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AP Calculus BC Free Response Questions

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  • View all (250)
  • Unit 1: Limits and Continuity (26)
  • Unit 2: Differentiation: Definition and Fundamental Properties (38)
  • Unit 3: Differentiation: Composite, Implicit, and Inverse Functions (20)
  • Unit 4: Contextual Applications of Differentiation (37)
  • Unit 5: Analytical Applications of Differentiation (24)
  • Unit 6: Integration and Accumulation of Change (32)
  • Unit 7: Differential Equations (22)
  • Unit 8: Applications of Integration (17)
  • Unit 9: Parametric Equations, Polar Coordinates, and Vector-Valued Functions (34)
Unit 1: Limits and Continuity

Analyzing a Composite Function Involving a Limit

Let $$f(x)=\sin(x)$$ and define the function $$g(x)=\frac{f(x)}{x}$$ for $$x\neq0$$, with the conven

Medium

Analyzing a Function with a Removable Discontinuity

Consider the function $$r(x)=\frac{x^2-9}{x-3}$$ for $$x\neq3$$ and $$r(3)=2.$$ Answer the follow

Easy

Application of the Squeeze Theorem

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

Hard

Drainage Rate with a Removable Discontinuity

A drainage system is modeled by the function $$R_{out}(t)=\frac{t^2-2\,t-15}{t-5}$$ liters per minut

Easy

Estimating Limits from Tabulated Data

A function $$g(x)$$ is experimentally measured near $$x=2$$. Use the following data to estimate $$\l

Easy

Evaluating a Rational Function Limit Using Algebraic Manipulation

Consider the function $$f(x)=\frac{x^2-9}{x-3}$$. Analyze the limit as $$x \to 3$$.

Easy

Evaluating Limits Involving Radical Expressions

Consider the function $$h(x)= \frac{\sqrt{4x+1}-3}{x-2}$$.

Medium

Graphical Analysis of a Continuous Polynomial Function

Consider the function $$f(x)=2*x^3-5*x^2+x-7$$ and its graph depicted below. The graph provided accu

Easy

Graphical Analysis of Water Tank Volume

The water volume in a tank over time is recorded and displayed in the graph provided. Due to a senso

Medium

Implicitly Defined Function and Differentiation

Consider the curve defined implicitly by the equation $$x*y + \sin(x) + y^2 = 10$$. Answer the follo

Medium

Intermediate Value Theorem Application with a Cubic Function

A function f(x) is continuous on the interval [-2, 2] and its values at certain points are given in

Medium

Intermediate Value Theorem in a Continuous Function

Consider the continuous function $$p(x)=x^3-3*x+1$$ on the interval $$[-2,2]$$. Answer the followi

Medium

Limits in Composite Functions Involving Absolute Values

Define the function $$f(x)=\frac{x^2-1}{x-1}$$ for x \neq 1 and let $$g(x)=|x-1|.$$ Consider the com

Medium

Limits Involving Absolute Value Functions

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

Easy

Limits Involving Radicals

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

Hard

Limits Involving Trigonometric Functions and the Squeeze Theorem

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

Medium

Manufacturing Process Tolerances

A manufacturing company produces components whose dimensional errors are found to decrease as each c

Medium

One-Sided Limits and Jump Discontinuity Analysis

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

Medium

One-Sided Limits for a Piecewise Inflow

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

Easy

Piecewise Inflow Function and Continuity Check

A water tank's inflow is measured by a piecewise function due to a change in sensor calibration at \

Easy

Pollution Level Analysis and Removable Discontinuity

A function $$f(x)$$ represents the concentration of a pollutant (in mg/L) in a river as a function o

Easy

Rational Function Analysis of a Drainage Rate

A drain’s outflow rate is given by $$R_{out}(t)=\frac{3\,t^2-12\,t}{t-4}$$ for \(t\neq4\). Answer th

Medium

Related Rates: Changing Shadow Length

A streetlight is mounted at the top of a 12 m tall pole. A person 1.8 m tall walks away from the pol

Hard

Removable Discontinuity in a Rational Function

Consider the function given by $$f(x)= \frac{(x+3)*(x-1)}{(x-1)}$$ for $$x \neq 1$$. Answer the foll

Easy

Squeeze Theorem with a Log Function

Let $$f(x)= x\,\ln\Bigl(1+\frac{1}{x}\Bigr)$$ for $$x > 0$$. Use the Squeeze Theorem to determine $$

Easy

Squeeze Theorem with an Oscillating Function

Let $$f(x)=x * \cos(\frac{1}{x})$$ for $$x \neq 0$$, and define $$f(0)=0$$. Answer the following:

Medium
Unit 2: Differentiation: Definition and Fundamental Properties

Analysis of a Piecewise Function's Differentiability

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

Easy

Analysis of Derivatives: Tangents and Normals

Consider the curve defined by $$y = x^3 - 6*x^2 + 9*x + 2.$$ (a) Compute the derivative $$y'$$ an

Easy

Analyzing a Function's Rate of Change from Graphical Data

A function has been experimentally measured and its values are represented by the following graph. U

Medium

Analyzing a Removable Discontinuity in a Rational Function

Consider the function defined by $$f(x)= \begin{cases} \frac{x^2-1}{x-1} & x \neq 1 \\ 3 & x = 1 \e

Medium

Chemical Reaction Rate Control

During a chemical reaction in a reactor, reactants enter at a rate of $$R_{in}(t)=\frac{10*t}{t+2}$$

Extreme

Derivative of a Composite Function Using the Limit Definition

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

Hard

Derivative via the Limit Definition: A Rational Function

Consider the function $$f(x)=\frac{1}{x+2}$$. Use the limit definition of the derivative to find $$f

Hard

Derivatives of a Rational Function

Consider the function $$g(x)= \frac{2*x^3 - 1}{x^2+4}$$. Use differentiation rules to answer the fol

Medium

Differentiation from First Principles

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

Medium

Differentiation in Exponential Growth Models

A population is modeled by $$P(t)=P_0e^{r*t}$$ with the initial population $$P_0=500$$ and growth ra

Easy

Differentiation in Polar Coordinates

Consider the polar curve defined by $$r(\theta)= 1+\cos(\theta).$$ (a) Use the formula for polar

Hard

Drug Concentration in Bloodstream: Differentiation Analysis

A drug's concentration in the bloodstream is modeled by $$C(t)= 50e^{-0.25t} + 5$$, where t is in ho

Hard

Electricity Consumption: Series and Differentiation

A household's monthly electricity consumption increases geometrically due to seasonal variations. Th

Easy

Epidemiological Rate Change Analysis

In epidemiology, the spread of a disease is sometimes modeled by a combination of logarithmic and ex

Medium

Finding the Derivative of a Logarithmic Function

Consider the function $$g(x)=\ln(3*x+1)$$. Answer the following:

Medium

Higher-Order Derivatives

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

Easy

Instantaneous Rate of Change of a Trigonometric Function

Consider the function $$h(t)=\sin(2*t) + \cos(t)$$ which models the displacement (in centimeters) of

Medium

Instantaneous Velocity from a Displacement Function

A particle moves along a straight line with its position at time $$t$$ (in seconds) given by $$s(t)

Easy

Investment Return Rates: Continuous vs. Discrete Comparison

An investment's value grows continuously according to $$V(t)= 5000e^{0.07t}$$, where t is in years.

Medium

Logarithmic Differentiation: Equating Powers

Consider the equation $$y^x = x^y$$ that relates $$x$$ and $$y$$ implicitly.

Extreme

Maclaurin Series for arctan(x) and Error Estimate

An engineer in signal processing needs the Maclaurin series for $$g(x)=\arctan(x)$$ and an understan

Hard

Piecewise Function and Discontinuity Analysis

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

Medium

Polar Coordinates and Tangent Lines

Consider the polar curve $$r(\theta)=1+\cos(\theta)$$. Answer the following:

Hard

Population Growth Rate

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

Medium

Population Growth Rates

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

Medium

Population of a Colony: Sum and Derivative Analysis

A colony of cells grows such that the number of cells on the nth day is given by $$a_n= 100(1.2)^{n-

Medium

Profit Rate Analysis in Economics

A firm’s profit function is given by $$\Pi(x)=-x^2+10*x-20$$, where $$x$$ (in hundreds) represents t

Medium

Radioactive Decay and Derivative

A radioactive substance decays according to $$M(t)= 50e^{-0.03t}$$, where t is in years and M(t) is

Hard

Rates of Change in Economics

A company’s demand function for a product is given by $$D(p) = 120 - 3*p^2,$$ where \(p\) is the

Easy

Renewable Energy Storage

A battery storage system experiences charging at a rate of $$C(t)=50+10\sin(0.5*t)$$ kWh and dischar

Medium

Secant and Tangent Slope Analysis

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

Medium

Tangent and Normal Lines to a Curve

Given the function $$p(x)=\ln(x)$$ defined for $$x > 0$$, analyze its rate of change at a specific p

Medium

Temperature Change Rate

The temperature in a chemical reactor is modeled by $$T(t)=\frac{\sin(2*t)}{t}$$ for \(t>0\), where

Hard

Temperature Function Analysis

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

Medium

Traffic Flow Analysis

A highway on-ramp has vehicles entering at a rate of $$V_{in}(t)=30+2*t$$ vehicles per minute and ve

Medium

Velocity Function from a Cubic Position Function

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

Medium

Water Reservoir Depth Analysis

The depth of water (in meters) in a reservoir is modeled by $$d(t)=10+3*t-0.5*t^2$$, where $$t$$ is

Easy

Water Treatment Plant Simulator

A water treatment plant receives contaminated water at a rate of $$R_{in}(t)=50e^{-0.1*t}$$ liters p

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

Bacterial Culture: Nutrient Inflow vs Waste Outflow

In a bioreactor, the nutrient inflow rate is given by $$N(t)=\ln(2*t+1)$$ (in units/min). The waste

Hard

Composite Function with Hyperbolic Sine

A cable's displacement over time is modeled by $$s(t)= \sinh(\ln(t+1))$$, where $$t$$ is in seconds.

Medium

Composite Functions in Population Growth

Consider a population $$P(t) = f(g(t))$$ modeled by the functions $$g(t) = 2 + t^2$$ and $$f(u) = 10

Easy

Composite Population Growth Function

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

Medium

Continuity and Differentiability of a Piecewise Function

Consider the function defined by $$ f(x)= \begin{cases} x^2, & x < 1, \\ 2*x + c, & x \ge 1. \end{ca

Easy

Geometric Context: Sun Angle and Shadow Length Inverse Function

Consider the function $$f(\theta)=\tan(\theta)+\theta$$ for $$0<\theta<\frac{\pi}{2}$$, which models

Medium

Implicit Differentiation in a Circle

Consider the circle defined by $$ x^2+y^2=49 $$.

Easy

Implicit Differentiation in a Radical Equation

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

Medium

Implicit Differentiation Involving Product and Logarithm

Consider the curve defined by $$x*y + \ln(y) = x^2$$. Answer the following parts:

Medium

Implicit Differentiation with Trigonometric Equation

Consider the curve defined implicitly by $$\sin(x*y) + x^2 = y^3$$. Answer the following parts:

Hard

Inverse Analysis via Implicit Differentiation for a Transcendental Equation

Consider the equation $$e^{x*y}+x-y=0$$ defining y implicitly as a function of x near a point where

Hard

Inverse Function Analysis for Exponential Functions

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

Medium

Inverse Function Differentiation Basics

Let $$f$$ be a one-to-one differentiable function with $$f(3)=5$$ and $$f'(3)=2$$, and let $$g$$ be

Easy

Inverse Function Differentiation in Economics

A product’s demand is modeled by a one-to-one differentiable function $$Q = f(P)$$, where $$P$$ is t

Easy

Inverse Function Differentiation in Exponential-Linear Model

Let $$f(x)= x + e^{-x}$$, which is invertible with inverse function $$g(x)$$. Use the inverse functi

Hard

Inverse Functions in Economic Modeling

Let the cost function be given by $$f(x)= 4*x + \sqrt{x}$$, where x represents the number of items p

Easy

Inverse Trigonometric Differentiation

Differentiate the function $$ y= \arctan\left(\frac{2*x}{1-x}\right) $$.

Hard

Rocket Fuel Consumption Analysis

A rocket’s fuel consumption rate is modeled by the composite function $$C(t)=n(m(t))$$, where $$m(t)

Extreme

Tangent Line to an Ellipse

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

Easy

Trigonometric Composite Inverse Function Analysis

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

Easy
Unit 4: Contextual Applications of Differentiation

Air Pressure Change in a Sealed Container

The air pressure in a sealed container is modeled by $$P(t)=\frac{100}{1+e^{-0.5*(t-10)}}$$, where $

Hard

Analyzing Pollutant Concentration in a River

The concentration of a pollutant in a river is modeled by $$C(t)=50-5*t+0.5*t^2$$, where C is in mg/

Easy

Area Under a Curve: Definite Integral Setup

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

Medium

Bacterial Population Growth

The population of a bacterial culture is modeled by $$P(t)=1000e^{0.3*t}$$, where $$P(t)$$ is the nu

Medium

Circular Motion and Angular Rate

A point moves along a circle of radius 5 meters. Its angular position is given by $$\theta(t)=2*t^2-

Medium

City Population Migration

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

Medium

Concavity and Acceleration in Motion

A car’s position is modeled by $$s(t)= t^3 - 6*t^2 + 9*t+5$$ with time $$t$$ in seconds. Analyze the

Medium

Critical Points and Inflection Analysis

Consider the function $$f(x)= (x-2)^2*(x+5)$$ which models a physical quantity. Answer the following

Medium

Cubic Function with Parameter and Its Inverse

Examine the family of functions given by $$f(x)=x^3+k*x$$, where $$k$$ is a constant.

Hard

Curvature Analysis in the Design of a Bridge

A bridge's vertical profile is modeled by $$y(x)=100-0.5*x^2+0.05*x^3$$, where $$y$$ is in meters an

Extreme

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

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

Easy

Differentiation of a Product Involving Exponentials and Logarithms

Consider the function $$f(t)=e^{-t}\ln(t+2)$$, defined for t > -2. Answer the following:

Hard

Estimating Rates from Experimental Position Data

The table below represents experimental measurements of the position (in meters) of a moving particl

Medium

Exponential Relation

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

Hard

Graphical Interpretation of Slope and Instantaneous Rate

A graph (provided below) displays a linear function representing a physical quantity over time. Use

Easy

Implicit Differentiation in a Tank Filling Problem

A tank's volume and liquid depth are related by $$V=10y^3$$, where y (in meters) is the depth. Water

Easy

Implicit Differentiation on a Circle

Consider the circle defined by $$x^2+y^2=25$$, where both $$x$$ and $$y$$ are functions of time $$t$

Easy

Infrared Sensor Distance Analysis

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

Medium

Interpreting Position Graphs: From Position to Velocity

A graph of position (in meters) versus time (in seconds) is provided in the stimulus. The graph show

Medium

Linear Account Growth in Finance

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

Easy

Linearization in Engineering Load Estimation

In an engineering project, the load on a beam is modeled by $$L(x)=100*x^2+50*x+200$$, where $$L(x)$

Medium

Linearization in Finance

The value of an investment is modeled by $$V(x)=1000x^{0.5}$$ dollars, where x represents a market i

Easy

Linearization in Inverse Function Approximation

Let $$f(x)=x^5+2*x+1$$ be a one-to-one function. Although its inverse cannot be found explicitly, li

Extreme

Marginal Analysis in Economics

The cost function for producing $$x$$ items is given by $$C(x)= 0.1*x^3 - 2*x^2 + 20*x + 100$$ dolla

Medium

Marginal Cost Analysis

A company’s cost function is given by $$C(x)=100+25*x+4*x^2$$, where $$x$$ represents the number of

Easy

Modeling Cooling: Coffee Temperature with Logarithmic Decline

A cup of coffee cools according to the model $$T(t)= 90 - 20\ln(1+t)$$, where $$T$$ is in degrees Ce

Easy

Motion Model Inversion

Suppose that the position of a particle moving along a line is given by $$f(t)=t^3+t$$. Analyze the

Medium

Motion on a Straight Line with a Logarithmic Position Function

A particle moves along a straight line with its position given by $$s(t)=\ln(t+2)+t^2$$ (in meters),

Medium

Optimization of Material Cost for a Pen

A rectangular pen is to be built against a wall, requiring fencing on only three sides. The area of

Hard

Popcorn Sales Growth Analysis

A movie theater observes that the number of popcorn servings sold increases by 15% each week. Let $$

Medium

Population Growth and Change: A Nonlinear Model

The population of a bacterial culture is modeled by $$P(t)=\frac{500e^{0.3*t}}{1+e^{0.3*t}}$$, where

Extreme

Revenue Concavity Analysis

A company’s revenue from sales is modeled by the function $$R(x)= 300*x - 2*x^2$$, where \(x\) repre

Easy

Revenue Concavity Analysis

A company's revenue over time is modeled by $$R(t)=100\ln(t+1)-2t$$. Answer the following:

Medium

Series Analysis in Profit Optimization

A company's profit function near a break-even point is approximated by $$\pi(x)= 1000 + \sum_{n=1}^{

Hard

Series Approximation of a Temperature Function

The temperature in a chemical reaction is modeled by $$T(t)= 100 + \sum_{n=1}^{\infty} \frac{(-1)^n

Easy

Series Integration in Fluid Flow Modeling

The flow rate of a fluid is modeled by $$Q(t)= \sum_{n=0}^{\infty} (-1)^n \frac{(0.1t)^{n+1}}{n+1}$$

Hard

Temperature Change in Coffee Cooling

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

Medium
Unit 5: Analytical Applications of Differentiation

Analysis of a Logarithmic Function

Consider the function $$q(x)=\ln(x)-\frac{1}{2}*x$$ defined on the interval [1,8]. Answer the follow

Medium

Analysis of an Absolute Value Function

Consider the function $$f(x)=|x^2-4|$$. Answer the following parts:

Hard

Analysis of Relative Extrema and Increasing/Decreasing Intervals

A particle moves along a line with position given by $$s(x)=x^3-6*x^2+9*x+4$$, where $$x$$ represent

Easy

Analyzing a Function with Implicit Logarithmic Differentiation

Consider the implicit equation $$x\,\ln(y) + y\,e^x = 10$$. Analyze this function by differentiating

Extreme

Area Between a Curve and Its Tangent

Consider the curve $$f(x)=x^2$$ and its tangent line at \(x=1\). Investigate the region bounded by t

Hard

Car Depreciation Analysis

A new car is purchased for $$30000$$ dollars. Its value depreciates by 15% each year. Analyze the de

Medium

Epidemic Infection Model

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

Hard

Evaluating an Improper Integral using Series Expansion

The function $$I(x)=\sum_{n=0}^\infty (-1)^n * \frac{(2*x)^{n}}{n!}$$ converges to a known function.

Extreme

Exploring Inverses of a Trigonometric Transformation

Consider the function $$f(x)= 2*\tan(x) + x$$ defined on the interval $$(-\pi/4, \pi/4)$$. Answer th

Extreme

Extreme Value Theorem in Temperature Variation

A metal rod’s temperature (in °C) along its length is modeled by the function $$T(x) = -2*x^3 + 12*x

Medium

Garden Design Optimization

A gardener wants to design a rectangular garden adjacent to a river, so that fencing is required for

Easy

Implicit Differentiation in Economic Context

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

Easy

Inverse Derivative Analysis of a Quartic Polynomial

Consider the function $$f(x)= x^4 - 4*x^2 + 2$$ defined for $$x \ge 0$$. Answer the following.

Medium

Inverse Function Derivative in a Constrained Domain

Let $$f(x)= \frac{x}{x+2}$$ defined for $$x > -2$$. Analyze the function and its inverse.

Hard

Investigation of a Series with Factorials and Its Operational Calculus

Consider the series $$F(x)=\sum_{n=0}^\infty \frac{x^n}{n!}$$, which represents an exponential funct

Easy

Mean Value Theorem on a Quadratic Function

Consider the function $$f(x)=x^2-4*x+3$$ defined on the closed interval $$[1, 5]$$. Verify that the

Easy

Mean Value Theorem with Trigonometric Function

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

Easy

Population Growth Model Analysis

A population of organisms is modeled by the function $$P(t)= -2*t^2+20*t+50$$, where $$t$$ is measur

Easy

Projectile Trajectory: Parametric Analysis

A projectile is launched with its trajectory given in parametric form by $$x(t) = 10*t$$ and $$y(t)

Hard

Real-World Modeling: Radioactive Decay with Logarithmic Adjustment

A radioactive substance decays according to $$N(t)= N_0\,e^{-0.03t}$$. In an experiment, the recorde

Easy

Retirement Savings with Diminishing Deposits

Alex starts a retirement savings plan where the deposit each month forms a geometric sequence. In th

Medium

Taylor Series for $$\cos(2*x)$$

Consider the function $$f(x)=\cos(2*x)$$. Construct its 4th degree Maclaurin polynomial, determine t

Easy

Taylor Series for $$e^{-x^2}$$

Consider the function $$f(x)=e^{-x^2}$$. In this problem, you will derive its Maclaurin series up to

Hard

Trigonometric Function and its Inverse

Consider the function $$f(x)= \sin(x) + x$$ defined on the interval $$[-\pi/2, \pi/2]$$. Answer the

Medium
Unit 6: Integration and Accumulation of Change

Accumulation Function and the Fundamental Theorem of Calculus

Let $$F(x) = \int_{2}^{x} \sqrt{1 + t^3}\, dt$$. Answer the following parts regarding this accumulat

Medium

Accumulation Function from a Rate Function

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

Easy

Application of the Fundamental Theorem

Consider the function $$f(x)=x^2+2*x$$ defined on the interval $$[1,4]$$. Evaluate the definite inte

Easy

Arc Length of an Architectural Arch

An architectural arch is described by the curve $$y=4 - 0.5*(x-2)^2$$ for $$0 \le x \le 4$$. The len

Hard

Area Under a Parametric Curve

Consider the parametric curve defined by $$x(t)=t$$ and $$y(t)=t^2$$ for $$t \in [0,3]$$. The area u

Hard

Average Temperature from a Continuous Function

Along a metal rod, the temperature is modeled by $$f(t)= t^3 - 3*t^2 + 2*t$$ (in $$^\circ C$$) for

Easy

Center of Mass of a Rod with Variable Density

A thin rod of length 10 m has a linear density given by $$\rho(x)= 2 + 0.3*x$$ (in kg/m), where x is

Hard

Comparing Riemann Sums with Definite Integral in Estimating Distance

A vehicle's velocity (in m/s) is recorded at discrete times during a trip. Use these data to estimat

Medium

Definite Integral Involving an Inverse Function

Evaluate the definite integral $$\int_{1}^{4} \frac{1}{\sqrt{x}}\,dx$$ and explain the significance

Easy

Definite Integral via U-Substitution

Evaluate the definite integral $$\int_{1}^{3} (2*x-1)^6\,dx$$ using u-substitution.

Medium

Determining Antiderivatives and Initial Value Problems

Suppose that $$F(x)$$ is an antiderivative of the function $$f(x)=5*x^4 - 2*x + 3$$, and that it is

Easy

Estimating Chemical Production via Riemann Sums

In a laboratory experiment, the reaction rate of a chemical process is recorded at various times. Th

Medium

Flow of Traffic on a Bridge

Cars cross a bridge at a rate modeled by $$R(t)=300+50*\cos\left(\frac{\pi*t}{6}\right)$$ vehicles p

Hard

Integration of a Piecewise Function

A function representing a rate is defined piecewise by $$f(t)= \begin{cases} 2*t, & 0 \le t \le 3 \

Medium

Integration of a Rational Function

Consider the function $$f(x)=\frac{1}{x^2+4}$$ on the interval $$[0,2]$$. Evaluate the area under th

Hard

Integration Using U-Substitution

Evaluate the integral $$\int (3*x+2)^5\,dx$$ using u-substitution.

Medium

Midpoint Approximation Analysis

Let $$f(x)=\sqrt{x}$$ on the interval [0, 9]. Answer the following:

Easy

Modeling Bacterial Growth Through Accumulated Change

A bacteria population's growth rate is given by $$r(t)=\frac{2*t}{1+t^{2}}$$ (in thousands per hour)

Hard

Particle Motion and the Fundamental Theorem of Calculus

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

Medium

Region Bounded by a Parabola and a Line: Area and Volume

Consider the region bounded by the curves $$y=x^{2}$$ and $$y=2*x+3$$. Answer the following:

Medium

Reservoir Water Level

A reservoir experiences a net water inflow modeled by $$W(t)=40*\sin\left(\frac{\pi*t}{12}\right)-5$

Medium

Revenue Estimation Using the Trapezoidal Rule

A company recorded its daily revenue (in thousands of dollars) over four days. Use the data in the t

Medium

Sandpile Accumulation

At an industrial site, sand is continuously added to and removed from a pile. The addition rate is g

Medium

Taylor/Maclaurin Series Approximation and Error Analysis

Consider the function $$f(x)=\ln(1+x)$$. This function is infinitely differentiable at x = 0 and has

Hard

Total Cost from a Marginal Cost Function

A company’s marginal cost function is given by $$MC(x)= 4*x+7$$ (in dollars per unit), where x repre

Easy

Transportation Model: Distance and Inversion

A transportation system is modeled by $$f(t)= (t-1)^2+3$$ for $$t \ge 1$$, where \(t\) is time in ho

Medium

Trapezoidal and Riemann Sums from Tabular Data

A scientist collects data on the concentration of a chemical over time as given in the table below.

Medium

Trapezoidal Sum Approximation for $$f(x)=\sqrt{x}$$

Consider the function $$f(x)=\sqrt{x}$$ on the interval $$[0,4]$$. Use a trapezoidal sum with 4 equa

Easy

U-Substitution Integration

Evaluate the definite integral $$\int_1^5 (2*x-3)^4 dx$$ using the method of u-substitution.

Medium

Vehicle Distance Estimation from Velocity Data

A vehicle's velocity over time is recorded in the table provided. Use Riemann sums to estimate the v

Medium

Volume by Cross-Section: Squares on a Parabolic Base

A solid has a base in the xy-plane bounded by the curves $$y=x^2$$ and $$y=4$$. Cross-sections perpe

Hard

Work Done by a Variable Force

A force acting along a displacement is given by $$F(x)=5*x^2-2*x$$ (in Newtons), where x is measured

Medium
Unit 7: Differential Equations

Combined Differential Equations and Function Analysis

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

Extreme

Cooling and Mixing Combined Problem

A container holds 2 L of water initially at 80°C. Cold water at 20°C flows into the container at a r

Extreme

Cooling of a Smartphone Battery

A smartphone battery cools according to Newton’s law: $$\frac{dT}{dt} = -k*(T-T_{room})$$. Initially

Easy

Environmental Modeling Using Differential Equations

The concentration $$C(t)$$ of a pollutant in a lake is modeled by the differential equation $$\frac{

Extreme

Estimating Total Change from a Rate Table

A car's velocity (in m/s) is recorded at various times according to the table below:

Easy

Euler's Method Approximation

Let the differential equation be $$\frac{dy}{dx} = x+y$$ with the initial condition $$y(0)=1$$. Usin

Hard

FRQ 2: Separable Differential Equation with Initial Condition

Consider the differential equation $$\frac{dy}{dx} = \frac{4*x}{y}$$ with the initial condition $$y(

Medium

FRQ 11: Linear Differential Equation via Integrating Factor

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

Hard

FRQ 14: Dynamics of a Car Braking

A car braking is modeled by the differential equation $$\frac{dv}{dt} = -k*v$$, where the initial ve

Easy

FRQ 15: Cooling of a Beverage in a Fridge

A beverage cools according to Newton's Law of Cooling, described by $$\frac{dT}{dt}=-k(T-A)$$, where

Easy

Interpreting Slope Fields for a Differential Equation

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

Medium

Logistic Growth Model

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

Medium

Mixing Problem in a Tank

A tank initially contains 50 liters of pure water. A brine solution with a salt concentration of $$3

Medium

Mixing Problem in a Tank

A tank initially contains $$100$$ liters of water with $$5$$ kg of dissolved salt. Brine with a salt

Medium

Mixing Problem with Differential Equations

A tank initially holds 100 L of a salt solution containing 5 kg of salt. Brine with a salt concentra

Medium

Modeling Disease Spread with Differential Equations

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

Hard

Modeling Temperature in a Biological Specimen

A biological specimen initially at $$37^\circ C$$ is cooling in an environment where the ideal ambie

Medium

Population Dynamics with Harvesting

Consider a population model that includes constant harvesting, given by the differential equation $$

Hard

Separable Differential Equation and Slope Field Analysis

Consider the differential equation $$\frac{dy}{dx} = x*y$$ with the initial condition $$y(0)=2$$. A

Easy

Series Solution for a Second-Order Differential Equation

Consider the differential equation $$y'' - y = 0$$ with the initial conditions $$y(0)=1$$ and $$y'(0

Extreme

Slope Field and Sketching a Solution Curve

The differential equation $$\frac{dy}{dx}=x-y$$ has been represented by a slope field. Answer the fo

Medium

Temperature Change and Differential Equations

A hot liquid cools in a room at $$20^\circ C$$ according to the differential equation $$\frac{dT}{dt

Medium
Unit 8: Applications of Integration

Arc Length of a Curve

Consider the curve defined by $$y= \ln(\cos(x))$$ for $$0 \le x \le \frac{\pi}{4}$$. Determine the l

Hard

Average Temperature Over a Day

A research team studies the variation in water temperature in a lake over a 24‐hour period. The temp

Easy

Average Value and Critical Points of a Trigonometric Function

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

Hard

Average Value of a Piecewise Function

A function $$f(x)$$ is defined piecewise over the interval $$[0,6]$$ as follows: $$f(x)=\begin{case

Easy

Average Value of a Trigonometric Function

Let $$f(x)=C+\cos(2*x)$$ be defined on the interval $$[0,\pi]$$. Answer the following:

Easy

Balloon Inflation Related Rates

A spherical balloon is being inflated such that its radius $$r(t)$$ (in centimeters) increases at a

Easy

Center of Mass of a Lamina

A thin lamina occupies the region under the curve $$y=\sqrt{x}$$ on the interval $$[0,4]$$ and has a

Hard

Center of Mass of a Non-uniform Rod

A thin rod of length 10 m has a linear density given by $$\lambda(x)= 3 + 0.5*x$$ (in kg/m) for $$0

Medium

Center of Mass of a Plate

A lamina occupies the rectangular region defined by $$0 \le x \le 2$$ and $$0 \le y \le 3$$, with a

Hard

Implicit Differentiation with Exponential Terms

Consider the equation $$e^{x * y} + x^2 * y = y^3$$. Answer the following:

Hard

Inflow Rate to a Reservoir

The inflow rate of water into a reservoir is given by $$R(t)=\frac{100*t}{5+t}$$ (in cubic meters pe

Medium

Rainfall Accumulation Analysis

The rainfall rate (in cm/hour) at a location is modeled by $$r(t)=0.5+0.1*\sin(t)$$ for $$0 \le t \l

Easy

Solid of Revolution using Washer Method

The region bounded by the curves $$y = x^2$$ and $$y = 2 * x$$ is rotated about the x-axis. Answer t

Medium

Volume of a Rotated Region via Washer Method

Consider the region bounded by the curves $$y=x$$ and $$y=\sqrt{x}$$ along with the vertical line $$

Extreme

Volume of a Solid using the Washer Method

Consider the region bounded by the curves $$y= x$$ and $$y= \sqrt{x}$$ for $$0 \le x \le 1$$. This r

Medium

Volume with Equilateral Triangle Cross Sections

The region bounded by $$y=4-x^2$$ and $$y=0$$ for $$x\in[-2,2]$$ serves as the base of a solid. Cros

Hard

Work Done by a Variable Force

A variable force applied to move an object along a straight line is given by $$F(x)=3*x^2$$ (in newt

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

Arc Length and Speed from Parametric Equations

Consider the curve defined by $$x(t)=e^t$$ and $$y(t)=e^{-t}$$ for $$-1 \le t \le 1$$. Analyze the a

Medium

Arc Length of a Cycloid

Consider the cycloid defined by the parametric equations $$x(t)= t - \sin(t)$$ and $$y(t)= 1 - \cos(

Medium

Arc Length of a Parametric Curve

Consider the curve defined by $$x(t)=t^3-3*t$$ and $$y(t)=t^2+2$$ for $$t \in [0,2]$$.

Medium

Arc Length of a Quarter-Circle

Consider the circle defined parametrically by $$x(t)=\cos(t)$$ and $$y(t)=\sin(t)$$ for $$0 \le t \l

Easy

Area Between Polar Curves: Annulus with a Hole

Two polar curves are given by \(R(\theta)=3\) and \(r(\theta)=2+\cos(\theta)\) for \(0\le\theta\le2\

Hard

Area Enclosed by a Polar Curve

Consider the polar curve defined by $$r=2+2\sin(\theta)$$. This curve is a cardioid. Answer the foll

Hard

Average Position from a Vector-Valued Function

A particle moves in the plane according to the vector function $$\mathbf{r}(t)=\langle \sin(t), \cos

Easy

Continuity Analysis of a Discontinuous Parametric Curve

Consider the parametric curve defined by $$x(t)= \begin{cases} t^2, & t < 1 \\ 2*t - 1, & t \ge 1 \

Hard

Curvature and Oscillation in Vector-Valued Functions

Given the vector function $$\vec{r}(t)=\langle \ln(t+1), \cos(t) \rangle$$ for $$t \ge 0$$, answer t

Hard

Equivalence of Parametric and Polar Circle Representations

A circle is represented by the parametric equations $$x(t)=3\cos(t)$$ and $$y(t)=3\sin(t)$$ for $$0\

Easy

Finding the Slope of a Tangent to a Parametric Curve

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

Medium

Implicit Differentiation and Curves in the Plane

The curve defined by $$x^2y + xy^2 = 12$$ describes a relation between $$x$$ and $$y$$.

Medium

Inflow and Outflow in a Water Tank

A water tank has water entering at a rate given by $$I(t)=5+\sin(t)$$ (liters per minute) and water

Medium

Intersection of Two Parametric Curves

Two curves are represented parametrically as follows: Curve A is given by $$x(t)=t^2, \; y(t)=2*t+1$

Medium

Length of a Polar Spiral

For the polar spiral defined by $$r=\theta$$ for $$0 \le \theta \le 2\pi$$, answer the following:

Hard

Limit Evaluation of a Trigonometric Piecewise Function

Define the function $$f(θ)= \begin{cases} \frac{1-\cos(θ)}{θ}, & θ \neq 0 \\ 0, & θ=0 \end{cases}$$

Easy

Modeling Circular Motion with Vector-Valued Functions

An object moves along a circle of radius $$3$$ with its position given by $$\mathbf{r}(t)=\langle 3\

Easy

Numerical Integration Techniques for a Parametric Curve

A curve is defined by the parametric equations $$x(t)=\ln(t+1)$$ and $$y(t)=\sqrt{t}$$ for $$0 \le t

Medium

Optimization on a Parametric Curve

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

Hard

Parametric and Polar Conversion Challenge

Consider the parametric equations $$x(t)= \frac{1-t^2}{1+t^2}$$ and $$y(t)= \frac{2*t}{1+t^2}$$ for

Extreme

Parametric Curve: Intersection with a Line

Consider the parametric curve defined by $$ x(t)=t^3-3*t $$ and $$ y(t)=2*t^2 $$. Analyze the proper

Hard

Parametric Equations and Tangent Slopes

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

Medium

Parametric Intersection and Enclosed Area

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

Hard

Parametric Slope and Arc Length

Consider the parametric curve defined by $$x(t)= t-\ln(t)$$ and $$y(t)= t\cdot\ln(t)$$ for $$t > 1$$

Medium

Parametric Tangent Line and Curve Analysis

For the curve defined by the parametric equations $$x(t)=t^{2}$$ and $$y(t)=t^{3}-3t$$, answer the f

Easy

Particle Motion in the Plane

A particle's position is given by the parametric equations $$x(t)=3*t^2-2*t$$ and $$y(t)=4*t^2+t-1$$

Hard

Polar Boundary Conversion and Area

A region in the polar coordinate plane is defined by $$1 \le r \le 3$$ and $$0 \le \theta \le \frac{

Hard

Reparameterization between Parametric and Polar Forms

A curve is described by the parametric equations $$x(t)=a*t$$ and $$y(t)=b*t^2$$, where $$a$$ and $$

Hard

Satellite Orbit: Vector-Valued Functions

A satellite’s orbit is modeled by the vector function $$\mathbf{r}(t)=\langle \cos(t)+0.1*\cos(6*t),

Extreme

Taylor/Maclaurin Series: Approximation and Error Analysis

Let $$f(x)=\ln(1+x)$$. Without using a calculator, generate the third-degree Maclaurin polynomial fo

Hard

Vector-Valued Functions and 3D Projectile Motion

An object's position in three dimensions is given by $$\mathbf{r}(t)=\langle 3t, 4t, 10t-5t^2 \rangl

Medium

Vector-Valued Functions and Kinematics

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

Medium

Vector-Valued Functions: Position, Velocity, and Acceleration

Let $$\textbf{r}(t)= \langle e^t, \ln(t+1) \rangle$$ represent the position of a particle in the pla

Medium

Velocity and Acceleration of a Particle

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

Easy

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FAQWe thought you might have some questions...
Where can I find practice free response questions for the AP Calculus BC exam?
The free response section of each AP exam varies slightly, so you’ll definitely want to practice that before stepping into that exam room. Here are some free places to find practice FRQs :
  • Of course, make sure to run through College Board's past FRQ questions!
  • Once you’re done with those go through all the questions in the AP Calculus BCFree Response Room. You can answer the question and have it grade you against the rubric so you know exactly where to improve.
  • Reddit it also a great place to find AP free response questions that other students may have access to.
How do I practice for AP AP Calculus BC Exam FRQs?
Once you’re done reviewing your study guides, find and bookmark all the free response questions you can find. The question above has some good places to look! while you’re going through them, simulate exam conditions by setting a timer that matches the time allowed on the actual exam. Time management is going to help you answer the FRQs on the real exam concisely when you’re in that time crunch.
What are some tips for AP Calculus BC free response questions?
Before you start writing out your response, take a few minutes to outline the key points you want to make sure to touch on. This may seem like a waste of time, but it’s very helpful in making sure your response effectively addresses all the parts of the question. Once you do your practice free response questions, compare them to scoring guidelines and sample responses to identify areas for improvement. When you do the free response practice on the AP Calculus BC Free Response Room, there’s an option to let it grade your response against the rubric and tell you exactly what you need to study more.
How do I answer AP Calculus BC free-response questions?
Answering AP Calculus BC free response questions the right way is all about practice! As you go through the AP AP Calculus BC Free Response Room, treat it like a real exam and approach it this way so you stay calm during the actual exam. When you first see the question, take some time to process exactly what it’s asking. Make sure to also read through all the sub-parts in the question and re-read the main prompt, making sure to circle and underline any key information. This will help you allocate your time properly and also make sure you are hitting all the parts of the question. Before you answer each question, note down the key points you want to hit and evidence you want to use (where applicable). Once you have the skeleton of your response, writing it out will be quick, plus you won’t make any silly mistake in a rush and forget something important.