjacent to a river, using the river as one side of

Easy

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

Extrema in a Cost Function

A company's cost function is given by $$C(x) = x^3 - 6*x^2 + 12*x + 7$$, where $$x$$ represents the

Medium

FRQ 8: Mean Value Theorem and Non-Differentiability

Examine the function $$f(x)=|x|$$ on the interval [ -1, 1 ].

Easy

FRQ 10: First Derivative Test for a Cubic Profit Function

A company’s profit function is given by $$P(x)= x^3 - 9*x^2 + 24*x + 1$$, where $$x$$ represents the

Medium

FRQ 12: Optimization in Manufacturing: Minimizing Cost

A company’s cost function is given by $$C(x)= 0.5*x^2 - 10*x + 125$$ (in dollars), where $$x$$ repre

Medium

FRQ 20: Profit Analysis Combining MVT and Optimization

A company’s profit function is given by $$P(x)= -2*x^3 + 18*x^2 - 48*x + 40$$, where $$x$$ (in thous

Hard

Increasing/Decreasing Behavior in a Financial Model

A financial analyst models the performance of an investment with the function $$f(x)= \ln(x) - \frac

Medium

Instantaneous Velocity Analysis via the Mean Value Theorem

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

Medium

Inverse Analysis of a Piecewise Function

Consider the piecewise function $$f(x)=\begin{cases}2*x+1 & x\le 0,\\ 3*x+1 & x>0\end{cases}$$. Ans

Easy

Jump Discontinuity in a Piecewise Linear Function

Consider the piecewise function $$ f(x) = \begin{cases} 2x + 1, & x < 3, \\ 2x - 4, & x \ge 3. \end

Easy

Mean Value Theorem for a Cubic Function

Consider the function $$f(x)= x^3 - 2*x^2 + x$$ on the closed interval $$[0,2]$$. In this problem, y

Medium

Minimizing Average Cost in Production

A company’s cost function is given by $$C(x)= 0.5*x^3 - 6*x^2 + 20*x + 100$$, where $$x$$ represents

Hard

Monotonicity and Inverse Function Analysis

Consider the function $$f(x)= x + e^{-x}$$ defined for all real numbers. Investigate its monotonicit

Easy

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

Optimization in Production with Exponential Cost Function

A manufacturer’s cost function is modeled by $$C(x)= 200 + 50*x + 100*e^{-0.1*x}$$ where $$x$$ repre

Hard

Optimizing an Open-Top Box from a Metal Sheet

A rectangular sheet of metal with dimensions 24 cm by 18 cm is used to create an open-top box by cut

Hard

Pharmacokinetics: Drug Concentration Decay

A drug in the bloodstream decays according to $$D(t)= D_0*e^{-k*t}$$, where $$t$$ is in hours. Answe

Hard

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

Rational Function Optimization

Consider the rational function $$f(x)= \frac{x^2 + 1}{x - 1}$$ defined on the interval $$[2,6]$$. An

Hard

Related Rates in an Evaporating Reservoir

A reservoir’s volume decreases due to evaporation according to $$V(t)= V_0*e^{-a*t}$$, where $$t$$ i

Extreme

Reservoir Sediment Accumulation

A reservoir experiences sediment deposition from rivers and sediment removal via dredging. The sedim

Medium

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

Accumulated Chemical Concentration

A scientist observes that the rate of change of chemical concentration in a solution is given by $$r

Easy

Accumulated Water Volume in a Tank

A water tank is being filled at a rate given by $$R(t) = 4*t$$ (in cubic meters per minute) for $$0

Easy

Accumulation and Inflection Points

Suppose a function's rate of change is given by $$f'(x)=3*x^2-12*x+9.$$ Answer the following parts:

Medium

Analyzing Work Done by a Variable Force

An object is acted on by a force given by $$F(x)= 3*x^2 - x + 2$$ (in newtons), where $$x$$ is the d

Medium

Area Between Curves

Consider the curves defined by $$f(x)=x^2$$ and $$g(x)=2*x$$. The region enclosed by these curves is

Medium

Area Under a Parabola

Consider the quadratic function $$f(x)=x^2 - 4*x + 3$$. Analyze the function on the interval $$[1,4]

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

Bacterial Growth Modeling with Antibiotic Administration

A bacterial culture is subject to both growth and treatment simultaneously. The bacterial growth rat

Hard

Chemical Accumulation in a Reactor

A chemical reactor has a net accumulation rate given by $$R(t)=5*\cos(t) + 2$$ (in kg/hour), where $

Hard

Comparing Riemann Sum and the Fundamental Theorem

Let $$f(x)=3*x^2$$ on the interval $$[1,4]$$.

Hard

Convergence of Riemann Sum Estimations

Consider the function $$f(x)=\sqrt{x}$$ on the interval $$[1,4]$$. Answer the following questions re

Hard

Cooling of a Liquid Mixture

In a tank, the cooling rate is given by $$C(t)=20e^{-0.3t}$$ J/min while an external heater adds a c

Easy

Economic Accumulation of Revenue

The marginal revenue (MR) for a company is given by $$MR(x)=50*e^{-0.1*x}$$ (in dollars per item), w

Medium

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 Definite Integral Using U-Substitution

Compute the integral $$\int_{0}^{3} (2*t+1)^5\,dt$$ using u-substitution.

Medium

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

FRQ5: Inverse Analysis of a Non‐Elementary Integral Function

Consider the function $$ P(x)=\int_{0}^{x} e^{t^2}\,dt $$ for x ≥ 0. Answer the following parts.

Extreme

FRQ6: Inverse Analysis of a Displacement Function

Let $$ S(t)=\int_{0}^{t} (6-2*u)\,du $$ for t in [0, 3], representing displacement in meters. Answer

Easy

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

FRQ16: Inverse Analysis of an Integral Function via U-Substitution

Let $$ U(x)=\int_{0}^{x} 2*(t-3)^2\,dt $$ for x ≥ 3. Answer the following parts.

Hard

FRQ20: Inverse Analysis of a Function with a Piecewise Continuous Integrand

Consider the function $$ I(x)= \begin{cases} \int_{0}^{x}\cos(t)\,dt, & 0 \le x \le \pi/2 \\ \int_{0

Hard

Function Transformations and Their Integrals

Let $$f(x)= 2*x + 3$$ and consider the transformed function defined as $$g(x)= f(2*x - 1)$$. Analyze

Medium

Medication Concentration and Absorption Rate

A patient's blood concentration of a drug (in mg/L) is monitored over time before reaching its peak.

Medium

Medication Infusion in Bloodstream

A patient receives medication through an IV at a rate $$I(t)= 5\sqrt{t+1}$$ mg/min, while the body m

Medium

Motion Analysis from Velocity Data

A particle moves along a straight line with the following velocity data (in m/s) recorded at specifi

Medium

Net Change in Population Growth

A town's population grows at a rate given by $$P'(t)=0.1*t+50$$ (in individuals per year) where $$t$

Easy

Net Change in Salaries: An Accumulation Function

A company models its annual bonus savings with the rate function $$B'(t)= 500*(1+\sin(t))$$ dollars

Medium

Particle Motion with Variable Acceleration

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

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

Related Rates: Expanding Balloon

A spherical balloon is being inflated such that its volume increases at a constant rate of $$\frac{d

Medium

Riemann Sum Approximation of f(x) = 4 - x^2

Consider the function $$f(x)=4-x^2$$ on the interval $$[0,2]$$. Use Riemann sums to approximate the

Easy

Seismic Data Analysis: Ground Acceleration

A seismograph records ground acceleration (in m/s²) during an earthquake. Use the data in the table

Hard

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

Tabular Riemann Sums for Electricity Consumption

A household's daily electricity consumption (in kWh) over 5 consecutive days is recorded in the tabl

Medium

Trigonometric Integral with U-Substitution

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

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

U-Substitution in a Trigonometric Integral

Evaluate the integral $$\int \sin(2*x) * \cos(2*x)\,dx$$ using u-substitution.

Easy

Water Flow in a Tank

Water flows into a tank at a rate given by $$R(t)=3*t+2$$ (in liters per minute) for $$0 \le t \le 6

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

Bernoulli Differential Equation via Substitution

Consider the differential equation $$\frac{dy}{dx}=y+x*y^2$$. Recognize that this is a Bernoulli equ

Hard

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

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

Cooling with a Time-Dependent Coefficient

A substance cools according to $$\frac{dT}{dt} = -k(t)(T-25)$$ where the cooling coefficient is give

Extreme

Differential Equation in Business Profit

A company's profit $$P(t)$$ changes over time according to $$\frac{dP}{dt} = 100\,e^{-0.5t} - 3P$$.

Medium

Falling Object with Air Resistance

A falling object experiences air resistance proportional to the square of its velocity. Its velocity

Hard

Implicit Solution for $$\frac{dy}{dx}=\frac{x+2}{y+1}$$

Solve the differential equation $$\frac{dy}{dx} = \frac{x+2}{y+1}$$ with the initial condition $$y(0

Medium

Investment Growth with Continuous Contributions

An investment account grows continuously with an annual interest rate of 5% while continuous deposit

Medium

Investment Growth with Continuous Deposits

An investment account accrues interest continuously at an annual rate of 0.05 and receives continuou

Easy

Linear Differential Equation using Integrating Factor

Solve the linear differential equation $$\frac{dy}{dx} + 2y = x$$ with the initial condition $$y(0)=

Medium

Logistic Growth Model

A population is modeled by the logistic differential equation $$\frac{dP}{dt}=0.5*P\left(1-\frac{P}{

Medium

Mixing Problem in a Salt Solution Tank

A 100-liter tank initially contains a solution with 10 kg of salt. Brine with a salt concentration o

Hard

Mixing Problem in a Tank

A tank initially contains 200 L of water with 10 kg of dissolved salt. Brine with a salt concentrati

Medium

Mixing Problem with Evaporation and Drainage

A tank initially contains 200 L of water with 20 kg of pollutant. Water enters the tank at 2 L/min w

Extreme

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

Population Dynamics with Harvesting

A wildlife population (in thousands) is monitored over time and is subject to harvesting. The popula

Medium

Radioactive Decay

A radioactive substance decays according to $$\frac{dy}{dt} = -0.05\,y$$ with an initial mass of $$y

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

Sand Pile Dynamics

Sand is added to a pile at a constant rate of $$15$$ kg/min while some sand is simultaneously lost d

Medium

Seasonal Temperature Variation

The temperature $$T(t)$$ in a region is modeled by the differential equation $$\frac{dT}{dt} = -0.2\

Extreme

Separable Differential Equation involving $$y^{1/3}$$

Consider the differential equation $$\frac{dy}{dx} = y^{1/3}$$ with the initial condition $$y(8)=27$

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 Exploration

Consider the differential equation $$\frac{dy}{dx} = \sin(x)$$. The provided slope field (see stimul

Easy

Vehicle Deceleration

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

Easy
Unit 8: Applications of Integration

Analysis of a Rational Function's Average Value

Consider the rational function $$f(x)=\frac{2*x}{x^2+1}$$ defined on the interval $$[-1,1]$$. Analyz

Medium

Area Between a Parabola and a Line

Consider the functions $$f(x)=5*x-x^2$$ and $$g(x)=x$$. A set of experimental data provided the foll

Medium

Area Between a Parabolic Curve and a Line

Consider the functions $$f(x)=5*x-x^2$$ and $$g(x)=x$$ on the interval $$[0,4]$$. The table below sh

Medium

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: 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

Area Between Parabolic Curves

Consider the curves $$f(x)=x^2$$ and $$g(x)=4*x-x^2$$. Determine the area of the region bounded by t

Easy

Average of a Logarithmic Function

Let $$f(x)=\ln(x+2)$$ represent a measured quantity over the interval $$[0,6]$$.

Medium

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 Speed from Variable Acceleration

A car accelerates along a straight road with acceleration given by $$a(t)=2*t-1$$ (in m/s²) for $$t\

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 Value of a Polynomial Function

Consider the function $$f(x)= 3*x^2 - 2*x + 1$$ defined on the interval $$[0,4]$$. Use the concept o

Easy

Cell Phone Battery Consumption

A cell phone’s battery life degrades over time such that the effective battery life each month forms

Medium

Center of Mass of a Lamina with Variable Density

A thin lamina occupies the interval $$[0,4]$$ along the x-axis and has a variable density $$\delta(x

Extreme

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

Designing an Open-Top Box

An open-top box with a square base is to be constructed with a fixed volume of $$5000\,cm^3$$. Let t

Hard

Determining Field Area from Intersection of Curves

A farmer's field is bounded by the curves $$y=0.5*x^2$$ and $$y=4*x$$. Find the area of the field wh

Medium

Discontinuities in a Piecewise Function

Consider the function $$f(x)=\begin{cases} \frac{\sin(x)}{x} & \text{if } x \neq 0 \\ 2 & \text{if }

Easy

Discounted Cash Flow

A company projects that its annual cash flow will grow according to a geometric sequence. The initia

Extreme

Economics: Consumer Surplus Calculation

Given the demand function $$d(p)=100-2p$$ and the supply function $$s(p)=20+3p$$, determine the cons

Medium

Hollow Rotated Solid

Consider the region in the first quadrant bounded by the curves $$y=x$$ and $$y=x^2$$. This region i

Medium

Ice Rink Design: Volume and Area

An ice rink is designed with a cross-sectional profile given by $$y=4-x^2$$ (with y=0 as the base).

Extreme

Implicit Differentiation in an Electrical Circuit

In an electrical circuit, the voltage $$V$$ and current $$I$$ are related by the equation $$V^2 + (3

Hard

Inverse for a Quadratic Function

Consider the function $$f(x)=x^2+4$$ defined for $$x \ge 0$$. Analyze its inverse function.

Easy

Optimization of Average Production Rate

A manufacturing process has a production rate modeled by the function $$P(t)=50e^{-0.1*t}+20$$ (unit

Hard

Particle Motion on a Parametric Path

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

Hard

Pipeline Installation Cost Analysis

The cost to install a pipeline along a route is given by $$C(x)=100+5*\sin(x)$$ (in dollars per mete

Medium

Population Growth and Average Rate

A town's population is modeled by the function $$P(t)=1000*e^{0.03*t}$$, where $$t$$ is measured in

Medium

Rebounding Ball

A ball is dropped from a height of $$16$$ meters. Each time the ball bounces, its maximum height is

Medium

Resource Consumption in an Ecosystem

The rate of consumption of a resource in an ecosystem is given by $$C(t)=50*\ln(1+t)$$ (in units per

Easy

Solid of Revolution: Water Tank

A water tank is formed by rotating the region bounded by the curve $$y=\sqrt{x}$$, the x-axis, and t

Medium

Volume Calculation via Cross-Sectional Areas

A solid has cross-sectional areas perpendicular to the x-axis that are circles with radius given by

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 is formed over the region under the line $$f(x)=4-x$$ from $$x=0$$ to $$x=4$$ in the x-y pla

Medium

Work Calculation from an Exponential Force Function

An object is acted upon by a force modeled by $$F(x)=5*e^{-0.2*x}$$ (in newtons) along a displacemen

Medium

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Need to review before working on AP Calculus AB FRQs?

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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.