AP Calculus BC FRQ Room

Analyzing Limits of a Composite Function

Let $$f(x)=\frac{\sin(\sqrt{4+x}-2)}{x}$$ for $$x \neq 0$$. Answer the following:

Continuity Analysis in Road Ramp Modeling

A highway ramp is modeled by the function $$y(x)= \frac{(x-3)(x+2)}{x-3}$$ for $$x\neq3$$, where x i

Continuity Assessment of a Rational Function with a Redefined Value

Consider the function $$r(x)= \begin{cases}\frac{x^2-9}{x-3}, & x \neq 3 \\ 7, & x=3\end{cases}$$.

Continuity in Piecewise-Defined Functions

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

Defining a Function with a Unique Limit Behavior

Construct a function $$f(x)$$ that meets the following conditions: - It is defined and continuous fo

Determining Continuity via Series Expansion

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

Determining Limits for a Function with Absolute Values and Parameters

Consider the function $$ f(x)= \begin{cases} \frac{|x-2|}{x-2}, & x \neq 2 \\ c, & x = 2 \end{cases

Examining Continuity with an Absolute Value Function

Consider the function defined by $$f(x)=\frac{|x-2|}{x-2}$$ for $$x \neq 2$$. (a) Evaluate $$\lim_{x

Factorization and Limits

Consider the function $$f(x)=\frac{x^2-4 * x}{x-4}$$ defined for $$x \neq 4$$. Answer the following:

Higher‐Order Continuity in a Log‐Exponential Function

Define $$ f(x)= \begin{cases} \frac{e^x - 1 - \ln(1+x)}{x^3}, & x \neq 0 \\ D, & x = 0, \end{cases}

Intermediate Value Theorem Application

Let $$f(x)=x^3-4*x+1$$, which is continuous on the real numbers. Answer the following:

Intermediate Value Theorem in Engineering Context

In a structural analysis, the stress on a beam is modeled by a continuous function $$S(x)$$ on the i

Limit Behavior in a Container Optimization Problem

A manufacturer designs a closed cylindrical container with a fixed volume $$V$$ (in cubic units). Th

Limit Definition of the Derivative for a Polynomial Function

Let $$f(x)=3*x^2-2*x+1$$. Use the limit definition of the derivative to find $$f'(2)$$.

Limits Involving Exponential Functions

Consider the function $$f(x)= \frac{e^{2*x}-1}{x}$$ defined for $$x\neq0$$.

Limits Involving Radicals

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

Limits with Composite Logarithmic Functions

Consider the function $$t(x)=x*\ln(x)$$ defined for x > 0.

Manufacturing Process Tolerances

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

Non-Existence of a Limit due to Oscillation

Consider the function $$h(x)= \sin(\frac{1}{x})$$. Answer the following regarding its limit as x app

One-Sided Limits for a Piecewise Function

Consider the piecewise function $$f(x)= \begin{cases} 2*x+1 & \text{if } x< 3 \\ x^2-5*x+8 & \text{i

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

Population Growth and Limits

The population $$P(t)$$ of a small town is recorded every 10 years as shown in the table below. Assu

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

Rational Functions and Limit at Infinity

Consider the rational function $$r(x)= \frac{2x^2+3x-1}{x^2-4}$$.

Real-World Temperature Sensor Analysis

A temperature sensor is modeled by the function $$T(t)=\frac{t^2-9}{t-3}$$ for t ≠ 3 (with t in minu

Removable Discontinuity in a Cubic Function

Consider the function $$f(x)=\frac{x^3-8}{x-2}$$ defined for $$x\neq2$$. Answer the following:

Analysis of Higher-Order Derivatives

Let $$f(x)=x*e^{-x}$$ model the concentration of a substance over time. Analyze both the first and s

Biochemical Reaction Rates: Derivative from Experimental Data

The concentration of a reactant in a chemical reaction is modeled by $$C(t)= 8 - 5t + t^2$$ (in M) w

Car Acceleration: Secant and Tangent Slope

A car's position along a straight road is given by $$s(t)= 2t^3 - 9t^2 + 12t$$, where s is in meters

Chemical Reaction Rate Analysis

The concentration of a reactant in a chemical reaction (in M) is recorded over time (in seconds) as

Derivative from the Limit Definition: Function $$f(x)=\sqrt{x+2}$$

Consider the function $$f(x)=\sqrt{x+2}$$ for $$x \ge -2$$. Using the limit definition of the deriva

Differentiation and Linear Approximation for Error Estimation

Let $$f(x) = \ln(x)*x^2$$. Use differentiation and linear approximation to estimate changes in the f

Differentiation of Implicitly Defined Functions

An ellipse is defined by the equation $$\frac{x^2}{4} + \frac{y^2}{9} = 1$$. Use implicit differenti

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

Graph Behavior of a Log-Exponential Function

Let $$f(x)=e^{-x}+\ln(x)$$, where the domain is $$x>0$$.

Higher Order Derivatives: Concavity and Inflection Points

Consider the function $$f(x)= x^4 - 4*x^3+6*x^2.$$ (a) Find the first derivative \(f'(x)\) and th

Implicit Differentiation and Tangent Line Slope

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

Implicit Differentiation in Circular Motion

A particle moves along the circle defined by $$x^2 + y^2 = 25$$. Answer the following parts.

Implicit Differentiation: Cost Allocation Model

A company's cost allocation between two departments is modeled by the equation $$x^2 + x*y + y^2 = 1

Irrigation Reservoir Analysis

An irrigation reservoir has an inflow rate modeled by $$I(t)=\frac{100}{1+e^{-0.5*(t-10)}}$$ liters

Linearization and Tangent Approximations

Let $$f(x)=e^{-x}$$ represent a cost decay function over time. Use linear approximation near $$x=0$$

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

Population Growth Approximation using Taylor Series

A biologist models population growth with the exponential function $$P(t)=e^{0.05*t}$$. To estimate

Population Growth Rates

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

Product and Quotient Rule Application

Consider the function $$f(x)=\frac{x*\ln(x)}{e^{x}+2}$$, defined for $$x>0$$. Analyze its behavior u

Projectile Motion Analysis

A projectile is launched and its height in feet at time $$t$$ seconds is given by $$h(t)=-16*t^2+32*

Rate Function Involving Logarithms

Consider the function $$h(x)=\ln(x+3)$$.

Related Rates in Circle Expansion

A circular oil spill is expanding such that its radius increases at a constant rate of $$0.5\,m/s$$.

Related Rates: Sweeping Spotlight

A spotlight located at the origin rotates at a constant rate of $$2 \text{ rad/s}$$. A wall is posit

River Flow and Differentiation

The rate of water flow in a river is modeled by $$Q(t)= 5t^2 + 8t + 3$$ in cubic meters per second,

Secant Line Approximations and Instantaneous Slopes

The function $$g(x)=e^{x} - 2*x$$ models the mass (in grams) of a chemical in a reaction over time,

Secant to Tangent Convergence

Consider the natural logarithm function $$f(x)=\ln(x)$$ for \(x>0\). Answer the following:

Temperature Change with Provided Data

The temperature at different times after midnight is modeled by $$T(t)=5*\ln(t+1)+20$$, with $$t$$ i

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

Widget Production Rate

A widget manufacturing plant produces widgets according to the function $$P(t)=4*t^2 - 3*t + 10$$ wh

Chain Rule with Exponential Function

Consider the function $$h(x)= e^{\sin(4*x)}$$ which models a process with exponential growth modulat

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 Analysis

Consider the function $$f(x)= \sqrt{3*x^2+2*x+1}$$ which arises in an experimental study of motion.

Composite Function Rates in a Chemical Reaction

In a chemical reaction, the concentration of a substance at time $$t$$ is given by $$C(t)= e^{-k*(t+

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.

Composite Functions in a Biological Model

In a biological model, the concentration of a substance is given by $$P(x)=e^{-\sqrt{x^2+1}}$$, wher

Differentiation of a Logarithmic-Square Root Composite Function

Let $$f(x)= \ln(\sqrt{1+ 3*x^2})$$. Differentiate the function with respect to $$x$$, simplify your

Differentiation of an Inverse Exponential Function

Let $$f(x)=e^{2*x}-7$$, and let g denote its inverse function. Answer the following parts.

Higher-Order Derivatives via Implicit Differentiation

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

Ice Cream Storage Dynamics

An ice cream storage facility receives ice cream at a rate given by the composite function $$I(t)=d(

Implicit Differentiation in Exponential Equation

Consider the equation $$e^{x*y}+x^2-y^3=0$$ that relates x and y. Answer the following parts:

Implicit Differentiation with Exponential and Trigonometric Mix

Consider the equation $$e^{x*y} + \sin(x) - y = 0$$. Differentiate implicitly with respect to $$x$$

Implicit Differentiation with Logarithmic Functions

Let $$x$$ and $$y$$ be related by the equation $$\ln(x*y) + x - y = 0$$.

Implicit Differentiation: Circle and Tangent Line

The equation $$x^2 + y^2 = 25$$ represents a circle. Use implicit differentiation to find the deriva

Inverse Derivative via Chain Rule for a Logarithmic-Exponential Function

Let $$f(x)=\ln(1+e^x)$$. Analyze its inverse derivative.

Inverse Differentiation of a Trigonometric Function

Consider the function $$f(x)=\arctan(2*x)$$ defined for all real numbers. Analyze its inverse functi

Navigation on a Curved Path: Boat's Eastward Velocity

A boat's location in polar coordinates is described by $$r(t)= \sqrt{4*t+1}$$ and its direction by $

Parametric Curve Analysis with Composite Functions

A curve is defined by the parametric equations $$x(t)=\ln(1+t^2)$$ and $$y(t)=\sqrt{t+4}$$, where t

Reservoir Levels and Evaporation Rates

A reservoir is being filled with water from an inflow while losing water through controlled release

Water Tank Composite Rate Analysis

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

Applying L'Hospital's Rule to a Transcendental Limit

Evaluate the limit $$\lim_{x\to 0}\frac{e^{2*x}-1}{\sin(3*x)}$$.

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

Draining Conical Tank

Water is draining from a conical tank at a rate of $$5$$ m³/min. The tank has a height of $$10$$ m a

Estimation Error with Differentials

Let $$f(x)=x^3$$. Use differentials to estimate the value of $$f(2.05)$$ and determine the approxima

Fuel Consumption Rate Analysis

The fuel consumption of a car (in gallons per 100 miles) is modeled by $$C(v)=0.05*v^2+1$$, where $$

Horizontal Tangents on Cubic Curve

Consider the curve defined by $$x^3 + y^3 - 6*x*y = 0$$.

Inflating Balloon

A spherical balloon is being inflated. The volume $$V$$ and the radius $$r$$ are related by $$V = \f

Instantaneous vs. Average Rate of Change in Temperature

A rod's temperature along its length is modeled by $$T(x)=20\ln(x+1)+e^{-x}$$, where x (in meters) i

Inverse Trigonometric Composition

Consider the function $$f(x)=2*\sin(x)-1$$ defined on $$\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$$.

L'Hôpital’s Rule in Chemical Reaction Rates

In a chemical reaction, the ratio of certain concentrations is modeled by $$R(x)=\frac{3*x^2-2*x+1}{

Limits and L'Hôpital's Rule Application

Consider the function $$f(x)=\frac{5*x^3-4*x^2+1}{7*x^3+2*x-6}$$. Answer the following:

Linearization for Approximating Function Values

Let $$f(x)= \sqrt{x}$$. Use linearization at $$x=10$$ to approximate $$\sqrt{10.1}$$. Answer the fol

Maclaurin Series for ln(1+x)

Consider the function $$f(x)= \ln(1+x)$$. Its Maclaurin series may be used to approximate values of

Maximizing the Area of an Inscribed Rectangle

A rectangle is inscribed in a semicircle of radius $$R$$, with its base along the diameter. The rect

Optimization of a Rectangular Enclosure

A rectangular enclosure is to be built adjacent to a river. Only three sides of the enclosure requir

Optimizing a Cylindrical Can Design

A manufacturer wants to design a cylindrical can with a fixed surface area of $$600\pi$$ cm² in orde

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

Rational Function Inversion

Consider the rational function $$f(x)=\frac{2*x+3}{x-1}$$. Analyze its inverse.

Related Rates: Expanding Circular Oil Spill

In a coastal region, an oil spill is spreading uniformly and forms a circular region. The area of th

Related Rates: Inflating Spherical Balloon

A spherical balloon is being inflated so that its volume, given by $$V= \frac{4}{3}\pi*r^3$$, increa

Revenue Function and Marginal Revenue

A company’s revenue (in thousands of dollars) is modeled as a function of units sold (in thousands)

Road Trip Distance Analysis

During a road trip, the distance traveled by a car is given by $$s(t)=3*t^2+2*t+5$$, where $$t$$ is

Temperature Change of Cooling Coffee

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

Temperature Conversion Model Inversion

The temperature conversion function is given by $$f(x)=\frac{9}{5}*x+32$$, which converts Celsius to

Trigonometric Implicit Relation

Consider the implicit equation $$\sin(x*y) + x - y = 0$$.

Varying Acceleration and Particle Motion

A particle moves along a straight line with acceleration given by $$a(t)=4-2*t$$ (in m/s²) for $$t\g

Analysis of a Cubic Function

Consider the function $$f(x)=x^3-6*x^2+9*x+2$$. Using this function, answer the following parts.

Analysis of an Exponential-Linear Function

Consider the function $$p(x)=e^x-4*x$$. Answer the following parts:

Analysis of Total Distance Traveled

A particle moves along a line with a velocity function given by $$v(t)=t^2-4*t+3$$ for $$t\in[0,5]$$

Analyzing Extrema for a Rational Function

Let $$f(x)= \frac{x^2+2}{x+1}$$ be defined on the interval $$[0,4]$$. Use calculus methods to analyz

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

Area Between Curves and Rates of Change

An irrigation canal has a cross-sectional shape described by \( y=4-x^2 \) for \( |x| \le 2 \). The

Candidate’s Test for Absolute Extrema in Projectile Motion

A projectile is launched such that its height at time $$t$$ is given by $$h(t)= -16*t^2+32*t+5$$ (in

Concavity of an Integral Function

Let $$F(x)= \int_0^x (t^2-4*t+3)\,dt$$. Analyze the concavity of $$F(x)$$.

Curve Sketching with Second Derivative

Consider the function $$f(x) = x^4 - 4*x^3 + 6*x^2 - 4*x + 1$$.

Differentiability of a Piecewise Function

Consider the piecewise function $$r(x)=\begin{cases} x^2, & x \le 2 \\ 4*x-4, & x > 2 \end{cases}$$.

Discounted Cash Flow Analysis

A project is expected to return cash flows that decrease by 10% each year from an initial cash flow

Dynamic Analysis Under Time-Varying Acceleration in Two Dimensions

A particle moves in the plane with acceleration given by $$\vec{a}(t)=\langle3\cos(t),-2\sin(t)\rang

Echoes in an Auditorium

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

Economic Equilibrium and Implicit Differentiation

An economic equilibrium is modeled by the implicit equation $$e^{p}*q + p^2 = 100$$, where \( p \) r

Elasticity Analysis of a Demand Function

The demand function for a product is given by $$Q(p) = 100 - 5*p + 0.2*p^2$$, where p (in dollars) i

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

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

Function Behavior Analysis

Consider the function \( f(x) = x^4 - 4*x^3 + 6*x^2 - 4*x + 1 \). Answer the following parts:

Inverse Function Analysis in an Optimization Scenario

Consider the cost function $$f(x)= x^4 + 2*x^2 + 1$$ defined for $$x \ge 0$$, where f(x) represents

Investigation of Extreme Values on a Closed Interval

For a particle moving along a path given by $$f(x)=x^3-6*x^2+9*x+5$$ where $$x\in[0,5]$$, analyze it

Parameter-Dependent Concavity Conditions

Consider the function $$ f(x)=x^3+a*x^2+2x,$$ where $$a$$ is a real parameter. Answer the following

Population Growth Modeling

A region's population (in thousands) is recorded over a span of years. Use the data provided to anal

Series Manipulation and Transformation in an Economic Forecast Model

A forecast model is given by the series $$F(x)=\sum_{n=0}^\infty \frac{(-1)^n}{(n+1)^2} * x^n$$. Ans

Volume by Cross Sections Using Squares

A region in the xy-plane is bounded by $$y=x$$, $$y=0$$, and $$x=3$$. Perpendicular to the x-axis, c

Volume of a Solid of Revolution Using the Washer Method

Find the volume of the solid obtained by revolving the region bounded by $$y=\sqrt{x}$$, $$y=\frac{x

Water Tank Rate of Change

The volume of water in a tank is modeled by $$V(t)= t^3 - 6*t^2 + 9*t$$ (in cubic meters), where $$t

Accumulation Function in an Investment Model

An investment has an instantaneous rate of return given by $$r(t)=0.05*t+0.02$$ (per year). The accu

Approximating Water Volume Using Riemann Sums

A storm causes a varying inflow rate f(t) (in m³/h) into a reservoir. The inflow rate was recorded a

Area Between a Curve and Its Tangent

For the function $$f(x)=x^3-3*x^2+2*x$$, analyze the area between the curve and its tangent line at

Area Between the Curves f(x)=x² and g(x)=2x+3

Given the two functions $$f(x)= x^2$$ and $$g(x)= 2*x+3$$ on the interval where they intersect, dete

Area Estimation with Riemann Sums

A water flow rate function f(x) (in m³/s) is measured at various times. The table below shows the me

Chemical Reactor Concentration

In a chemical reactor, a reactant enters at a rate of $$C_{in}(t)=5+t$$ grams per minute and is simu

Convergence of an Improper Integral

Consider the function $$f(x)=\frac{1}{x*(\ln(x))^2}$$ for $$x > 1$$.

Cost and Inverse Demand in Economics

Consider the cost function representing market demand: $$f(x)= x^2 + 4$$ for $$x\ge0$$. Answer the f

Cyclist's Distance Accumulation Function

A cyclist’s total distance traveled is modeled by $$D(t)= \int_{0}^{t} (5+\sin(u))\, du + 2$$ kilom

Differentiation and Integration of a Power Series

Consider the function given by the power series $$f(x)=\sum_{n=0}^\infty \frac{x^n}{2^n}$$.

Estimating Area Under a Curve Using Riemann Sums

Consider the function $$f(x)$$ whose values on the interval $$[0,10]$$ are given in the table below.

Estimating Chemical Production via Riemann Sums

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

Exploring Riemann Sums and Discontinuities from Graphical Data

A graph of a function f(x) is provided that shows a smooth curve with a removable discontinuity (a h

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

Fuel Consumption Estimation with Midpoint Riemann Sums

A vehicle’s fuel consumption rate (in liters per hour) over a trip is recorded at various times. The

Integration Using U-Substitution

Evaluate the indefinite integral $$\int (4*x+2)^5\,dx$$ using u-substitution.

Modeling Water Inflow Using Integration

Water flows into a tank at a rate given by $$R(t)=4-0.5*t$$ (in liters per minute) for $$t\in[0,8]$$

Population Growth: Rate to Accumulation

A population's growth rate (in thousands of individuals per year) is modeled by $$P'(t)=2*t - 1$$ fo

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

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

Trapezoidal Approximation of a Definite Integral from Tabular Data

The table below shows the height H(t) (in meters) of a liquid in a tank at specific times. Use a tra

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

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

Water Tank Inflow and Outflow

A water tank begins operation at t = 0 with an initial volume of 0 liters. Water flows in through an

Analysis of a Nonlinear Differential Equation

Consider the nonlinear differential equation $$\frac{dy}{dx} = y^3-3*y$$.

Autocatalytic Reaction Dynamics

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

Braking of a Car

A car decelerates according to the differential equation $$\frac{dv}{dt} = -k*v$$, where k is a posi

Car Engine Temperature Dynamics

The temperature $$T(t)$$ (in °C) of a car engine is modeled by the differential equation $$\frac{dT}

Chemical Reaction Rate and Series Approximation

A chemical reaction is modeled by the differential equation $$\frac{dC}{dt} = -0.2 * C^2$$ with the

Differential Equation in a Gravitational Context

Consider the differential equation $$\frac{dv}{dt}= -G\,\frac{M}{(R+t)^2}$$, which models a simplifi

Estimating Total Change from a Rate Table

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

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

FRQ 4: Newton's Law of Cooling

A cup of coffee cools according to Newton's Law of Cooling, where the temperature $$T(t)$$ satisfies

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

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

Infectious Disease Spread Model

In a closed population of N individuals, the number of infected individuals $$I(t)$$ is modeled by t

Integrating Factor for a Non-Exact Differential Equation

Consider the differential equation $$ (y - x)\,dx + (y + 2*x)\,dy = 0 $$. This equation is not exact

Inverse Function Analysis Derived from a Differential Equation Solution

Consider the function $$f(x)=x^3+2$$. Although this function is provided outside of a differential e

Logistic Equation with Harvesting

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

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

Logistic Population Growth Model

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

Mixing Problem in a Tank

A tank initially contains 100 L of water with 5 kg of dissolved salt. Brine containing 0.1 kg of sal

Mixing Problem with Constant Flow Rate

A tank holds 500 L of water and initially contains 10 kg of dissolved salt. Brine with a salt concen

Modeling Disease Spread with Differential Equations

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

Modeling Free Fall with Air Resistance

An object falls under gravity while experiencing air resistance proportional to its velocity. The mo

Newton's Law of Cooling

An object cools according to Newton's Law of Cooling, which is modeled by the differential equation

Newton's Law of Cooling

An object cooling in a room follows Newton's law of cooling described by $$\frac{dT}{dt} = -k*(T-A)$

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

Piecewise Differential Equation with Discontinuities

Consider the following piecewise differential equation defined for a function $$y(x)$$: For $$x < 2

Population Growth with Harvesting

A fish population in a lake is modeled by the differential equation $$\frac{dP}{dt}= rP - H$$, where

Predator-Prey Model with Harvesting

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

Projectile Motion with Air Resistance

A projectile is launched with an initial speed $$v_0$$ at an angle $$\theta$$ relative to the horizo

RC Circuit Differential Equation

In an RC circuit, the capacitor charges according to the differential equation $$\frac{dQ}{dt}=\frac

RL Circuit Analysis

An RL circuit is described by the differential equation $$L\frac{di}{dt} + R*i = V$$, where $$L=0.5\

Solution Curve Sketching Using Slope Fields

Given the differential equation $$\frac{dy}{dx} = x - y$$, a slope field is provided. Use the field

Tank Draining Problem

A tank with a variable cross-sectional area is being emptied. The height \(h(t)\) of the water satis

Verification of a Candidate Solution

Consider the candidate solution $$y(x)= \sqrt{4*x^2+3}$$ proposed for the differential equation $$\f

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

Analyzing a Motion Graph from Data

The following table represents the instantaneous velocity (in m/s) of a vehicle over a 6-second inte

Arc Length of a Parabolic Curve

Find the arc length of the curve defined by $$y = x^2$$ for $$x$$ in the interval $$[0,3]$$.

Arc Length of the Logarithmic Curve

For the function $$f(x)=\ln(x)$$ defined on the interval $$[1,e]$$, determine the arc length of the

Area and Volume: Rotated Region

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

Area Between Exponential Curves

Consider the functions $$f(x)=e^{-x}$$ and $$g(x)=e^{-2*x}$$ for $$x\ge0$$. Answer the following:

Area Calculation: Region Under a Parabolic Curve

Let $$f(x)=4-x^2$$. Consider the region bounded by the curve $$f(x)$$ and the x-axis.

Area Under a Curve with a Discontinuity

Consider the function $$f(x)=\frac{1}{x+2}$$ defined on $$[0,3]$$.

Average Concentration of a Drug in Bloodstream

The concentration of a drug in the bloodstream is modeled by $$C(t)=3e^{-0.9*t}+2$$ mg/L, where $$t$

Average Cost Function in Production

A factory’s cost function for producing $$x$$ units is modeled by $$C(x)=0.5*x^2+3*x+100$$, where $$

Average Fuel Consumption and Optimization

A vehicle's fuel consumption rate is modeled by the function $$f(x)=2*x^2-8*x+10$$, where $$x$$ repr

Average Population Density

In an urban study, the population density (in thousands per km²) of a city is modeled by the functio

Average Speed from a Variable Acceleration Scenario

A particle moves along the x-axis under an acceleration given by $$a(t)= 3*t - 2$$ (in m/s²) and has

Average Value of a Velocity Function

The velocity of a car is modeled by $$v(t)=3*t^2-12*t+9$$ (m/s) for $$t\in[0,5]$$ seconds. Answer th

Average Value of a Velocity Function

A particle moves along a line with its velocity given by $$v(t)= 2*\cos(t) + \sin(t)$$ for $$t \in [

Balloon Inflation Related Rates

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

Car Braking and Stopping Distance

A car decelerates with an acceleration given by $$a(t)=-2*t$$ (in m/s²) and has an initial velocity

Comparing Average and Instantaneous Rates of Change

For the quadratic function $$f(x)= 3*x^2 - 4*x + 1$$ on the interval $$[1,3]$$, investigate both its

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 $

Electric Charge Distribution Along a Rod

A rod of length 10 m has a linear charge density given by $$\lambda(x) = 3e^{-0.5*x}$$ coulombs per

Force on a Submerged Plate

A vertical rectangular plate is submerged in water. The plate is 3 m wide and extends from a depth o

Implicit Differentiation with Exponential Terms

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

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

Particle Motion with Velocity Reversal

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

Power Series Representation for ln(x) about x=4

The function $$f(x)=\ln(x)$$ is to be expanded as a power series centered at $$x=4$$. Find this seri

Projectile Motion with Constant Acceleration

A ball is thrown upward and moves under the constant acceleration due to gravity $$a(t)=-9.8$$ (in m

Savings Account with Decreasing Deposits

An individual opens a savings account with an initial deposit of $1000 in the first month. Every sub

Surface Area of a Rotated Parabolic Curve

The curve $$y = x^2$$ is rotated about the x-axis for $$x$$ in the interval $$[0,3]$$ to form a surf

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 by the Washer Method: Between Curves

Consider the region between the curves $$y=\sqrt{x}$$ and $$y=\frac{x}{2}$$ for $$x$$ between their

Volume of a Hollow Cylinder Using the Washer Method

A manufacturer designs a hollow cylindrical container. The outer surface is modeled by $$y=10-\sqrt{

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 $$

Volume of a Solid with Square Cross Sections

The base of a solid is the region in the plane bounded by $$y=x$$ and $$y=x^2$$ (with $$x$$ between

Volume of a Solid: ln(x) Region Rotated

Consider the region in the $$xy$$-plane bounded by $$y=\ln(x)$$, $$y=0$$, $$x=1$$, and $$x=e$$. This

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

Work Done by a Variable Force

A variable force acting along a straight line is given by $$F(x)=5*x$$ (in Newtons), where $$x$$ is

Work Done by a Variable Force

A force acting along a straight line is given by $$F(x)=10 - 0.5*x$$ newtons for $$0 \le x \le 12$$

Analyzing the Concavity of a Parametric Curve

A curve is defined by $$x(t)=t-\sin(t)$$ and $$y(t)=1-\cos(t)$$.

Arc Length and Curvature Comparison

Consider two curves given by: $$C_1: x(t)=\ln(t),\, y(t)=\sqrt{t}$$ for $$1\leq t\leq e$$, and $$C_2

Arc Length of a Parametric Curve

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

Area Between Polar Curves

Consider the polar curves $$ r_1=2+\cos(\theta) $$ and $$ r_2=1+\cos(\theta) $$. Although the curves

Circular Motion in Vector-Valued Form

A particle moves along a circle of radius 5 with its position given by $$ r(t)=\langle 5*\cos(t),\;

Converting Polar to Cartesian and Computing Slope

The polar curve is given by the equation $$r=4\cos(\theta)$$. Answer the following:

Curve Analysis and Optimization in a Bus Route

A bus follows a route described by the parametric equations $$x(t)=t^3-3*t$$ and $$y(t)=2*t^2-t$$, w

Distance Traveled in a Turning Curve

A curve is defined by the parametric equations $$x(t)=4*\sin(t)$$ and $$y(t)=4*\cos(t)$$ for $$0\le

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\

Exponential Growth in Parametric Representation

A model for population growth is given by the parametric equations $$x(t)=t$$ and $$y(t)=e^{0.3t}$$,

Inner Loop of a Limaçon in Polar Coordinates

The polar curve given by \(r=1+2\cos(\theta)\) forms a limaçon with an inner loop. Answer the follow

Intersection Analysis with the Line y = x

Given the parametric equations $$x(t)=\ln(t+2)$$ and $$y(t)=t^2-1$$ for $$t \ge 0$$, answer the foll

Motion Along a Helix

A particle moves along a helix defined by $$\mathbf{r}(t)=\langle \cos(t), \sin(t), t \rangle$$.

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

Parametric Curves and Intersection Points

Two curves are defined by $$C_1: x(t)=t^2,\, y(t)=2*t+1$$ and $$C_2: x(s)=4-s^2,\, y(s)=3*s$$. Find

Parametric Slope and Arc Length

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

Parametric to Polar and Integration

The spiral curve is given in parametric form by $$x(t)=t*\cos(t)$$ and $$y(t)=t*\sin(t)$$ for $$t\ge

Particle Motion in the Plane

A particle moves in the plane with its position described by the parametric equations $$x(t)=3*\cos(

Particle Motion on an Elliptical Arc

A particle moves along a curve described by the parametric equations $$x(t)= 2*cos(t)$$ and $$y(t)=

Polar Coordinates and Dynamics

A point moves along a spiral defined by the polar equation $$r=3\theta$$, where $$\theta$$ is given

Polar Coordinates: Area Between Curves

Consider two polar curves: the outer curve given by $$R(\theta)=4$$ and the inner curve by $$r(\thet

Polar Plots and Intersection Points in Design

A designer creates a pattern using the polar equations $$r=5\cos(θ)$$ and $$r=5\sin(θ)$$. Analyze th

Projectile Motion in Parametric Form

A projectile is launched with an initial speed of $$20\,m/s$$ at an angle of $$30^\circ$$ above the

Projectile Motion with Parametric Equations

A ball is launched from ground level with an initial speed of $$20 \text{ m/s}$$ at an angle of $$\f

Real-World Data Analysis from Tabular Measurements

A vehicle's distance (in meters) along a straight road is recorded at various times (in seconds) as

Slope of a Tangent Line for a Polar Curve

For the polar curve defined by \(r=3+\sin(\theta)\), determine the slope of the tangent line at \(\t

Vector-Valued Integrals in Motion

A particle's acceleration is given by the vector function $$\vec{a}(t)=<\ln(t),\, t^{-1},\, e^{t}>$$