AP Calculus BC FRQ Room

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

Analyzing Limits Using Tabular Data

A function $$f(x)$$ is described by the following table of values: | x | f(x) | |------|------|

Application of the Squeeze Theorem with Trigonometric Oscillations

Consider the function $$f(x)= x^2*\sin(1/x)$$ for $$x \neq 0$$ and $$f(0)=0$$. Answer the following

Asymptotic Behavior in Rational Functions

Consider the rational function $$g(x)=\frac{2*x^3-5*x^2+1}{x^3-3*x+4}.$$ Answer the following parts

Composite Function and Continuity

Consider the piecewise function $$ g(x)=\begin{cases} x^2 & \text{if } x<2, \\ 3x-2 & \text{if } x\

Composite Function in Water Level Modeling

Suppose the water volume in a tank is given by a composite function \(V(t)=f(g(t))\) where $$g(t)=\f

Composite Function Involving Logarithm and Rational Expression

Consider the piecewise function $$ f(x)=\begin{cases} \frac{1}{x-1} & \text{if } x<2, \\ \ln(x-1) &

Continuity in Piecewise-Defined Functions

Consider the piecewise function $$f(x)=\begin{cases} x^2 + 1 & \text{if } x < 2, \\ k * x - 3 & \tex

Economic Model of Depreciating Car Value

A car purchased for $$30,000$$ dollars depreciates in value by $$15\%$$ each year. The value of the

End Behavior and Horizontal Asymptote Analysis

Consider the function $$f(x)=\frac{3*x^3-5*x+2}{2*x^3+4*x^2-1}$$. Answer the following:

Finding a Parameter in a Limit Involving Logs and Exponentials

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

Fuel Efficiency and Speed Graph Analysis

A car manufacturer records data on fuel efficiency (in mpg) as a function of speed (in mph). A graph

Graph Analysis of Discontinuities

A function $$q(x)$$ is defined piecewise as follows: $$q(x)=\begin{cases} x+2, & x<1, \\ 4, & x=1,

Graphical Analysis of Discontinuities

A graph of a function is provided that shows multiple discontinuities, including a removable discont

Graphical Analysis of Limits and Asymptotic Behavior

A graphical study titled 'Graph of Experimental Data' shows the measured concentration of a chemical

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

Horizontal and Vertical Asymptotes of a Rational Function

Let $$h(x)=\frac{2*x^2-3*x+1}{x^2-1}.$$ Answer the following:

Implicitly Defined Curve and Its Tangent Line

Consider the circle defined by the equation $$x^2+y^2=16$$. Answer the following:

Left-Hand and Right-Hand Limits for a Sign Function

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

Limits Involving Absolute Value

Let $$h(x)=\frac{|x^2-9|}{x-3}.$$ Answer the following parts.

Limits Involving Radicals

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

Limits with Infinite Discontinuities

Consider the function $$k(x)=\frac{1}{x-2}$$.

Manufacturing Cost Sequence

A company's per-unit manufacturing cost decreases by $$50$$ dollars each year due to economies of sc

Physical Applications: Temperature Continuity

A temperature sensor records temperature (in °C) over time according to the function $$T(t)=\frac{t^

Piecewise Function Continuity

Consider the function defined by $$ f(x)=\begin{cases} \frac{x^2-4}{x-2}, & x\neq2\\ k, & x=2 \en

Rational Function Analysis with Removable Discontinuities

Consider the function $$f(x)=\frac{(x+3) * (x-1)}{(x-1)}$$ for $$x \neq 1$$. This function exhibits

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

Removing a Removable Discontinuity in a Piecewise Function

Examine the function $$g(x)= \begin{cases} \frac{x^2-9}{x-3}, & x \neq 3 \\ m, & x=3 \end{cases}$$.

Saturation of Drug Concentration in Blood

A patient is given a drug with each dose containing 50 mg. However, due to metabolism, only 20% of t

Trigonometric Limits

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

Trigonometric Rate Function Analysis

A pump’s output is modified by a trigonometric factor. The outflow rate is recorded as $$R(t)=\frac{

Vertical Asymptote Analysis in a Rational Function

Consider the function $$g(x)=\frac{x+1}{x-3}$$, which is undefined at $$x=3$$. Answer the following:

Water Tank Inflow with Oscillatory Variation

A water tank is equipped with a sensor that records the inflow rate with a slight oscillatory error.

Analyzing Car Speed from a Distance-Time Table

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

Chain Rule Verification with a Power Function

Let $$f(x)= (3*x+2)^4$$.

Composite Function Differentiation and Taylor Series for $$e^{\sin(x)}$$

Consider the composite function $$f(x)=e^{\sin(x)}$$. A physicist needs to approximate this function

Compound Exponential Rate Analysis

Consider the function $$f(t)=\frac{e^{2*t}}{1+t}$$, which arises in compound growth models. Analyze

Continuous Compound Interest Analysis

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

Cost Optimization in Production: Derivative Application

A company's cost function is given by $$K(x)= 0.1x^3 - 2x^2 + 15x + 100$$, where x represents the nu

Differentiation from First Principles

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

Differentiation of a Trigonometric Function

Let $$f(x)=\sin(x)+x*\cos(x)$$. Differentiate the function using the sum and product rules.

Estimating Instantaneous Acceleration from Velocity Data

An object's velocity (in m/s) is recorded over time as shown in the table below. Use the data to ana

Heat Transfer in a Rod: Modeling and Differentiation

The temperature distribution along a rod is given by $$T(x)= 100 - 2x^2 + 0.5x^3$$, where x is in me

Implicit Differentiation: Elliptic Curve

Consider the curve defined by $$2*x^2 + 3*x*y + y^2 = 20$$.

Motion Along a Line

An object moves along a line with its position given by $$s(t)=4*t^3 - 12*t^2 + 9*t$$, where $$s$$ i

Motion Model with Logarithmic Differentiation

A particle moves along a track with its displacement given by $$s(t)=\ln(2*t+3)*e^{-t}$$, where $$t$

Particle Motion on a Straight Line: Average and Instantaneous Rates

A particle moving along a straight line has its position given by $$s(t)=t^3 - 6*t^2 + 9*t + 4$$ for

Rate of Change Analysis in a Temperature Model

A temperature model is given by $$T(t)=25+4*t-0.5*t^2$$, where $$t$$ is time in hours. Analyze the t

Secant Line Approximation in an Experimental Context

A temperature sensor records the following data over a short experiment:

Secants and Tangents in Profit Function

A firm’s profit is modeled by the quadratic function $$f(x)=-x^2+6*x-8$$, where $$x$$ (in thousands)

Second Derivative Test and Stability

Consider the function $$f(x)=x^4-8*x^2+16$$.

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

Tangent Line to a Logarithmic Function

Consider the function $$f(x)= \ln(x+1)$$.

Velocity and Acceleration Analysis

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

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

Chain Rule and Higher-Order Derivatives

Given the function $$f(x)= \ln(\sqrt{1 + e^{3*x}})$$, answer the following parts:

Chain Rule Application: Differentiating a Nested Trigonometric Function

Consider the function $$f(x) = \sin(\cos(2*x))$$. Analyze its derivative using the chain rule.

Chain Rule in a Trigonometric Light Intensity Model

A light sensor records the intensity of light according to the function $$I(x) = \cos(\sqrt{3*x + 2}

Combined Differentiation: Inverse and Composite Function

Let $$f(x)= \ln(2*x+1)$$ and let $$g$$ be the inverse function of $$f$$. Answer the following parts:

Composite Function: Polynomial Exponent

Consider the function $$ f(x)= (2*x^2+3*x-5)^3 $$. Analyze the function's derivative and behavior.

Differentiation in a Logistic Population Model

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

Differentiation of an Inverse Trigonometric Composite Function

Let $$f(x)= \arctan(e^{2*x})$$. Answer the following parts:

Implicit Differentiation in Circular Motion

Consider the circle described by $$x^2+y^2=49$$, representing a particle's path. Answer the followin

Implicit Differentiation on a Trigonometric Curve

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

Implicit Differentiation with Logarithms and Products

Consider the equation $$ \ln(x+y) + x*y = \ln(4)+4 $$.

Implicit Differentiation with Trigonometric Equation

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

Inverse Function Differentiation for Cubic Functions

Let $$f(x)= x^3 + 2*x$$, and let $$g(x)$$ be its inverse function. Answer the following:

Inverse of a Composite Function

Let $$f(x)=\sqrt{3*x+1}$$ and $$g(x)=x^2-1$$, and define $$h(x)=f(g(x))$$. Analyze the invertibility

Rainwater Harvesting System

A rainwater harvesting system collects water in a reservoir. The inflow rate is given by the composi

Second Derivative of an Implicit Function

The curve is defined implicitly by the equation $$x^2*y+y^3=4$$. Answer the following parts:

Temperature Modeling and Composite Functions

A weather balloon ascends and the temperature at altitude x (in kilometers) is modeled by $$T(x) = \

Water Tank Composite Rate Analysis

A water tank receives water from an inflow pipe where the inflow rate is given by the composite func

Analyzing Runner's Motion

A runner's displacement is modeled by the function $$s(t)=-t^3+9t^2+1$$, where s(t) is in meters and

Approximating Levels in a Chemical Reaction using Differentials

The concentration of a chemical substance in a reaction vessel is given by $$C(t)=100*e^{-0.2*t}+5$$

Bacterial Culture Dynamics

In a bioreactor, bacteria are introduced at a rate given by $$I(t)=200e^{-0.1t}$$ (cells per minute)

Biological Growth Rate

A bacterial culture grows according to the model $$P(t)= 500*e^{0.8*t}$$, where \(P(t)\) is the popu

Chemical Reaction Rate Model

A chemical reaction has its reactant concentration modeled by $$C(t)= 0.5*t^2 - 3*t + 4$$, where \(C

Differentiation and Concavity for a Non-Motion Problem: Water Filling a Tank

The volume of water in a tank is given by $$V(t)=4*t^3-12*t^2+9*t+15$$, where $$V$$ is in gallons an

Economic Marginal Cost Analysis

A manufacturer’s total cost for producing $$x$$ units is given by $$C(x)= 0.01*x^3 - 0.5*x^2 + 10*x

Ellipse Tangent Line Analysis

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

Expanding Rectangle: Related Rates

A rectangle has a length $$l$$ and width $$w$$ that are changing with time. At a certain moment, the

Exponential Function Inversion

Consider the function $$f(x)=e^{2*x}+3$$ which models the growth of a certain variable. Analyze the

Graphical Data and Derivatives

A set of experimental data is provided below, showing the concentration (in moles per liter) of a ch

Parametric Motion with Logarithmic and Radical Components

A particle’s motion is described by the vector function $$\mathbf{r}(t)=\langle \ln(t+1),\sqrt{t} \r

Population Growth Rate Analysis

A population grows exponentially according to $$P(t)=1200e^{0.15t}$$, where t is measured in months.

Related Rates: Inflating Spherical Balloon with Exponential Volume Rate

A spherical balloon is being inflated so that its volume changes at a rate of $$\frac{dV}{dt}=8e^{0.

Seasonal Reservoir Dynamics

A reservoir receives water inflow influenced by seasonal variations, modeled by $$I(t)=50+30\sin\Big

Series Approximation for Investment Growth

An investment accumulation function is modeled by $$A(t)= 1 + \sum_{n=1}^{\infty} \frac{(0.07t)^n}{n

Series Approximation in Population Dynamics

A population function is given by $$P(t)= 500 \times \sum_{n=0}^{\infty} \frac{(0.03t)^n}{n!}$$. Ans

Series Approximation with Center Shift

Consider the function $$f(x)= \sum_{n=0}^{\infty} \frac{(-1)^n (3x-1)^n}{n+1}$$. Answer the followin

Series Expansion in Vibration Analysis

A vibrating system has its displacement modeled by $$y(t)= \sum_{n=0}^{\infty} \frac{(-1)^n (2t)^{2*

Spherical Balloon Inflation

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

Absolute Extrema via Candidate's Test

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

Analysis of a Function with Oscillatory Behavior

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

Analysis of a Motion Function Incorporating a Logarithm

A particle's position is given by $$s(t)= \ln(t+1)+ t$$, where $$t$$ is in seconds. Analyze the moti

Analysis of a Rational Function and Its Inverse

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

Application in Motion: Approximate Velocity using Taylor Series

A particle’s position is given by $$s(t)=e^{-t}+t^2$$. Using Taylor series approximations near $$t=0

Application of Rolle's Theorem

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

Application of Rolle's Theorem to a Trigonometric Function

Consider the function $$f(x)=\cos(x)$$ on the interval [0,π]. Answer the following parts:

Application of the Extreme Value Theorem in Economics

A company's revenue is modeled by $$R(x)= -2*x^2+40*x+100$$, where $$x$$ is the number of units sold

Application of the Mean Value Theorem

Consider the function $$f(t)=t^3-3*t^2+2*t+5$$ representing the position (in meters) of a car along

Derivative Sign Chart and Function Behavior

Given the function $$ f(x)=\frac{x}{x^2+1},$$ answer the following parts:

Differentiability and Critical Points of a Piecewise Function

Consider the piecewise function $$ f(x)= \begin{cases} x^2 & \text{if } x \le 2, \\ 4*x-4 & \text{i

Echoes in an Auditorium

In an auditorium, an audio signal produces echoes. The first echo has an intensity that is 70% of th

Epidemic Infection Model

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

Extreme Value Theorem in a Polynomial Function

Consider the function $$h(x)=x^4-8*x^2+16$$ defined on the closed interval $$[-3,3]$$. Answer the fo

Extremum Analysis Using the Extreme Value Theorem

Given the function $$f(x)= \cos(x)-x$$ defined on the interval $$\left[0, \frac{\pi}{2}\right]$$, an

Finding and Interpreting Inflection Points in a Complex Function

Analyze the function $$f(x)= e^{-x}\,\ln(x)$$ for $$x > 0$$. Investigate the points of inflection an

Graph Analysis of a Logarithmic Function

Consider the function $$g(x)= \ln(x) - \frac{1}{x}$$ defined for $$x>0$$. Analyze its behavior and g

Linear Approximation of a Radical Function

For the function $$f(x)= \sqrt{x+1}+x$$, find its linear approximation at $$x=3$$ and use it to appr

Optimization in a Log-Exponential Model

A firm's profit is given by the function $$P(x)= x\,e^{-x} + \ln(1+x)$$, where x (in thousands) repr

Optimization in Particle Routing

A delivery robot’s distance along its predetermined path is described by $$s(t)=\sin(t)+0.5*t$$, whe

Particle Motion on a Curve

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

Radiocarbon Dating in Artifacts

An archaeological artifact contains a radioactive isotope with an initial concentration of 100 units

Relative Motion in Two Dimensions

A point moves in the plane along the path defined by $$x(t)=t^2-4*t+3$$ and $$y(t)=\ln(t+1)$$ for $$

Road Trip Analysis

A car's speed (in mph) during a road trip is recorded at various times. Use the table provided to an

Second Derivative Test for Critical Points

Consider the function $$f(x)=x^3-9*x^2+24*x-16$$.

Series Convergence and Differentiation in Thermodynamics

In a thermodynamic process, the temperature $$T(x)=\sum_{n=0}^\infty \frac{(-2)^n}{n+1} * (x-5)^n$$

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

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

Taylor Series for $$\ln(1+3*x)$$

Let $$f(x)=\ln(1+3*x)$$. Develop its Maclaurin series up to the 3rd degree, determine the radius of

Taylor Series for $$\sqrt{x}$$ Centered at $$x=4$$

For the function $$f(x)=\sqrt{x}$$, find the Taylor series expansion centered at $$x=4$$ including t

Wastewater Treatment Reservoir

At a wastewater treatment reservoir, wastewater enters at a rate of $$W_{in}(t)=12+2*t$$ m³/min and

Water Tank Dynamics

A water tank receives water from a pipe at a rate of $$R(t)=3*t+5$$ liters/min and loses water throu

Accumulated Displacement from a Piecewise Velocity Function

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

Analyzing and Integrating a Function with a Removable Discontinuity

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

Antiderivatives and the Constant of Integration

Consider the rate function $$ r(t)= 2*t + 3 $$ where t represents time in seconds.

Area Estimation with Riemann Sums

Consider the function $$f(x)=x^2-4*x+3$$ on the interval $$[1,5]$$. Using a partition of 4 equal sub

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

Bacteria Population Accumulation

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

Bacterial Growth Accumulation

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

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

Distance vs. Displacement from a Velocity Function

A runner's velocity is modeled by $$v(t)=5-0.5*t$$ (in m/s) for $$0\le t\le10$$. The runner may chan

Drug Absorption Modeling

The rate of drug absorption into the bloodstream is modeled by $$C'(t)= 2*e^{-0.5*t}$$ mg/hr, with a

Economic Applications: Consumer and Producer Surplus

In a market, the demand function is given by $$D(p)=100-2p$$ and the supply function by $$S(p)=20+3p

Energy Consumption in a Household

A household's power usage is modeled by $$P(t)= 3\sin((\pi/12)*t)+3$$ kW for $$t \in [0,24]$$ hours.

Estimating Integral from Tabular Data

Given the following table of values for $$F(t)$$ over time, estimate the integral $$\int F(t)\,dt$$

Evaluating an Integral Involving an Exponential Function

Evaluate the definite integral $$\int_{0}^{\ln(4)} e^{x}\,dx$$.

Graphical Analysis of Riemann Sums

A graph titled 'Graph of Experimental Data' shows a curve representing the height function $$h(t)$$

Integration of a Piecewise-Defined Function

Define the function $$f(x)$$ as follows: $$f(x)= \begin{cases} 2*x, & 0\le x < 3 \\ 12, & 3 \le x \

Mechanical Systems: Total Change and Inverse Analysis

Consider the function \(f(x)= x^3 + 3*x\) defined for all real \(x\), modeling a mechanical system.

Motion and Accumulation: Particle Displacement

A particle moving along a straight line has a velocity function given by $$v(t)=3*t^{2} - 12*t + 5$$

Net Change in Drug Concentration

The rate of change of a drug's concentration in the bloodstream is given by $$R(t)=8*e^{-0.5*t}$$ (i

Parametric Integral and Its Derivative

Let $$I(a)= \int_{0}^{a} \frac{t}{1+t^2}dt$$ where a > 0. This integral is considered as a function

Particle Displacement and Total Distance

A particle moves along a straight line with a velocity function $$v(t)=3*t^2-2*t+1$$ for $$0\le t\le

Probability Density Function and Expected Value

Let the probability density function (pdf) be defined by $$f(x)=k*x*e^{-x}$$ for $$x\ge0$$.

Radioactive Decay: Accumulated Decay

A radioactive substance decays according to $$m(t)=50 * e^(-0.1*t)$$ (in grams), with time t in hour

Temperature Change Analysis

A series of temperature readings (in °C) are recorded over the day as shown in the table. Analyze th

Temperature Function Analysis with Inverses

A temperature profile over time is given by $$f(t)= \ln(2*t + 3)$$ for $$t \ge 0$$ (with temperature

U-Substitution Integration Challenge

Evaluate the integral $$\int_0^2 (2*x+1)\,(x^2+x+3)^5\,dx$$ using an appropriate u-substitution.

Volume of a Solid with Known Cross-sectional Area

A solid extends from $$x=0$$ to $$x=5$$, and its cross-sectional area perpendicular to the x-axis is

Volume of a Solid: Cross-Sectional Area

A solid has cross-sectional area perpendicular to the x-axis given by $$A(x)= (4-x)^2$$ for $$0 \le

Analysis of a Piecewise Function with Potential Discontinuities

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

Analyzing a Rational Differential Equation

Consider the differential equation $$\frac{dy}{dx} = \frac{x^2-1}{y}$$.

Autocatalytic Reaction Dynamics

Consider an autocatalytic reaction described by the differential equation $$\frac{dy}{dt} = k*y*\ln|

Capacitor Discharge in an RC Circuit

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

Coffee Cooling: Differential Equation Application

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

Combined Differential Equations and Function Analysis

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

Direction Fields and Phase Line Analysis

Consider the autonomous differential equation $$\frac{dy}{dt}=(y-2)(3-y)$$. Answer the following par

Euler's Method and Differential Equations

Consider the initial value problem $$\frac{dy}{dx} = x*y$$ with the initial condition $$y(0)=1$$. Eu

Exponential Growth via Slope Field Analysis

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

Exponential Growth with Variable Rate

A bacterial culture grows according to the differential equation $$\frac{dP}{dt}=k(t)P$$, where the

Flow Rate in River Pollution Modeling

A river system is modeled to study pollutant concentration $$C(t)$$ (in mg/L). Polluted water with c

Free-Fall with Air Resistance

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

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

Interpreting Slope Fields for a Differential Equation

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

Logistic Equation with Harvesting

A fish population in a lake follows a logistic growth model with the addition of a constant harvesti

Logistic Growth in Population Dynamics

The population of a small town is modeled by the logistic differential equation $$\frac{dP}{dt}=rP\l

Logistic Growth Model in Population Dynamics

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

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

Mixing Problem: Salt Water Tank

A tank initially contains $$1000$$ liters of pure water with $$50$$ kg of salt dissolved in it. Brin

Newton's Law of Cooling

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

Newton's Law of Cooling: Temperature Change

A hot object is cooling in a room with an ambient temperature of 20°C. Measurements of the object's

Particle Motion with Variable Acceleration

A particle moves along a straight line with an acceleration given by $$a(t)=6-4*t$$. At time t = 0,

Phase-Plane Analysis of a Nonlinear Differential Equation

Consider the logistic differential equation $$\frac{dy}{dt} = y(1-y)$$, which models a normalized po

Population Saturation Model

Consider the differential equation $$\frac{dy}{dt}= \frac{k}{1+y^2}$$ with the initial condition $$y

Predator-Prey Model with Harvesting

Consider a simplified model for the prey population in a predator-prey system that includes constant

Radioactive Decay Data Analysis

A radioactive substance is decaying over time. The following table shows the measured mass (in grams

Radioactive Decay with Constant Source

A radioactive material is produced at a constant rate S while simultaneously decaying. This process

Separable Differential Equation and Maclaurin Series Approximation

Consider the differential equation $$\frac{dy}{dx} = e^{x} * \sin(y)$$ with the initial condition $$

Separable Differential Equation and Slope Field Analysis

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

Separable Differential Equation with Initial Condition

Solve the differential equation $$\frac{dy}{dx} = \frac{x}{y}$$ subject to the initial condition $$y

Series Solution for a Differential Equation

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

Slope Field Analysis and DE Solutions

Consider the differential equation $$\frac{dy}{dx} = x$$. The equation has a slope field as represen

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

Water Pollution with Seasonal Variation

A river receives a pollutant with a time-varying influx modeled by $$I(t)=20+5\cos(0.5*t)$$ kg/day,

Accumulated Change in a Population Model

A population of insects grows at a rate given by $$P'(t)=10e^{-0.2*t}$$, where $$t$$ is in days and

Analysis of a Function with a Removable Discontinuity

Consider the function $$f(x)=\frac{x^2-4}{x-2}$$ for $$x\neq2$$, with an assigned value of $$f(2)=3$

Analyzing Convergence of a Taylor Series

Consider the function $$g(x)= e^{-x^2}$$. Analyze the Maclaurin series for this function.

Area and Volume: Rotated Region

Consider the region bounded by $$y=\ln(x)$$, $$y=0$$, and $$x=e^2$$.

Area Between a Function and Its Tangent Line

Let $$f(x)=x^3-x$$. At the point $$x=1$$, find the tangent line to the curve and determine the area

Area Between Curves: Parabolic & Linear Regions

Consider the curves $$f(x)=5*x-x^2$$ and $$g(x)=x$$. Answer the following questions regarding the re

Area Between Economic Curves

In an economic model, two cost functions are given by $$C_1(x)=100-2*x$$ and $$C_2(x)=60-x$$, where

Area Under an Exponential Decay Curve

Consider the function $$f(x)=e^{-x}$$ on the interval $$[0,1]$$. Answer the following:

Average Value and Monotonicity of an Oscillatory Function

Consider the function $$f(x)=\sin(2*x)+1$$ defined on the interval $$[0,\pi]$$.

Average Value of a Temperature Function

A region’s temperature throughout a day is modeled by the function $$T(t)=10+5*\sin(\frac{\pi}{12}*t

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

Chemical Reaction Rate Analysis

During a chemical reaction, the concentration of a reactant is modeled by $$C(t)=20e^{-0.3*t}$$ (in

Designing a Bridge Arch

A bridge arch is modeled by the curve $$y = 10 - 0.25*x^2$$, where $$x$$ is measured in meters and $

Determining Average Value of a Velocity Function

A runner’s velocity is given by $$v(t)=2*t+3$$ (in m/s) for $$t\in[0,5]$$ seconds.

Economic Analysis: Consumer and Producer Surplus

In a market analysis, the demand and supply for a product are modeled by the equations: Demand: $$D(

Electric Charge Accumulation

A circuit has a current given by $$I(t)=4e^{-t/3}$$ A for $$t$$ in seconds. Analyze the charge accum

Electric Current and Charge

An electric current in a circuit is defined by $$I(t)=4*\cos\left(\frac{\pi}{10}*t\right)$$ amperes,

Flow Rate into a Tank

Water flows into a tank at a rate given by $$Q(t)=\frac{100}{1+t^2}$$ liters per hour on the interva

Fluid Flow in a River

The rate of water flow in a river is given by $$Q(t)=50+10*\sin\left(\frac{\pi}{6}*t\right)$$ cubic

Fluid Flow Rate and Total Volume

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

Implicit Differentiation with Exponential Terms

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

Integral Approximation Using Taylor Series

Approximate the integral $$\int_{0}^{0.2} \sin(2*x)\,dx$$ by using the Taylor series expansion of $$

Moment of Inertia of a Thin Plate

A thin plate occupies the region bounded by the curves $$y= x$$ and $$y= x^2$$ for $$0 \le x \le 1$$

Motion Analysis on a Particle with Variable Acceleration

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

Movement Under Variable Acceleration

A car accelerates along a straight road with acceleration given by $$a(t)=2*t - 3$$ (in m/s²) and ha

Optimization of Material Usage in a Container

A container's volume is given by $$V(h)=\int_0^h \pi*(3-0.5*\ln(1+x))^2dx$$, where $$h$$ is the heig

Population Growth: Cumulative Increase

A bacterial culture grows at a rate given by $$r(t)=3*e^{0.2*t}$$ (in thousands per hour), where $$t

Rainfall Accumulation Analysis

A meteorological station recorded the following rainfall data (in mm) over a 24‐hour period. The rai

Volume by Shell Method: Rotating a Region

Consider the region bounded by the curves $$y=x$$ and $$y=x^2$$. This region is rotated about the y-

Volume of a Solid by Disc Method

The region bounded by $$y=\sqrt{x}$$ and $$y=x$$ for $$x\in[0,1]$$ is rotated about the $$x$$-axis t

Volume of a Solid Rotated about y = -1

The region bounded by $$y=\sqrt{x}$$ and $$y=0$$ for $$x\in[0,9]$$ is rotated about the line $$y=-1$

Volume of a Solid via Shell Method

Consider the region bounded by $$y=x^2$$ and $$y=4$$ in the first quadrant. This region is revolved

Volume of a Water Tank with Varying Cross-Sectional Area

A water tank has a cross-sectional area given by $$A(x)=3*x^2+2$$ in square meters, where $$x$$ (in

Volume of an Arch Bridge Support

The arch of a bridge is modeled by $$y=12-\frac{x^2}{4}$$ for $$x\in[-6,6]$$. Cross-sections perpend

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

Work Done by a Variable Force

A force acting on an object moving along a straight line is given by $$F(x)= 6 - x$$ (in Newtons) as

Work Done by a Variable Force

A variable force acting along the x-axis is given by $$F(x) = 2 * x + 3$$ (in Newtons). An object mo

Acceleration Analysis in a Vector-Valued Function

Consider the vector function describing an object's motion: $$\textbf{r}(t)= \langle \ln(t+2), \sqrt

Arc Length of a Polar Curve

Consider the polar curve $$r(\theta)= 1+\cos(\theta)$$ for \(0 \le \theta \le \pi\).

Area between Two Polar Curves

Given two polar curves: $$r_1 = 1+\cos(\theta)$$ and $$r_2 = 2\cos(\theta)$$, consider the region wh

Area Between Two Polar Curves

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

Area Enclosed by a Polar Curve

Let the polar curve be defined by $$r=3\sin(\theta)$$ with $$0\le \theta \le \pi$$. Answer the follo

Component-Wise Integration of a Vector-Valued Function

Given the acceleration vector $$\textbf{a}(t)= \langle 3\cos(t), -3\sin(t) \rangle$$, answer the fol

Comprehensive Motion Analysis Using Parametric and Vector-Valued Functions

A particle moves with position given by $$ r(t)=\langle t*e^{-t},\;\ln(1+t) \rangle $$ for $$ t\ge0

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 \

Conversion of Polar to Parametric Form

A particle’s motion is given in polar form by the equations $$r = 4$$ and $$\theta = \sqrt{t}$$ wher

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

Curvature of a Vector-Valued Function

Let $$\vec{r}(t)=\langle t, t^2, \ln(t) \rangle$$ for \(t>0\). The curvature \(\kappa(t)\) is given

Designing a Roller Coaster: Parametric Equations

The path of a roller coaster is modeled by the equations $$x(t)=t-\sin(t)$$ and $$y(t)=1-\cos(t)$$ f

Displacement from a Vector-Valued Velocity Function

A particle's velocity is given by $$\vec{v}(t)=\langle \cos(t), \sin(t), t \rangle$$ for $$t \in [0,

Implicit Differentiation with Implicitly Defined Function

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

Intersection of Polar Curves

Consider the polar curves given by $$r=2\sin(\theta)$$ and $$r=1+\cos(\theta)$$. Answer the followin

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\

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

Parametric Equations of a Cycloid

A cycloid is generated by a point on the circumference of a circle of radius $$r$$ rolling along a s

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

Polar to Parametric Conversion and Arc Length

A curve is defined in polar coordinates by $$r= 1+\sin(\theta)$$. Convert and analyze the curve.

Projectile Motion Modeled by Vector-Valued Functions

A projectile is launched with an initial velocity vector $$\vec{v}_0=\langle 10, 20 \rangle$$ (in m/

Tangent Line Analysis through Polar Conversion

Consider the polar curve defined by $$r(θ)= 4\sin(θ)$$. Answer the following:

Tangent Lines to Polar Curves

Consider the polar curve $$r(\theta)= 3\sin(\theta)$$. Analyze the tangent line at a point correspo

Vector Functions and Work Done Along a Path

A force field is given by $$\mathbf{F}(x,y)=\langle x*y, x^2 \rangle$$. A particle moves along the p

Vector-Valued Function with Constant Acceleration

A particle moves in the plane with its position given by $$\vec{r}(t)=\langle 5*t, 3*t+2*t^2 \rangle

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

Velocity and Acceleration of a Particle

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

Work Done by a Force along a Vector Path

A force field is given by $$\mathbf{F}(t)=\langle2*t,\;3\sin(t)\rangle$$ and an object moves along a