AP Calculus BC FRQ Room

Analysis of Rational Function Asymptotes and Removable Discontinuities

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

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

Bacterial Growth Experiment

A laboratory experiment involves a bacterial culture whose population at hour $$n$$ is modeled by a

Continuity Across Piecewise‐Defined Functions with Mixed Components

Let $$ f(x)= \begin{cases} e^{\sin(x)} - \cos(x), & x < 0, \\ \ln(1+x) + x^2, & 0 \le x < 2, \\

Continuity Analysis from Table Data

The water level (in meters) in a reservoir is recorded at various times as shown in the table below.

Continuity in a Piecewise Function with Polynomial and Trigonometric Components

Consider the function $$f(x)= \begin{cases} x^2-1 & \text{if } x < \pi \\ \sin(x) & \text{if } x \ge

Economic Model of Depreciating Car Value

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

Epsilon-Delta Style (Conceptual) Analysis

Consider the function $$f(x)=\begin{cases} 3*x+2, & x\neq1, \\ 6, & x=1. \end{cases}$$ Answer the

Evaluating a Complex Limit for Continuous Extension

Consider the function $$ f(x)= \begin{cases} \frac{\ln(1+x+e^x) - (x+e^x-1)}{x^2}, & x \neq 0 \\ C,

Factorable Discontinuity Analysis

Let $$q(x)=\frac{x^2-x-6}{x-3}.$$ 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

Identifying and Removing a Discontinuity

Consider the function $$g(x)=\frac{x^2-4}{x-2}$$ for $$x\neq2$$, which is undefined at $$x=2$$.

Intermediate Value Theorem Application

Suppose $$f(x)$$ is a continuous function on the interval $$[1, 5]$$ with $$f(1) = -2$$ and $$f(5) =

Intermediate Value Theorem in Temperature Analysis

A city's temperature during a day is modeled by a continuous function $$T(t)$$, where t (in hours) l

Investigating Limits Involving Nested Rational Expressions

Evaluate the limit $$\lim_{x\to3} \frac{\frac{x^2-9}{x-3}}{x-2}$$. (a) Simplify the expression and e

Investment Portfolio Rebalancing

An investment portfolio is rebalanced periodically, yielding profits that form a geometric sequence.

Limits Involving Exponential Functions

Consider the function $$p(x)=\frac{e^x}{e^x+5}$$.

Limits Involving Trigonometric Ratios

Consider the function $$f(x)= \frac{\sin(2*x)}{x}$$ for $$x \neq 0$$. A table of values near $$x=0$$

Maclaurin Polynomial Approximation and Error Analysis for $$\ln(1+x)$$

Consider the function $$f(x)=\ln(1+x)$$. You are asked to approximate $$f(0.5)$$ using its Maclaurin

Modeling Temperature Change with Continuity

A city’s temperature throughout the day is modeled by the continuous function $$T(t)=\frac{1}{2}t^2-

One-Sided Limits and Jump Discontinuities

Consider the piecewise function $$j(x)=\begin{cases}x+2 & \text{if } x< 3,\\ 5-x & \text{if } x\ge 3

Oscillatory Behavior and Squeeze Theorem

Consider the function $$h(x)= x^2 \cos(1/x)$$ for $$x \neq 0$$ with $$h(0)=0$$.

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 Limit and Continuity

Consider the function $$f(x)=\frac{x^2+5*x+6}{x+3}$$ defined for $$x\neq -3$$. Notice that the funct

Rational Function with Removable Discontinuity

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

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

Resistor Network Convergence

A resistor network is constructed by adding resistors in a ladder configuration. The resistance adde

Trigonometric Rate Function Analysis

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

Using Power Series to Estimate a Trigonometric Function

The power series for $$\sin(x)$$ is $$Q(x)=\sum_{n=0}^{\infty} \frac{(-1)^n*x^{2*n+1}}{(2*n+1)!}.$$

Using the Squeeze Theorem for Trigonometric Limits

Let the function $$f(x)=x^2*\sin(1/x)$$ for x \neq 0 and define f(0)=0. Use the Squeeze Theorem to a

Water Treatment Plant Discontinuity Analysis

A water treatment plant monitors the inflow to a reservoir. Due to sensor calibration, the inflow ra

Acceleration and Jerk in Motion

The position of a car is given by $$s(t)=t^3-6*t^2+9*t+2$$, where $$t$$ is time in seconds and $$s(t

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

Average and Instantaneous Growth Rates in a Bacterial Culture

A bacterial population is modeled by the function $$P(t)= e^{0.3*t} + 10$$, where $$t$$ is measured

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

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

Determining Rates of Change with Secant and Tangent Lines

A function is graphed by the equation $$f(t)=t^3-3*t$$ and its graph is provided. Use the graph to a

Differentiation of Implicitly Defined Functions

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

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

Efficiency Ratio Rate Change Using the Quotient Rule

An efficiency ratio is modeled by $$E(x) = \frac{x^2+2}{3*x-1}$$, where x represents an input variab

Implicit Differentiation with Trigonometric Functions

Consider the curve defined by $$\sin(x*y) = x + y$$.

Instantaneous Versus Average Rates: A Comparative Study

Examine the function $$f(x)=\ln(x)$$. Analyze its average and instantaneous rates of change over a g

Irrigation Reservoir Analysis

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

Marginal Cost Analysis Using Composite Functions and the Chain Rule

A company's cost function is given by $$C(x)= e^{2*x} + \sqrt{x+5}$$, where x (in hundreds) represen

Optimization Using Derivatives

Consider the quadratic function $$f(x)=-x^2+4*x+5$$. Answer the following:

Population Growth Rate

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

Position Recovery from a Velocity Function

A particle moving along a straight line has a velocity function given by $$v(t)=6-3*t$$ (in m/s) for

Product Rule in Differentiation

Suppose the cost function is given by $$Q(x)=(3*x^2 - x)*e^{x}$$, which represents a cost (in dollar

Production Output Rate Analysis Using a Product Function

A factory's production output (in items per hour) is modeled by $$P(t) = t^2*(20 - t)$$, where t (in

Rate Function Involving Logarithms

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

Reconstructing Position from a Velocity Graph

A velocity versus time graph for a moving object is provided in the stimulus. Use the graph to answe

Tangent Line Approximation

Consider the function $$f(x)=\cos(x)$$. Answer the following:

Tracking a Car's Velocity

A car moves along a straight road according to the position function $$s(t)=t^3 - 6*t^2 + 9*t + 2$$,

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

Chain Rule and Quotient Rule for a Rational Composite Function

Let $$f(x)= \frac{(3*x^2 + 2)^4}{(1+x)^{1/2}}$$. Answer the following:

Chain Rule in Oscillatory Motion

A mass-spring system has its displacement modeled by $$ s(t)= e^{-0.5*t}\cos(3*t) $$.

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 Temperature Change in a Chemical Reaction

A chemical reaction in a laboratory is modeled by the composite temperature function $$R(t)= f(g(t))

Composite, Implicit, and Inverse: A Multi-Method Analysis

Let $$F(x)=\sqrt{\ln(5*x+9)}$$ for all x such that $$5*x+9>0$$, and let y = F(x) with g as the inver

Drug Concentration in the Bloodstream

A drug is infused into a patient's bloodstream at a rate given by the composite function $$R(t)=k(m(

Graphical Analysis of a Composite Function

Let $$f(x)=\ln(4*e^{x}+1)$$. A graph of f is provided. Answer the following parts.

Implicit Differentiation in a Chemical Reaction

In a chemical process, the concentrations of two reactants, $$x$$ and $$y$$, satisfy the relation $$

Implicit Differentiation in a Nonlinear Trigonometric Equation

Consider the equation $$ x^2+\sin(x*y)= y^2 $$.

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 Involving a Mixed Function

Consider the equation $$x*e^{y}+y*\ln(x)=10$$, where x > 0 and y is defined implicitly as a function

Implicit Differentiation of an Ellipse

The ellipse is given by $$4*x^2 + 9*y^2 = 36$$.

Implicit Differentiation of an Ellipse

Consider the ellipse defined by $$4*x^2+9*y^2=36$$. Use implicit differentiation to determine the sl

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

Inverse Analysis of a Composite Exponential-Trigonometric Function

Let $$f(x)=e^x+\cos(x)$$. Analyze the behavior of its inverse function under appropriate domain rest

Inverse Function Derivative with Logarithms

Let $$f(x)= \ln(x+2) + x$$ with inverse function $$g(x)$$. Find the derivative $$g'(y)$$ in terms of

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 Radical Function with Domain Restrictions

Consider the function $$f(x)=\sqrt{1-x^2}$$. Analyze its invertibility.

Modeling with Composite Functions: Pollution Concentration

The pollutant concentration in a lake is modeled by $$C(t) = \sqrt{100 - 2*e^{-0.1*t}}$$, where $$t$

Particle Motion with Composite Position Function

A particle moves along a line with its position given by $$s(t)= \sin(t^2)$$, where $$s$$ is in mete

Related Rates in an Inflating Balloon

The volume of a spherical balloon is given by $$V= \frac{4}{3}*\pi*r^3$$, where r is the radius. Sup

Water Tank Composite Rate Analysis

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

Approximating Changes with Differentials

Given that the surface area of a sphere is $$A = 4\pi r^2$$, use differentials to approximate the ch

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

Bacterial Growth and Linearization

A bacterial population is modeled by $$P(t)=100e^{0.3*t}$$, where $$t$$ is in hours. Answer the foll

Biology: Logistic Population Growth Analysis

A population is modeled by the logistic function $$P(t)= \frac{100}{1+ 9e^{-0.5*t}}$$, where $$t$$ i

Cooling Coffee Temperature Change

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

Cubic Curve Linearization

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

Differentials and Function Approximation

Consider the function $$f(x)=x^{1/3}$$. At $$x=8$$, answer the following parts.

Drug Concentration Dynamics

The concentration of a drug in the bloodstream is modeled by $$C(t)= 50*e^{-0.2*t} + 10$$ (in mg/L),

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

Estimating the Rate of Change from Reservoir Data

A reservoir's water level h (in meters) was recorded at different times, as shown in the table below

Exponential Function Inversion

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

GDP Growth Analysis

A country's GDP (in billions of dollars) is modeled by the function $$G(t)=200e^{0.04*t}$$, where t

Graphical Analysis of Derivatives

A function $$f(x)$$ is plotted on the graph provided below. Using this graph, answer the following:

Graphical Interpretation of Slope and Instantaneous Rate

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

Hyperbolic Motion

A particle moves along a path given by the hyperbola $$x*y = 16$$. The particle's position changes w

Infrared Sensor Distance Analysis

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

L'Hospital's Rule in Indeterminate Form Computation

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

Linearization in Finance

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

Linearization of a Radical Function

Consider the function $$f(x)= x^{1/3}$$. Use linearization to approximate function values. Answer th

Linearization to Estimate Change in Electrical Resistance

The resistance of a wire is modeled by $$R(T)=R_0(1+\alpha*T)$$, where $$R_0=100$$ ohms and $$\alpha

Logarithmic Transformation and Derivative Limits

Consider the function $$f(x)=\ln\left(\frac{e^{3x}+1}{1+e^{-x}}\right)$$. Answer the following:

Minimizing Travel Time in Mixed Terrain

A hiker travels from point A to point B. On a flat plain the hiker walks at 5 km/h, but on an uphill

Optimizing Area of a Rectangular Field

A farmer has 100 meters of fencing to enclose three sides of a rectangular field (the fourth side be

Ozone Layer Recovery Simulation

In a simulation of ozone layer dynamics, ozone is produced at a rate of $$I(t)=\frac{25}{t+1}$$ (Dob

Polar Curve: Slope of the Tangent Line

Consider the polar curve defined by $$r(\theta)=10e^{-0.1*\theta}$$.

Revenue Function and Marginal Revenue

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

Analysis of a Decay Model with Constant Input

Consider the concentration function $$C(t)= 30\,e^{-0.5t} + \ln(t+1)$$, where t is measured in hours

Application of Rolle's Theorem

Consider the function $$f(x) = x^2 - 4*x + 4$$ on the interval $$[0,4]$$.

Bouncing Ball with Energy Loss

A ball is dropped from a height of 100 meters. Each time it bounces, it reaches 60% of the height fr

Concavity in an Economic Model

Consider the function $$f(x)= x^3 - 3*x^2 + 2$$, which represents a simplified economic trend over t

Determining the Meeting Point of Two Functions

Consider the functions $$f(x)= e^x$$ and $$g(x)= 3 + \ln(x)$$ representing two different processes.

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

Economic Equilibrium and Implicit Differentiation

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

Economic Production Optimization

A company’s cost function is given by $$C(x) = 0.5*x^3 - 3*x^2 + 4*x + 200$$, where x represents the

Extreme Value Analysis

Consider the function $$f(x) = x^3 - 3*x^2 + 4$$ on the closed interval $$[0,3]$$. Use the Extreme V

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

Implicit Differentiation and Tangent to an Ellipse

Consider the ellipse defined by the equation $$4*x^2 + 9*y^2 = 36$$. Answer the following parts:

Inverse Analysis for a Function with Multiple Transformations

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

Inverse Analysis with a Radical Expression

Let $$f(x)= 3*\sqrt{x+4} - 2$$, defined for $$x \ge -4$$, which models a physical process. Answer th

Investment Portfolio Dividends

A company pays annual dividends that form an arithmetic sequence. The dividend in the first year is

Lake Ecosystem Nutrient Dynamics

In a lake, nutrients (phosphorus) enter at a rate given by $$N_{in}(t)=5*\sin(t)+10$$ mg/min and are

Mean Value Theorem in Temperature Analysis

A city’s temperature is modeled by the function $$T(t)= t^3 - 6*t^2 + 9*t + 5$$ (in °C), where $$t$$

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

Modeling Population Growth: Rate of Change

A population is modeled by the function $$ P(t)=100e^{0.05t}-20t, \quad 0 \le t \le 10,$$ where $$t

Optimization in a Geometric Setting: Garden Design

A farmer is designing a rectangular garden adjacent to a river. No fence is needed along the river s

Optimization in Production Costs

In an economic context, consider the cost function $$C(x)=0.5*x^3-6*x^2+25*x+100$$, where C(x) repre

Parameter-Dependent Concavity Conditions

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

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

Piecewise Function Discontinuities Analysis

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

Projectile Trajectory: Parametric Analysis

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

Road Trip Analysis

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

Rolle's Theorem: Modeling a Car's Journey

An object moves along a straight line and its position is given by $$s(t)= t^3-6*t^2+9*t$$ for $$t$$

Second Derivative Test for Critical Points

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

Series Approximation in Engineering: Oscillation Amplitude

An engineer models the oscillation amplitude by $$A(t)=\sin(0.2*t)\,e^{-0.05*t}$$. Derive the Maclau

Series Convergence and Differentiation in Thermodynamics

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

Series Representation in a Biological Growth Model

A biological process is approximated by the series $$B(t)=\sum_{n=0}^\infty (-1)^n * \frac{(0.3*t)^n

Taylor Series in Economics: Cost Function

An economic cost function is modeled by $$C(x)=1000\,e^{-0.05*x}+50\,x$$, where x represents the pro

Antiderivatives and the Fundamental Theorem of Calculus

Given the function $$f(x)= 2*x+3$$, use the Fundamental Theorem of Calculus to evaluate the definite

Area Under a Piecewise Function

A function is defined piecewise as follows: $$f(x)=\begin{cases} x & 0 \le x \le 2,\\ 6-x & 2 < x \

Determining the Average Value via Integration

Find the average value of the function $$f(x)=3*x^2-2*x+1$$ on the interval $$[1,4]$$.

Determining Velocity and Displacement from Acceleration

A particle's acceleration is given by $$a(t)=4*t-8$$ (in m/s²) for $$0 \le t \le 3$$ seconds. The in

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.

Error Analysis in Riemann Sum Approximations

Consider approximating the integral $$\int_{0}^{2} x^3\,dx$$ using a left-hand Riemann sum with $$n$

Estimating Chemical Production via Riemann Sums

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

Implicit Differentiation Involving an Integral

Consider the relationship $$y^2 + \int_{1}^{x} \cos(t)\, dt = 4$$. Answer the following parts.

Improper Integral and the p-Test

Determine whether the improper integral $$\int_1^{\infty} \frac{1}{x^2}\,dx$$ converges, and if it c

Integration by Substitution and Inverse Functions

Consider the function $$f(x)= (x-4)^2 + 3$$ for $$x \ge 4$$. Answer the following questions about $$

Interpreting the Constant of Integration in Cooling

An object cools according to the differential equation $$\frac{dT}{dt}=-k*(T-20)$$ where $$T(t)$$

Inverse Functions in Economic Models

Consider the function $$f(x) = 3*x^2 + 2$$ defined for $$x \ge 0$$, representing a demand model. Ans

Midpoint Riemann Sum for $$f(x)=\frac{1}{1+x^2}$$

Consider the function $$f(x)=\frac{1}{1+x^2}$$ on the interval $$[-1,1]$$. Use the midpoint Riemann

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

Particle Motion with Variable Acceleration

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

Population Growth from Birth Rate

In a small town, the birth rate is modeled by $$B(t)= \frac{100}{1+t^2}$$ people per year, where $$t

Rainfall Accumulation and Runoff

During a storm, rainfall intensity is modeled by $$R(t)=3*t$$ inches per hour for $$0 \le t \le 4$$

Rainfall Accumulation Over Time

A storm produces rainfall at a rate modeled by the function $$r(t)=6 * t^(1/2)$$, where $$0 \le t \l

Reservoir Water Level

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

Riemann and Trapezoidal Sums with Inverse Functions

Consider the function $$f(x)= 3*\sin(x) + 4$$ defined on the interval \( x \in [0, \frac{\pi}{2}] \)

Riemann Sum Approximations: Midpoint vs. Trapezoidal

Consider the function $$f(x)=e^(-x)$$ on the interval [0, 3]. Compare the approximations for the def

Riemann Sums and Inverse Analysis from Tabular Data

Let $$f(x)= 2*x + 1$$ be defined on the interval $$[0,5]$$. Answer the following questions about $$f

Trapezoidal Rule Error Estimation

Given the function $$f(x)=\ln(x)$$ on the interval $$[1,4]$$, answer the following:

Analyzing a Rational Differential Equation

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

Capacitor Discharge in an RC Circuit

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

Chemical Reaction and Separable Differential Equation

In a particular chemical reaction, the concentration $$C(t)$$ of a reactant decreases according to t

City Population with Migration

The population $$P(t)$$ of a city changes as individuals migrate in at a constant rate of $$500$$ pe

Coffee Cooling: Differential Equation Application

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

Constructing and Interpreting a Slope Field

Consider the differential equation $$\frac{dy}{dx} = \sin(x) - y$$. Answer the following:

Direction Fields and Isoclines

Examine the differential equation $$\frac{dy}{dx} = \frac{x+y}{x-y}$$. A direction field displaying

Direction Fields and Phase Line Analysis

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

Direction Fields and Stability Analysis

Consider the autonomous differential equation $$\frac{dy}{dt}=y(1-y)$$. Answer the following parts.

Dye Dilution in a Stream

A river has dye added at a constant rate of $$0.5$$ kg/min, and the dye is removed at a rate proport

Epidemic Spread Modeling

An epidemic in a closed population of $$N=10000$$ individuals is modeled by the logistic equation $$

Falling Object with Air Resistance

An object of mass $$m$$ falls under gravity, experiencing air resistance proportional to its velocit

FRQ 13: Cooling of a Planetary Atmosphere

A planetary atmosphere cools according to Newton's Law of Cooling: $$\frac{dT}{dt}=-k(T-T_{eq})$$, w

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

Logistic Population Model

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

Mixing in a Chemical Reaction

A solution in a tank is undergoing a chemical reaction described by the differential equation $$\fra

Mixing Problem in a Tank

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

Modeling the Spread of a Disease Using Differential Equations

Suppose the spread of a disease in a population is modeled by the differential equation $$\frac{dI}{

Newton's Law of Cooling

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

Population Dynamics with Harvesting

A species population is modeled by the differential equation $$\frac{dP}{dt} = 0.2*P\left(1-\frac{P}

Population Growth with Harvesting

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

Related Rates: Conical Tank Overflow

A conical tank has a height of $$10\,m$$ and a base radius of $$4\,m$$. Water is draining from the t

RL Circuit Analysis

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

Saltwater Mixing Problem

A tank initially contains 1000 L of a salt solution with a concentration of 0.2 kg/L (thus 200 kg of

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

Temperature Change with Variable Ambient Temperature

A heated object is cooling in an environment where the ambient temperature changes over time. For $$

Traffic Flow on a Highway

A highway segment experiences an inflow of cars at a rate of $$200+10*t$$ cars per minute and an out

Variable Carrying Capacity in Population Dynamics

In a modified logistic model, the carrying capacity of a population is time-dependent and given by $

Analysis of Particle Motion in the Plane

A particle’s position is given by the vector function $$\mathbf{r}(t)=\langle e^{-t},\,\sin(t)\rangl

Arc Length in Polar Coordinates

Find the length of the curve defined in polar coordinates by $$r(θ)= 1+ \cos(θ)$$ for $$θ \in [0, 2\

Arc Length of a Cable

A cable hanging in a particular configuration follows the curve $$y=\ln(x+1)$$ for $$x\in[0,4]$$. De

Arc Length of a Parabolic Curve

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

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: Supply and Demand Analysis

In an economic model, the supply and demand functions for a product (in hundreds of units) are given

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 of One Petal of a Polar Curve

The polar curve defined by $$r = \cos(2\theta)$$ forms a rose with four petals. Find the area of one

Average Temperature Over a Day

The temperature in a city over a 24-hour period is modeled by $$T(t)=10+5\sin\left(\frac{\pi}{12}*t\

Average Value of a Piecewise Function

Consider the function $$g(x)$$ defined piecewise on the interval $$[0,6]$$ by $$g(x)=\begin{cases} x

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

Center of Mass of a Lamina with Constant Density

A thin lamina occupies the region in the first quadrant bounded by $$y=x^2$$ and $$y=4$$. The densit

Center of Mass of a Rod

A thin rod of length 10 m has a linear density given by $$\rho(x)=3+0.4*x$$ (in kg/m) where $$x$$ is

Consumer Surplus Analysis

The demand function for a product is given by $$D(p)=120-2*p$$, where \(p\) is the price in dollars.

Distance Traveled versus Displacement

A particle moves along a line with velocity given by $$v(t)=t^3 - 6*t^2 + 9*t$$ (in m/s) for $$t\in[

Drug Concentration Profile Analysis

The functions $$f(t)=5*t+10$$ and $$g(t)=2*t^2+3$$ (where t is in hours and concentration in mg/L) r

Economic Analysis: Consumer and Producer Surplus

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

Electrical Charge Distribution

A wire of length 3 meters carries a charge distribution given by $$\rho(x)=\frac{5}{1+x^2}$$ (in cou

Environmental Contaminant Spread Analysis

A contaminant enters a lake at a rate given by $$r(t)=4e^{-0.5*t}$$ kilograms per day, where $$t$$ i

Inverse Function Analysis

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

Mass of a Wire with Variable Density

A thin wire lies along the curve $$y= \sqrt{x}$$ for $$0 \le x \le 4$$. The wire has a linear densit

Polar Coordinates: Area of a Region

A region in the plane is described in polar coordinates by the equation $$r= 2+ \cos(θ)$$. Determine

Population Change via Rate Integration

A small town’s population changes at a rate given by $$P'(t)=100*e^{-0.3*t}$$ (persons per year) wit

Rainfall Accumulation Analysis

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

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 Solid of Revolution

Consider the curve $$y= \sqrt{x}$$ over the interval $$0 \le x \le 4$$. When this curve is rotated a

Temperature Modeling: Applying the Average Value Theorem

The temperature of a chemical solution in a tank is modeled by $$T(t)=20+5\cos(0.5*t)$$ (°C) for $$t

Volume by Cross‐Sectional Area in a Variable Tank

A tank has a variable cross‐section. For a water level at height $$y$$ (in cm), the width of the tan

Volume of a Region via Washer Method

The region in the first quadrant bounded by the curves $$y=x$$ and $$y=x^2$$ is rotated about the x-

Volume of a Solid Obtained by Rotation

Consider the region bounded by the curves $$y=\sqrt{x}$$ and $$y=\frac{x}{2}$$. This region is rotat

Volume of a Solid Using the Shell Method

The region in the first quadrant bounded by $$y=x$$, $$y=0$$, and $$x=4$$ is rotated about the y-axi

Work Done by a Variable Force

A force acting on an object along a displacement is given by $$F(x)=3*x^2 -2*x+1$$ (in Newtons), whe

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 Concavity for a Polar Function

Consider the polar function given by \(r=5-2\sin(\theta)\). Answer the following:

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 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

Arc Length of a Cycloid

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

Arc Length of a Parametric Curve

Consider the parametric curve defined by $$ x(t)=t^2 $$ and $$ y(t)=t^3 $$ for $$ 0 \le t \le 2 $$.

Arc Length of a Parametric Curve

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

Arc Length of a Parametric Curve

Consider the parametric equations $$x(t) = t^2$$ and $$y(t) = t^3$$ for $$0 \le t \le 2$$.

Arc Length of a Polar Curve

Consider the polar curve given by $$r(\theta)=1+\frac{\theta}{\pi}$$ for $$0 \le \theta \le \pi$$. A

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 of a Parametric Curve

Consider the curve defined by the parametric equations $$x(t)=\ln(t)$$ and $$y(t)=t^2$$ for \(t>0\).

Drone Altitude Measurement from Experimental Data

A drone’s altitude (in meters) is recorded at various times (in seconds) as shown in the table below

Exponential-Logarithmic Particle Motion

A particle moves in the plane with its position given by the parametric equations $$x(t)=e^{t}+\ln(t

Integrating a Vector-Valued Function

A particle has a velocity given by $$\vec{v}(t)= \langle e^t, \cos(t) \rangle$$. Its initial positio

Modeling Projectile Motion with Parametric Equations

A projectile is launched with an initial speed of \(20\) m/s at an angle of \(45^\circ\) above the h

Modeling with Polar Data

A researcher collects the following polar coordinate data for a phenomenon.

Optimization of Walkway Slope with Fixed Arc Length

A walkway is designed with its shape given by the parametric equations $$x(t)= t$$ and $$y(t)= c*t*(

Parameter Values from Tangent Slopes

A curve is defined parametrically by $$x(t)=t^2-4$$ and $$y(t)=t^3-3t$$. Answer the following:

Parametric Curve Intersection

Two curves are defined parametrically as follows: For curve C1, $$x(t) = t^2$$ and $$y(t) = 2*t + 1$

Parametric Equations and Intersection Points

Consider the curves defined parametrically by $$x_1(t)=t^2, \; y_1(t)=2t$$ and $$x_2(s)=s+1, \; y_2(

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

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 Differentiation and Tangent Lines

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

Projectile Motion via Vector-Valued Functions

A projectile is launched from the origin with an initial velocity given by \(\mathbf{v}(0)=\langle 5

Spiral Motion in Polar Coordinates

A particle moves in polar coordinates with \(r(\theta)=4-\theta\) and the angle is related to time b

Tangent Line to a Parametric Curve

Consider the circle parametrized by $$x(t)=3\sin(t)$$ and $$y(t)=3\cos(t)$$ for $$0\le t\le 2\pi$$.

Time of Nearest Approach on a Parametric Path

An object travels along a path defined by $$x(t)=5*t-t^2$$ and $$y(t)=t^3-6*t$$ for $$t\ge0$$. Answe

Vector-Valued Fourier Series Representation

The vector function $$\mathbf{r}(t)=\langle \cos(t), \sin(t), 0 \rangle$$ for $$t \in [-\pi,\pi]$$ c

Vector-Valued Function Analysis

Let the vector-valued function be given by $$\vec{r}(t)=<e^{t},\, \sin(t),\, \cos(t)>$$ for $$0\leq

Vector-Valued Functions and Kinematics

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

Vector-Valued Integrals in Motion

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