AP Calculus BC FRQ Room

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

Complex Rational Function and Continuity Analysis

Let $$R(x)= \frac{x^3-8}{x-2}$$ for $$x \neq 2$$.

Continuity and the Intermediate Value Theorem in Temperature Modeling

A temperature function $$T(t)$$, continuous for $$t\in[0,24]$$ (in hours), satisfies $$T(6)=10^\circ

Continuity of Log‐Exponential Function

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

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 $$f(0)=L$$.

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,

Evaluating Limits Involving Radical Expressions

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

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:

Graphical Analysis of Discontinuities

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

Implicitly Defined Function and Differentiation

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

Indeterminate Forms in Log‐Exponential Context

Consider the limit $$\lim_{x \to 0} \frac{e^{\sin(x)} - 1}{\ln(1+x)}.$$

Intermediate Value Theorem Application

Let $$g(x)=x^3+2*x-1$$ be defined on the interval [0, 1].

Investigating a Function with a Removable Discontinuity

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

Limits and Removable Discontinuity in Rational Functions

Consider the rational function $$g(x) = \frac{(x-2)(x+3)}{x-2}.$$ Use this expression to answer the

Limits from Table and Graph

A function $$g(x)$$ is studied in a lab experiment. The following table gives sample values of $$g(x

Limits Involving Exponential Functions

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

Logarithmic Function Limits

Consider the function $$f(x)=\frac{\ln(1+3*x)}{x}$$ for $$x \neq 0$$. Answer the following:

Modeling with a Removable Discontinuity

A water reservoir’s inflow rate is modeled by the function $$R(t)=\frac{t^2-16}{t-4}$$ liters per mi

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

Parameter Determination for Continuity

Let $$f(x)= \frac{x^2-1}{x-1}$$ for $$x \neq 1$$, and suppose that $$f(1)=m$$. Answer the following:

Parameter Determination for Continuity

Let $$h(x)= \begin{cases} \frac{e^{2x} - 1 - a\,\ln(1+bx)}{x} & x \neq 0 \\ c & x = 0 \end{cases}.$$

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

Removable Discontinuity in a Trigonometric Function

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

Sine over x Function with Altered Value

Consider the function $$ f(x)=\begin{cases} \frac{\sin(x)}{x} & \text{if } x \neq 0, \\ 3 & \text{i

Squeeze Theorem Application

Consider the function $$f(x)=\frac{\sin(3*x)}{x}$$ defined for x ≠ 0.

Water Filling a Leaky Tank

A water tank is initially empty. Every minute, 10 liters of water is added to the tank, but due to a

Water Treatment Plant Discontinuity Analysis

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

Applying the Quotient Rule

Let the function $$R(x)=\frac{x^2+1}{2*x-1}$$ represent a ratio used to gauge the rate of return on

Car Motion: Velocity and Acceleration

A car’s position along a straight road is given by $$s(t)=t^3-9*t$$, where $$t$$ is in seconds and $

Cooling Model Rate Analysis

The temperature of a cooling object is modeled by $$T(t)=e^{-2*t}+\ln(t+3)$$, where $$t$$ is time in

Derivative from First Principles: Reciprocal Function

Let $$f(x)= \frac{1}{x}$$.

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

Derivatives of a Rational Function

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

Differentiation in Polar Coordinates

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

Estimating Temperature Change

A scientist recorded the temperature of a liquid at different times (in minutes) as it was heated. U

Finding and Interpreting Critical Points and Derivatives

Examine the function $$f(x)=x^3-9*x+6$$. Determine its derivative and analyze its critical points.

Icy Lake Evaporation and Refreezing

An icy lake gains water from melting ice at a rate of $$M_{in}(t)=5+0.2*t$$ liters per hour and lose

Implicit Differentiation in a Geometric Context

Consider the circle defined by the equation $$x^2+y^2=25.$$ (a) Use implicit differentiation to f

Implicit Differentiation: Square Root Equation

Consider the curve defined by $$\sqrt{x} + \sqrt{y} = \sqrt{10}$$, where $$x, y \ge 0$$.

Instantaneous vs. Average Rate of Change

Consider the trigonometric function $$f(x)= \sin(x)$$.

Limit Definition of the Derivative for a Trigonometric Function

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

Logarithmic Differentiation in Temperature Modeling

The temperature distribution along a rod is modeled by the function $$T(x)=\ln(5*x^2+1)*e^{-x}$$. He

Logarithmic Differentiation: Equating Powers

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

Maclaurin Series for ln(1+x)

A scientist modeling logarithmic growth wishes to approximate the function $$\ln(1+x)$$ around $$x=0

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

Population Growth Approximation using Taylor Series

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

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

Projectile Trajectory: Rate of Change Analysis

The height of a projectile is given by $$h(t)= -4.9t^2 + 20t + 1.5$$ in meters, where t is in second

Radioactive Decay and Derivative

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

Rainwater Harvesting System

A rainwater harvesting system collects water with an inflow rate of $$R_{in}(t)=100+25\sin((\pi*t)/1

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,

Second Derivative Test and Stability

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

Tangent Line Estimation in Transportation Modeling

A vehicle's displacement along a highway is modeled by $$s(t)=\ln(3*t+1)*e^{t}$$, where $$t$$ denote

Temperature Change Rate

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

Testing Differentiability at a Junction Point

Consider the function $$f(x)= \begin{cases} x+2 & x < 3 \\ 5 & x = 3 \\ 2*x-1 & x > 3 \end{cases}$$.

Water Tank: Inflow-Outflow Dynamics

A water tank initially contains $$1000$$ liters of water. Water enters the tank at a rate of $$R_{in

Water Treatment Plant Simulator

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

Bacterial Culture: Nutrient Inflow vs Waste Outflow

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

Chain Rule and Implicit Differentiation in a Pendulum Oscillation Experiment

In a pendulum experiment, the angle \(\theta(t)\) in radians satisfies the relation $$\cos(\theta(t)

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 with Trigonometric Composite Function

Examine the function $$ h(x)= \sin((2*x^2+1)^2) $$.

Coffee Cooling Dynamics using Inverse Function Differentiation

A cup of coffee cools according to the model $$T=100-a\,\ln(t+1)$$, where $$T$$ is the temperature i

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))

Differentiation of a Product Involving Inverse Trigonometric Components

Let $$m(x)=\arctan(x)+x*\arcsin\left(\frac{x}{2}\right)$$, where the domain of arcsin requires $$-2\

Differentiation of an Inverse Function

Let f be a differentiable and one-to-one function with inverse $$f^{-1}$$. Suppose that $$f(3)=7$$ a

Differentiation of Composite Exponential and Trigonometric Functions

Let $$f(x) = e^{\sin(x^2)}$$ be a composite function. Differentiate $$f(x)$$ and simplify your answe

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 a Circle

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

Implicit Differentiation in an Economic Cost Function

A cost function \(C(q)\) satisfies the relation $$q^2 + q*\sqrt{C(q)} - C(q) = 0$$, where \(q\) repr

Implicit Differentiation of a Product and Composite Function

Consider the equation $$x^2*\sin(y)+e^{y}=x$$, which defines y implicitly as a function of x. Answer

Implicit Differentiation with an Exponential Function

Given the equation $$ e^{x*y}= x+y $$, use implicit differentiation.

Implicit Differentiation with Logarithmic Functions

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

Implicit Differentiation: Second Derivative of Exponential-Trigonometric Equation

Consider the equation $$e^{x*y} + \sin(y) - x^2 = 0$$ where $$y$$ is defined implicitly as a functio

Infinite Series in a Financial Deposit Model

An investor makes monthly deposits that follow a geometric sequence, with the deposit in the nth mon

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 Differentiation in a Sensor

A sensor produces a reading described by the function $$f(t)= \ln(t+1) + t^2$$, where $$t$$ is in se

Inverse Trigonometric Differentiation

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

Logarithmic and Composite Differentiation

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

Multi-step Differentiation of a Composite Logarithmic Function

Consider the function $$F(x)= \sqrt{\ln\left(\frac{1+e^{2*x}}{1-e^{2*x}}\right)}$$, defined for valu

Parameter Dependent Composite Function

The temperature in a metal rod is modeled by $$T(x)= e^{a*x}$$, where the parameter $$a$$ changes wi

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

Parametric Equations and the Chain Rule

A particle moves in the plane according to the parametric equations $$x(t)= e^{2*t}$$ and $$y(t)= \l

Polar and Composite Differentiation: Arc Slope for a Polar Curve

Consider the polar curve $$r(\theta)=2+\cos(\theta)$$. Answer the following parts:

Reservoir Level: Inverse Function Application

A reservoir's water level $$h$$ (in feet) is related to time $$t$$ (in minutes) through an invertibl

Taylor/Maclaurin Polynomial Approximation for a Logarithmic Function

Let $$f(x) = \ln(1+3*x)$$. Develop a second-degree Maclaurin polynomial, determine its radius of con

Analyzing Motion on an Inclined Plane

A sled moves along an inclined plane with displacement given by $$s(t)=5*t^2-0.5*t^3$$, where $$s$$

Circular Motion and Angular Rate

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

Complex Limit Analysis with L'Hôpital's Rule

Evaluate the limit $$\lim_{x \to 0} \frac{e^{2*x} - 1 - 2*x}{x^2}$$ using L'Hôpital's Rule. Answer t

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 Function with Parameter and Its Inverse

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

Differentials and Function Approximation

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

Differentials in Engineering: Beam Stress Analysis

The stress S (in Pascals) experienced by an engineering beam under load is modeled by $$S(x)=0.02*x^

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),

Estimating Rates from Experimental Position Data

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

Implicit Differentiation: Tangent to a Circle

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

Instantaneous vs. Average Speed in a Race

An athlete’s displacement during a 100 m race is modeled by $$s(t)=2*t^3-t^2+1$$, where $$s(t)$$ is

Interpreting Position Graphs: From Position to Velocity

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

Limit Evaluation Using L'Hôpital's Rule

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

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

Optimization with Material Costs

A company plans to design an open-top rectangular box with a square base that must have a volume of

Particle Motion Analysis

A particle moves along a straight line and its position at time $$t$$ seconds is given by $$s(t)= t^

Pollutant Scrubber Efficiency

A factory emits pollutants at a rate given by $$I(t)=100e^{-0.3t}$$ (kg per hour), and a scrubber re

Projectile Motion Analysis

A projectile is launched such that its horizontal and vertical positions are modeled by the parametr

Rational Function Inversion

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

River Flow Diversion

At a river junction, water flows in at a rate of $$I(t)=30+5t$$ (cubic feet per second) and exits at

Savings Account and Interest Accrual

A student starts with an initial savings account balance of $$B_0=1000$$ dollars and makes monthly d

Series Analysis in Acoustics

The sound intensity at a distance is modeled by $$I(x)= I_0 \sum_{n=0}^{\infty} \frac{(-1)^n (x-10)^

Series-Based Analysis of Experimental Data

An experiment models a measurement function as $$g(x)= \sum_{n=0}^{\infty} \frac{(-1)^n (x/4)^n}{n+1

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

Amusement Park Ride Braking Distance

An amusement park ride uses a sequence of friction pads to stop a roller coaster. The first pad diss

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

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

Arc Length Approximation

Let $$f(x) = \sqrt{x}$$ be defined on the interval [1,9].

Chemical Reaction Rate

During a chemical reaction, the concentration of a reagent (in M) is measured over time (in minutes)

Concavity and Inflection Points

Let $$f(x)=x^3-6x^2+9x+2.$$ Answer the following parts:

Concavity in an Economic Model

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

Exponential Growth and Logarithmic Transformation

A bacteria population is modeled by $$P(t)= A*e^{k*t}$$, where $$t$$ is measured in hours, A is the

Fuel Consumption in a Generator

A generator operates with fuel being supplied at a constant rate of $$S(t)=5$$ liters/hour and consu

Inverse Analysis of a Cubic Polynomial

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

Investment Portfolio Dividends

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

Linear Approximation and Differentials

Let \( f(x) = \sqrt{x} \). Use linear approximation to estimate \( \sqrt{10} \). Answer the followin

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

Rate of Change and Inverse Functions

Let $$f(x)=x^3 + 3*x + 1$$, which is one-to-one. Investigate the rate of change of \(f(x)\) and its

Roller Coaster Height Analysis

A roller coaster's height (in meters) as a function of time (in seconds) is given by $$h(t) = -0.5*t

Tangent Line and Linearization

Consider the function $$ f(x)=\sqrt{x+5}.$$ Answer the following parts:

Taylor Series for $$\sqrt{1+x}$$

Consider the function $$f(x)=\sqrt{1+x}$$. In this problem, compute its 3rd degree Maclaurin polynom

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

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

Wireless Signal Attenuation

A wireless signal, originally at an intensity of 80 units, passes through a series of walls. Each wa

Accumulation Function from a Rate Function

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

Advanced U-Substitution with a Quadratic Expression

Evaluate the indefinite integral $$\int \frac{2*x}{\sqrt{x^{2}+1}}\,dx$$ using u-substitution.

Area Between Curves

Consider the curves $$y=x^2$$ and $$y=4x-x^2$$.

Area Estimation Using Riemann Sums for $$f(x)=x^2$$

Consider the function $$f(x)=x^2$$ on the interval $$[1,4]$$. A table of computed values for the lef

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 Value and Accumulated Change

For the function $$f(x)= x^2+1$$ defined on the interval [0, 4], find the average value of the funct

Consumer Surplus in an Economic Model

For a particular product, the demand function is given by $$D(p)=100 - 5p$$ and the supply function

Consumer Surplus via Integration

In an economic model, the demand function is given by $$p(x)= 20 - 0.5*x$$, where p is the price in

Cost Accumulation from Marginal Cost Function

A company’s marginal cost function $$MC(q)$$ (in dollars per unit) for producing $$q$$ units is give

Estimating Integral from Tabular Data

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

Evaluating a Complex Integral

Evaluate the integral $$\int_{0}^{2} 2*x*(x+1)^3\,dx$$ using an appropriate substitution method.

Improper Integral Evaluation

Evaluate the integral $$\int_{1}^{\infty} \frac{1}{t^2}\, dt$$ by answering the following parts.

Integration by U-Substitution and Evaluation of a Definite Integral

Evaluate the definite integral $$\int_{0}^{1} \frac{2*t}{(t^2+1)^2}\, dt$$ by applying U-substitut

Integration of a Trigonometric Function by Two Methods

Evaluate the definite integral $$\int_0^{\frac{\pi}{2}} \sin(x)*\cos(x)\,dx$$ using two different me

Midpoint Approximation Analysis

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

Particle Motion with Changing Velocity Signs

A particle is moving along a line with its velocity given by $$v(t)= 6 - 4*t$$ (in m/s) for t betwee

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

Rainfall Accumulation and Runoff

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

Reservoir Water Level

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

Revenue Accumulation and Constant of Integration

A company's revenue is modeled by $$R(t) = \int_{0}^{t} 3*u^2\, du + C$$ dollars, where t (in years)

Riemann Sum Approximation with Irregular Intervals

A set of experimental data provides the values of a function $$f(x)$$ at irregularly spaced points a

Riemann Sum from a Table: Plant Growth Data

A function $$f(t)$$ describes the height (in cm) of a plant over time (in days). The measurements ar

Total Rainfall Accumulation from a Discontinuous Rate Function

Rain falls at a rate (in mm/hr) given by $$ R(t)= \begin{cases} 3t, & 0 \le t < 2, \\ 5, & t = 2, \\

Water Volume Accumulation with a Faulty Sensor Reading

Water flows into a container at a rate given by $$ r(t)= \begin{cases} 4t, & 0 \le t < 3, \\ 10, & t

Bacteria Growth with Antibiotic Treatment

A bacterial culture has a population $$N(t)$$ that grows at a rate proportional to its size, given b

Chain Reaction in a Nuclear Reactor

A simplified model for a nuclear chain reaction is given by the logistic differential equation $$\fr

Chemical Reaction in a Closed System

The concentration $$C(t)$$ of a reactant in a closed system decreases according to the differential

Chemical Reactor Mixing

In a chemical reactor, the concentration $$C(t)$$ (in M) of a chemical is governed by the equation $

Cooling and Mixing Combined Problem

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

Cooling of an Electronic Component

An electronic component overheats and its temperature $$T(t)$$ (in $$^\circ C$$) follows Newton's La

Economic Model: Differential Equation for Cost Function

A company’s marginal cost is represented by $$MC(x)=\frac{dC}{dx}=3+2*x$$, where $$x$$ is the number

Exponential Growth and Decay

A bacterial population grows according to the differential equation $$\frac{dy}{dt}=k\,y$$ with an i

Exponential Growth via Slope Field Analysis

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

FRQ 6: Radioactive Decay

A radioactive substance decays according to the differential equation $$\frac{dN}{dt}=-\lambda * N$$

FRQ 7: Projectile Motion with Air Resistance

A projectile is launched vertically upward with an initial velocity of 50 m/s. Its vertical motion i

FRQ 8: RC Circuit Analysis

In an RC circuit, the voltage across the capacitor, $$V(t)$$, satisfies the differential equation $$

FRQ 20: Epidemic Decay with Intervention

After strict intervention measures, the number of active cases in an epidemic decays according to th

Implicit Differential Equations and Slope Fields

Consider the implicit differential equation $$x\frac{dy}{dx}+ y = e^x$$. Answer the following parts.

Implicit Differentiation from an Implicitly Defined Relation

Consider the implicit equation $$x^3 + y^3 - 3*x*y = 0$$ which defines $$y$$ as a function of $$x$$

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 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

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

Maclaurin Series Solution for a Differential Equation

Given the differential equation $$\frac{dy}{dx} = y * \cos(x)$$ with initial condition $$y(0)=1$$, f

Mixing Problem in a Tank

A tank initially contains a certain amount of salt dissolved in 100 L of water. Brine with a known s

Mixing Problem with Constant Rates

A tank contains $$200\,L$$ of a well-mixed saline solution with $$5\,kg$$ of salt initially. Brine w

Modeling Ambient Temperature Change

The ambient temperature $$T(t)$$ of a city changes according to the differential equation $$\frac{dT

Modeling Cooling in a Variable Environment

Suppose the cooling of a heated object is modeled by the differential equation $$\frac{dT}{dt} = -k*

Motion Under Gravity with Air Resistance

An object falling under gravity experiences air resistance proportional to its velocity. Its motion

Oscillatory Behavior in Differential Equations

Consider the second-order differential equation $$\frac{d^2y}{dx^2}+y=0$$, which describes simple ha

Parametric Equations and Differential Equations

A particle moves in the plane along a curve defined by the parametric equations $$x(t)=\ln(t)$$ and

Phase-Plane Analysis of a Nonlinear Differential Equation

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

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

Rainfall in a Basin: Differential Equation Model

During a rainstorm, the depth of water $$h(t)$$ (in centimeters) in a basin is modeled by the differ

Reservoir Contaminant Dilution

A reservoir has a constant volume of 10,000 L and contains a pollutant with amount $$Q(t)$$ (in kg)

Salt Tank Mixing Problem

A tank contains $$100$$ L of water with $$10$$ kg of salt. Brine containing $$0.5$$ kg of salt per l

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

Consider the differential equation $$\frac{dy}{dx}= \frac{x}{1+y^2}$$, which is defined for all real

Separable Differential Equation and Slope Field Analysis

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

Temperature Control in a Chemical Reaction Vessel

In a chemical reactor, the temperature $$T(t)$$ is regulated by both natural cooling and an external

Area Between a Parabola and a Line

Consider the curves given by $$y=5*x-x^2$$ and $$y=x$$. These curves intersect at certain $$x$$-valu

Area Between Exponential Curves

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

Area Between Two Curves: Parabola and Line

Consider the functions $$f(x)= 5*x - x^2$$ and $$g(x)= x$$. These curves enclose a region in the pla

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 Analysis

A meteorological station recorded the temperature in a region as a function of time given by $$T(t)

Average Velocity and Displacement from a Polynomial Function

A car's velocity in m/s is given by $$v(t)=t^2-4*t+3$$ for $$t\in[0,5]$$ seconds. Answer the followi

Center of Mass of a Thin Rod

A thin rod extends from $$x=0$$ to $$x=4$$ m and has a density function $$\lambda(x)=1+\frac{\ln(x+2

Cost Analysis: Area Between Production Cost Curves

Suppose two cost functions for producing goods are given by $$f(x)=20+2*x$$ and $$g(x)=5*x-\frac{1}{

Cost Function from Marginal Cost

A manufacturing process has a marginal cost function given by $$MC(q)=3*\sqrt{q}$$, where $$q$$ (in

Electrical Charge Distribution

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

Particle Motion: Position, Velocity, and Acceleration

A particle moves along a straight line with acceleration $$a(t)=4-2*t$$ (in m/s²), initial velocity

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

Probability Density Function and Expected Value

A continuous random variable $$X$$ has a probability density function defined by $$f(x)=k*x$$ for $$

Salt Concentration in a Mixing Tank

A tank initially contains 50 L of water with 5 g of salt. A salt solution with a concentration of 0.

Solid of Revolution using Washer Method

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

Surface Area of a Rotated Curve

Consider the curve $$y=x^3$$ on the interval $$[0,2]$$. This curve is rotated about the x-axis, form

Volume by Shell Method: Rotated Parabolic Region

Consider the region in the first quadrant bounded by the curve $$y=x^2$$ and the horizontal line $$y

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 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 Using Washer Method

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

Volume with Square Cross Sections

The region in the $$xy$$-plane is bounded by $$y=\sqrt{x}$$, $$y=0$$, and $$x=4$$. A solid is formed

Work Done by a Variable Force

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

Work Done by a Variable Force

A variable force acting along a straight line is given by $$F(x) = \frac{10}{x+2}$$ (in Newtons). Fi

Work Done with a Discontinuous Force Function

A force acting on an object is defined piecewise by $$F(x)=\begin{cases} x^2, & 0 \le x < 4,\\ 16, &

Work to Pump Water from a Tank

A cylindrical tank of radius 2 ft and height 10 ft is partially filled with water to a depth of 8 ft

Arc Length of a Decaying Spiral

Consider the parametric equations $$x(t)= e^{-t}\cos(t)$$ and $$y(t)= e^{-t}\sin(t)$$ for $$t \ge 0$

Arc Length of a Parametric Curve

Consider the curve defined by the parametric equations $$x(t)=3*\cos(t)$$ and $$y(t)=3*\sin(t)$$ for

Arc Length of a Parametric Curve

Consider the parametric curve defined by $$x(t)= t^2$$ and $$y(t)= t^3$$ for $$0 \le t \le 1$$. Anal

Arc Length of a Polar Curve

Consider the polar curve given by $$r=2+\cos(\theta)$$ for $$0\le \theta \le \pi$$. Answer the follo

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

Area and Tangent for a Polar Curve

The polar curve is defined by $$r = 2+\cos(\theta)$$.

Area Enclosed by a Polar Curve

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

Area of a Region in Polar Coordinates with an Internal Boundary

Consider a region bounded by the outer polar curve $$R(\theta)=5$$ and the inner polar curve $$r(\th

Conversion of Parametric to Polar: Motion Analysis

An object moves along a path given by the parametric equations $$x(t)=t^2$$ and $$y(t)=2*t$$ for $$t

Curvature of a Parametric Curve

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

Differentiability of a Piecewise-Defined Vector Function

Consider the vector-valued function $$\textbf{r}(t)= \begin{cases} \langle t, t^2 \rangle & \text{i

Finding the Slope of a Tangent to a Parametric Curve

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

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 of Parametric Curves

Consider the parametric curves $$C_1$$ given by $$x(t)= t^2,\; y(t)= 2t$$ and $$C_2$$ given by $$x(s

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 Equations and Tangent Lines

A curve is defined parametrically by $$x(t)=t^3-3t$$ and $$y(t)=t^2+2$$, where $$t$$ is a real numbe

Parametric Equations from Real-World Data

A moving vehicle’s position is modeled by the parametric equations $$ x(t)=3*t+1 $$ and $$ y(t)=t^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

Parametric Motion Analysis

A particle moves with its position given by the parametric equations $$x(t) = t^2 - 4*t$$ and $$y(t)

Parametric Particle Motion

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

Parametric Plotting and Cusps

Let the parametric equations be $$ x(t)=t-\sin(t) $$ and $$ y(t)=1-\cos(t) $$ for $$ 0 \le t \le 2\p

Particle Motion in Circular Motion

A particle moves along a circular path given by the parametric equations $$x(t)= 5\cos(t)$$ and $$y(

Particle Motion in the Plane

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

Particle Motion in the Plane

Consider a particle whose motion in the plane is defined by the parametric equations $$x(t) = t^2 -

Periodic Motion: A Vector-Valued Function

A point moves on a circle with position given by $$\vec{r}(t)= \langle \cos(2t), \sin(2t) \rangle$$

Polar to Cartesian Conversion

Consider the polar curve defined by $$r = 4*\cos(\theta)$$.

Projectile Motion with Parametric Equations

An object is launched with projectile motion as described by the parametric equations $$x(t)=50*t$$

Tangent Line to a Parametric Curve

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

Tangent Line to a Polar Curve

Consider the polar curve $$r=5-2\cos(\theta)$$. Answer the following parts.

Vector Fields and Particle Trajectories

A particle moves in the plane with velocity given by $$\vec{v}(t)=\langle \frac{e^{t}}{t+1}, \ln(t+2

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: Velocity and Acceleration

A particle's position is given by the vector-valued function $$\vec{r}(t) = \langle \sin(t), \ln(t+1

Vector-Valued Kinematics

A particle follows a path in space described by the vector-valued function $$r(t) = \langle \cos(t),

Weather Data Analysis from Temperature Table

A meteorologist records the temperature (in $$^\circ C$$) at a weather station at various times (in