AP Calculus BC FRQ Room

Analysis of a Rational Inflow Function with a Discontinuity

A water tank is monitored by an instrument that records the inflow rate as $$R(t)=\frac{t^2-9}{t-3}$

Composite Function in Water Level Modeling

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

Compound Interest and Loan Repayment

A simplified model for a loan repayment assumes that a borrower owes $$10,000$$ dollars and the rema

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 Involving Logarithmic and Polynomial Expressions

Consider the function $$f(x)= \begin{cases} \frac{\ln(x+2)}{x} & \text{if } x<0 \\ (x+1)^2 & \text{i

Continuity in Composition of Functions

Let $$g(x)=\frac{x^2-4}{x-2}$$ for x ≠ 2 and undefined at x = 2, and let f(x) be a continuous functi

Epsilon-Delta Proof for a Polynomial Function

Let $$f(x)=x^2+3*x+2$$. Answer the following:

Factorization and Limits

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

Identifying and Removing Discontinuities

The function $$f(x)=\frac{x^2-9}{x-3}$$ is undefined at x = 3.

Implicitly Defined Curve and Its Tangent Line

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

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 a Continuous Function

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

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

Limits and Continuity in Particle Motion

A particle moves along a straight line with velocity given by $$v(t)=\frac{t^2-4}{t-2}$$ for t ≠ 2 s

Limits with Composite Logarithmic Functions

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

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 and Jump Discontinuities

Consider the piecewise function defined by: $$ f(x)=\begin{cases} 2-x, & x<1\\ 3*x-1, & x\ge1 \en

Piecewise Function Continuity and Differentiability

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

Piecewise Function Critical Analysis

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

Population Growth and Limits

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

Radical Function Limit via Conjugate Multiplication

Consider the function $$f(x)=\frac{\sqrt{2*x+9}-3}{x}$$ defined for $$x \neq 0$$. Answer the followi

Seasonal Temperature Curve Analysis

A graph represents the average daily temperature (in $$^\circ C$$) as a function of the day of the y

Water Treatment Plant Discontinuity Analysis

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

Analyzing Car Speed from a Distance-Time Table

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

Applying Product and Quotient Rules

For the function $$h(x)=\frac{(3*x^2+2)*(x-4)}{x+1}$$, determine its derivative by appropriately app

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

Average vs Instantaneous Rates

Consider the function $$f(x)=\frac{\sin(x)}{x}$$ for \(x\neq0\), with $$f(0)=1$$. Answer the followi

Bacteria Culturing in a Bioreactor

In a bioreactor, the bacterial inflow (growth) rate is given by $$B_{in}(t)=\frac{15}{1+e^{-0.3*(t-5

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

Car Motion and Critical Velocity

The position of a car is described by $$s(t)=t^3 - 12*t^2 + 36*t$$ (with $$s$$ in meters and $$t$$ i

Chain Rule Verification with a Power Function

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

Derivative of a Composite Function Using the Limit Definition

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

Derivative of a Function Involving an Absolute Value

Consider the function $$f(x)=|x-3|+2$$. Answer the following:

Derivative Using Limit Definition

Let $$f(x)=\frac{1}{x+2}$$. Using the definition of the derivative, find $$f'(x)$$.

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

Differentiating Composite Functions using the Chain Rule

Consider the function $$S(x)=\sin(3*x^2+2)$$ which might model the stress on a structure as a functi

Epidemic Spread Rate: Differentiation Application

The number of infected individuals in an epidemic is modeled by $$I(t)= \frac{200}{1+e^{-0.5(t-5)}}$

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.

Implicit Differentiation for a Rational Equation

Consider the curve defined by $$\frac{x*y}{x+y} = 3$$.

Maclaurin Series for arctan(x) and Error Estimate

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

Oil Spill Containment

Following an oil spill, containment efforts recover oil at a rate of $$O_{in}(t)=40-2*t$$ (accumulat

Optimization Problem via Derivatives

A manufacturer’s cost in dollars for producing $$x$$ units is modeled by the function $$C(x)= x^3 -

Pharmacokinetics: Drug Concentration Analysis

The concentration of a drug in the bloodstream is modeled by the function $$C(t)=10*\ln(t+2)*e^{-0.3

Radioactive Decay and Derivative

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

Radioactive Decay with Logarithmic Correction

A radioactive substance decays following the model $$A(t)=A_0*e^{-k*t}+\ln(t+1)$$, where $$t$$ is th

Rate Function Involving Logarithms

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

Related Rates: Changing Shadow Length

A 1.8 m tall man is walking away from a 5 m tall lamp at a constant speed of 1.2 m/s. The lamp casts

River Flow Dynamics

A river experiences seasonal variations. Its inflow is modeled by $$F_{in}(t)=30+10\cos((\pi*t)/12)$

Satellite Orbit Altitude Modeling

A satellite’s altitude (in kilometers) above Earth is modeled by $$a(t)= 500*\cos\left(\frac{\pi}{6}

Tangent Line Approximation vs. Taylor Series for ln(1+x)

An engineer studying the function $$f(x)=\ln(1+x)$$ is comparing the tangent line approximation with

Vibration Analysis: Rate of Change in Oscillatory Motion

The displacement of a vibrating string is given by $$f(t)= 3\sin(4t) - 2\cos(4t)$$, where t is in se

Chain Rule for a Multi-layered Composite Function

Let $$f(x)= \sqrt{\ln((3*x+2)^5)}$$. Answer the following:

Chain Rule in a Trigonometric Light Intensity Model

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

Composite Exponential Logarithmic Function Analysis

Consider the function $$f(x)=\ln(2*e^{3*x}+5)$$ which models a logarithmic transformation of an expo

Continuity and Differentiability of a Piecewise Function

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

Dam Water Release and River Flow

A dam releases water into a river at a rate given by the composite function $$R(t)=c(b(t))$$, where

Derivative of an Inverse Function with a Quadratic

Consider the function $$f(x) = x^2 + 6*x + 9$$ defined on $$x \ge -3$$. Let $$g$$ be the inverse of

Differentiation of a Log-Exponential Composition with Critical Points

Consider the function $$k(x)=x*\ln(e^{x}+3)$$. Answer the following parts.

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

Implicit Differentiation and Inverse Functions in a Trigonometric Equation

Consider the equation $$x*y + \sin(x+y)= 1$$ which defines y implicitly as a function of x.

Implicit Differentiation in a Circle

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

Implicit Differentiation in a Nonlinear Trigonometric Equation

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

Implicit Differentiation in Geometric Optics

A parabolic mirror used in a geometric optics experiment is described by the implicit equation $$x^2

Implicit Differentiation Involving Logarithms

Consider the equation $$\ln(x+y)=x*y$$ which relates x and y. Answer the following parts:

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 an Ellipse

Consider the ellipse defined by $$ 4*x^2+9*y^2=36 $$.

Implicit Differentiation: Circle and Tangent Line

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

Inverse Function Differentiation in Economics

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

Inverse Function Differentiation in Exponential-Linear Model

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

Inverse Trigonometric Differentiation

Consider the function $$y= \arctan(\sqrt{x+2})$$.

Lake Water Level Dynamics: Seasonal Variation

A lake's water inflow is modeled by the composite function $$I(t)=p(q(t))$$, where $$q(t)=0.5*t-1$$

Population Dynamics in a Fishery

A lake is being stocked with fish as part of a conservation program. The number of fish added per da

Related Rates: Temperature Change in a Moving Object

An object moves along a path where its temperature is given by $$T(x)= \ln(3*x + 2)$$ and its positi

Revenue Model and Inverse Analysis

A company's revenue (in millions) is modeled by $$R(x)=\sqrt{x+4}+3$$, where x represents production

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

Temperature Control: Heating Element Dynamics

A room's temperature is controlled by a heater whose output is given by the composite function $$H(t

Temperature Modeling and Composite Functions

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

Bacterial Population Growth

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

Chemistry: Rate of Change in a Reaction

In a chemical reaction, the concentration of a reactant is modeled by $$C(t)= 50e^{-0.2*t}+5$$, wher

Conical Tank Water Flow

Water is pumped into a conical tank at a rate of $$\frac{dV}{dt}=9\text{ ft}^3/\text{min}$$. The tan

Continuity in a Piecewise-Defined Function

Let $$g(x)= \begin{cases} x^2 - 1 & \text{if } x < 1 \\ 2*x + k & \text{if } x \ge 1 \end{cases}$$.

Data Table Inversion

A function $$f(x)$$ is represented by the following data table. Use the data to analyze the inverse

Deceleration of a Vehicle on a Straight Road

A vehicle travels along a straight road with velocity function $$v(t)=30-4*t$$ (m/s) for $$0 \le t \

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),

Industrial Mixer Flow Rates

In an industrial mixer, an ingredient is added at a rate of $$I(t)=7t$$ (kg per minute) and is consu

Inflating Balloon

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

Inflating Balloon: Related Rates

A spherical balloon is being inflated such that its volume increases at a constant rate of 10 in³/s.

Inflating Spherical Balloon: A Related Rates Problem

A spherical balloon is being inflated so that its volume increases at a constant rate of $$12\; in^3

Interpreting the Derivative in Straight Line Motion

A particle moves along a straight line with velocity given by $$ v(t)= 3*t^2 - 4*t + 2 $$. Analyze a

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 Inverse Function Context

Consider the function $$f(x)=x+e^{-x}$$. Although its inverse cannot be expressed in closed form, an

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 of Implicit Equation

Consider the implicit equation $$x^2 + y^2 - 2*x*y = 1$$, which defines $$y$$ as a function of $$x$$

Minimum Time to Cross a River

A person must cross a 100-meter-wide river. They can swim at 2 m/s and run along the bank at 5 m/s.

Optimization in Related Rates: Expanding Circular Oil Spill

An oil spill spreads out on a water surface forming a circle. At a certain moment, the area of the s

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 Using Cubic Position Function

Consider a particle moving along a straight line with its position given by $$x(t)=t^3 - 6*t^2 + 9*t

Polynomial Motion Analysis

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

Radical Function Inversion

Let $$f(x)=\sqrt{2*x+5}$$ represent a measurement function. Analyze its inverse.

Related Rates in a Conical Water Tank

Water is being pumped into a conical tank at a rate of $$2\;m^3/min$$. The tank has a height of 6 m

Revenue Function and Marginal Revenue

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

Series Approximation for Investment Growth

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

Series Approximation in Population Dynamics

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

Shadow Length Rate

A 6-foot lamp post casts a shadow from a 5-foot-tall person walking away from it. Let $$x$$ represen

Shadow Lengthening with a Lamp Post

A 2.5 m tall lamp post casts light on a 1.8 m tall man who walks away from the post at a constant sp

Tangent Line and Rate of Change Analysis

A scientist collected experimental data on the concentration of a chemical, and the graph below repr

Absolute Extrema and the Candidate’s Test

Let $$f(x)=x^3-3x^2-9x+5$$ be defined on the closed interval $$[-2,5]$$. Answer the following parts:

Analysis of a Motion Function Incorporating a Logarithm

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

Analysis of a Rational Function and Its Inverse

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

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

Analyzing The Behavior of a Log-Exponential Function Over a Specified Interval

Consider the function $$h(x)= \ln(x) + e^{-x}$$. A portion of its values is given in the following t

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

Average vs. Instantaneous Profit Rate

A company’s profit is modeled by the function $$P(t)= 0.2*t^3 - 3*t^2 + 10*t$$, where $$t$$ is the t

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

Chemical Reaction Rate

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

Curve Sketching Using Derivatives

For the function $$f(x)=\frac{x}{x^2+1}$$, use its derivatives to sketch a rough graph of the functi

Error Estimation in Approximating $$e^x$$

For the function $$f(x)=e^x$$, use the Maclaurin series to approximate $$e^{0.3}$$. Then, determine

Extreme Value Analysis

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

Graph Analysis of a Logarithmic Function

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

Increasing/Decreasing Intervals for a Rational Function

Consider the function $$f(x) = \frac{x^2}{x+2}$$, defined for $$x > -2$$ (with $$x \neq -2$$).

Interpreting a Velocity-Time Graph

A particle’s velocity over the interval $$[0,6]$$ seconds is depicted in the graph provided.

Inverse Function Derivative for a Piecewise Function

Suppose f is defined piecewise by $$f(x)= x^2$$ for $$x \ge 0$$ and $$f(x)= -x$$ for $$x < 0$$. Cons

Investigation of a Fifth-Degree Polynomial

Consider the function $$f(x)=x^5-5*x^4+10*x^3-10*x^2+5*x-1$$. Answer the following parts:

Linear Particle Motion Analysis

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

Logarithmic-Exponential Function Analysis

Consider the function $$f(x)= e^(x) + x$$ defined for all real numbers. Answer the following questio

Logistic Growth Model Analysis

Consider the logistic growth model given by $$P(t)=\frac{100}{1+9e^{-0.5*t}}$$. Answer the following

Maximizing the Area of an Inscribed Rectangle

A rectangle is inscribed in a circle of radius $$5$$. Determine the dimensions of the rectangle that

Motion Analysis: Particle’s Position Function

A particle moves along a straight line and its position is given by $$s(t)=t^4 - 8*t^2 + 16$$ (in me

Parameter Identification in a Log-Exponential Function

The function $$f(t)= a\,\ln(t+1) + b\,e^{-t}$$ models a decay process with t \(\geq 0\). Given that

Projectile Motion Analysis

A projectile is launched at a 45° angle with an initial speed of 20 m/s. Its motion is modeled by th

Projectile Trajectory: Parametric Analysis

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

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

Relative Motion in Two Dimensions

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

River Pollution Analysis

A river receives a pollutant through industrial discharge at a rate of $$P_{in}(t)=10*\exp(-0.1*t)$$

Ski Resort Snow Accumulation and Melting

At a ski resort, snow accumulates naturally at a rate given by $$S(t)=50*\exp(-0.1*t)$$ cm/hour due

Skier's Speed Analysis

A skier's speed (in m/s) on a slope was recorded at various time intervals. Use the data provided to

Stress Analysis in Engineering Structures

A beam experiences stress along its length given by $$S(x)=5*x^3-30*x^2+45*x$$ where x is the distan

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

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

Accumulated Displacement from Acceleration

A particle moving along a straight line has an acceleration of $$a(t)=6-4*t$$ (in m/s²), with an ini

Advanced Inflow/Outflow Dynamics

A reservoir receives water from a river at a rate given by $$f(t)=50*(1+0.1*t)$$ cubic meters per ho

Advanced U-Substitution with a Quadratic Expression

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

Area and Volume for an Exponential Function Region

Consider the curve $$y=e^{-x}$$ for $$x\ge0$$. Answer the following:

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 Under an Even Function Using Symmetry

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

Bacterial Growth Accumulation

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

Composite Functions and Inverses

Consider \(f(x)= x^2+1\) for \(x \ge 0\). Answer the following questions regarding \(f\) and its inv

Economic Applications: Consumer and Producer Surplus

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

Error Estimation in Riemann Sum Approximations

Consider the function $$f(x)=\sqrt{x}$$ on the interval $$[1,9]$$. When approximating the definite i

Estimating Rainfall Accumulation

Rainfall intensity measurements (in mm/hr) at various times are recorded in the table. Use Riemann s

Finding the Area Between Curves

Find the area of the region bounded by the curves $$y=4-x^2$$ and $$y=x$$.

Integration by Parts: Logarithmic Function

Evaluate the definite integral $$\int_{1}^{3} x*\ln(x) dx$$ using integration by parts. Answer the f

Integration of a Piecewise Function for Total Area

Consider the piecewise function $$f(x)$$ defined by: $$f(x)=3-x$$ for $$0 \le x \le 3$$, and $$f(x)=

Integration Using U-Substitution

Evaluate the definite integral $$\int_{0}^{2} (3*x+1)^{4} dx$$ using u-substitution. Answer the foll

Integration via Substitution and Numerical Methods

Evaluate the integral $$\int_0^2 \frac{2*x}{\sqrt{1+x^2}}\,dx$$.

Interpreting Area Under a Curve from a Graph

A graph displays the function $$f(x)=0.5*x+1$$ over the interval $$[0,6]$$.

Logistic Growth and Population Integration

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

Mechanical Systems: Total Change and Inverse Analysis

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

Midpoint Riemann Sum Approximation

Estimate the area under the curve $$f(x)=x^{3} - 2*x + 1$$ on the interval $$[0,3]$$ using a midpoin

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

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

Recovering Accumulated Change

A company’s revenue rate changes according to $$R'(t)=8*t-12$$ (in dollars per day). If the revenue

Taylor/Maclaurin Series Approximation and Error Analysis

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

Vehicle Motion and Inverse Time Function

A vehicle’s displacement (in meters) is modeled by the function $$f(t)= t^2 + 4$$ for $$t \ge 0$$ se

Area and Volume from a Differential Equation-derived Family

Consider the family of curves that are solutions to the differential equation $$\frac{dy}{dx} = 2*x$

Car Engine Temperature Dynamics

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

Chemical Reaction in a Closed System

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

Chemical Reaction Rate and Series Approximation

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

Euler's Method Approximation

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

Exact Differential Equation

Consider the differential equation $$ (2*x*y + y^2)\,dx + (x^2 + 2*x*y)\,dy = 0$$.

Free-Fall with Air Resistance

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

FRQ 5: Mixing Problem in a Tank

A tank initially contains 100 liters of water with 10 kg of dissolved salt. Brine with a salt concen

Growth and Decay with External Forcing Term

Consider the non-homogeneous differential equation $$\frac{dy}{dt} = k*y + f(t)$$ where $$f(t)$$ rep

Investment Growth with Nonlinear Dynamics

The rate of change of an investment amount $$I$$ is modeled by the nonlinear differential equation $

Logistic Growth in Populations

A population of organisms is modeled by the logistic differential equation $$\frac{dP}{dt} = rP \lef

Logistic Model with Harvesting

A fish population is modeled by a modified logistic differential equation that includes harvesting.

Medicine Infusion and Elimination Model

A patient receives an intravenous infusion of a drug such that the infusion rate is $$R(t)=e^{0.2*t}

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

Modeling Ambient Temperature Change

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

Modeling Exponential Growth

A population follows the differential equation $$\frac{dP}{dt} = k*P$$. Given that the population do

Newton’s Law of Cooling Application

An object is cooling in a room with ambient temperature $$T_a=20^\circ C$$. Its temperature $$T(t)$$

Nonlinear Differential Equation with Implicit Solution

Consider the nonlinear differential equation $$\frac{dy}{dx} = \frac{x^2+y^2}{2xy}$$. An implicit so

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 fired vertically upward with an initial velocity of $$50\,m/s$$. The projectile expe

Radioactive Decay

A radioactive substance decays according to the differential equation $$\frac{dA}{dt}=-kA$$, 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

Relative Motion with Acceleration

A car starts from rest and its velocity $$v(t)$$ (in m/s) satisfies the differential equation $$\fra

RL Circuit Analysis

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

Separable DE with Trigonometric Component

Solve the differential equation $$\frac{dy}{dx}=\sin(x)*\cos(y)$$ with the initial condition $$y(0)=

Separable Differential Equation and Slope Field Analysis

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

Series Solution for a Differential Equation

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

Slope Field Analysis and Solution Curve Sketching for $$\frac{dy}{dx}= x - y$$

Consider the differential equation $$\frac{dy}{dx} = x - y$$ with initial condition $$y(0)=1$$. You

Variable Carrying Capacity in Population Dynamics

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

Accumulated Interest in a Savings Account

An investor’s savings account experiences continuous deposits and withdrawals. The deposit rate is g

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\

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 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 Nonlinear Curves

Consider the functions $$f(x)=e^{-x}$$ and $$g(x)=\frac{x}{3}$$. Determine the area between these tw

Average and Instantaneous Analysis in Periodic Motion

A particle moves along a line with its displacement given by $$s(t)= 4*\cos(2*t)$$ (in meters) for $

Average Car Speed Analysis from Discrete Data

A car's speed (in km/h) is recorded at equal time intervals over a 1-hour journey. Analyze the car's

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 Temperature Analysis

A research team models the ambient temperature in a region over a 24‐hour period with the function $

Average Velocity of a Runner

A runner's velocity is modeled by $$v(t)=5+3\cos(0.5*t)$$ (m/s) for $$0\le t\le10$$ seconds. Answer

Bacterial Decay Modeled by a Geometric Series

A bacterial culture is treated with an antibiotic that reduces the bacterial population by 20% each

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

Determining Average Value of a Velocity Function

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

Displacement vs. Distance: Analysis of Piecewise Velocity

A particle moves along a line with velocity given by $$v(t)=\begin{cases} t^2, & 0 \le t < 2,\\ 8-t^

Electric Charge Accumulation

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

Net Change and Direction of Motion

A particle’s velocity is given by $$v(t)=\sin(t)-\frac{1}{2}*t$$ for $$0 \le t \le 6$$.

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

Series Convergence and Approximation

Consider the function defined by the infinite series $$f(x)=\sum_{n=0}^{\infty} \frac{(-1)^n*x^{2*n}

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

Volume by the Shell Method: Rotating a Region

Consider the region bounded by the curve $$y=\sqrt{x}$$, the line $$y=0$$, and the vertical line $$x

Volume of a Hollow Cylinder Using the Shell Method

A hollow cylindrical tube of height 5 m is formed by rotating the rectangular region bounded by $$x

Volume of a Solid of Revolution Using the Disc Method

Let R be the region bounded by $$y=\sqrt{x}$$, $$y=0$$, and $$x=4$$. This region is rotated about th

Volume of an Irregular Tank

A water tank has a varying cross-sectional profile described by $$y(x)=\sqrt{25 - (x-5)^2}$$, for $$

Work Done by a Variable Force

A force acting on an object is given by the function $$F(x)=3*x^2$$ (in Newtons). The object moves a

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

Acceleration in Polar Coordinates

An object moves in the plane with its position expressed in polar coordinates by $$r(t)= 4+\sin(t)$$

Analysis of a Cycloid

A cycloid is defined by the parametric equations $$x(t)=2*(t-\sin(t))$$ and $$y(t)=2*(1-\cos(t))$$ f

Analysis of a Vector-Valued Function

Consider the vector-valued function $$\mathbf{r}(t)= \langle t^2+1,\; t^3-3*t \rangle$$, where $$t$$

Analyzing a Cycloid

A cycloid is defined by the parametric equations $$x(t)= r*(t - \sin(t))$$ and $$y(t)= r*(1 - \cos(t

Analyzing a Looping Parametric Curve

The curve is defined by the equations $$x(t)=t^3-3t$$ and $$y(t)=t^2$$ for \(-2\le t\le 2\). Due to

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 Vector-Valued Function

Let $$ r(t)=\langle 3*t,\; 4*\sin(t) \rangle $$ for $$ 0 \le t \le \pi $$. Determine properties rela

Combined Motion Analysis

A particle’s path is described by the parametric equations $$x(t)= \ln(1+ t^2)$$ and $$y(t)= \sqrt{t

Conversion of Polar to Cartesian Coordinates

Consider the polar curve $$ r=4*\cos(\theta) $$. Analyze its Cartesian equivalent and some of its pr

Differentiation and Integration of a Vector-Valued Function

Let $$\mathbf{r}(t)=\langle e^{-t}, \sin(t), \cos(t) \rangle$$ for $$t \in [0,\pi]$$.

Dynamics in Polar Coordinates

A particle moves such that its polar coordinates are given by $$ r(\theta)=1+\theta $$, where $$ \th

Exploring Polar Curves: Spirals and Loops

Consider the polar curve $$r=θ$$ for $$0 \le θ \le 4\pi$$, which forms a spiral. Analyze the spiral

Exponential and Logarithmic Dynamics in a Polar Equation

Consider the polar curve defined by $$r=e^{\theta}$$. Answer the following:

Exponential Decay in Vector-Valued Functions

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

Implicit Differentiation and Curves in the Plane

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

Modeling with Polar Data

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

Motion Along a Helix

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

Motion Analysis of a Cycloid

Consider the parametric equations $$x(t)=t-\sin(t)$$ and $$y(t)=1-\cos(t)$$ for $$t \in [0,2\pi]$$,

Motion in the Plane: Logarithmic and Radical Components

A particle’s position in the plane is given by the vector-valued function $$\mathbf{r}(t)=\langle \l

Optimization in Garden Design using Polar Coordinates

A garden is to be designed in the shape of a circular sector with radius $$r$$ and central angle $$\

Parametric Curve Intersection

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

Parametric Curves and Concavity

Consider the parametric equations $$x(t)= \sin(t)$$ and $$y(t)= \cos(2*t)$$ for $$t \in [0, 2\pi]$$.

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 Particle with Acceleration and Jerk

A particle's motion is given by the parametric equations $$x(t)=t^4-6*t^2$$ and $$y(t)=2*t^3-9*t$$ f

Parametric Representation of an Ellipse

An ellipse is represented by the parametric equations $$x(t)=4\cos(t)$$ and $$y(t)=3\sin(t)$$ for $$

Polar Coordinates and Dynamics

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

Projectile Motion with Air Resistance: Parametric Analysis

A projectile is launched with air resistance, and its motion is modeled by the parametric equations:

Projectile Motion: Rocket Launch Tracking

A rocket is launched with its horizontal position given by $$x(t)=100*t$$ (in meters) and its vertic

Relative Motion of Two Objects

Two objects A and B move in the plane with positions given by the vector functions $$\vec{r}_A(t)= \

Spiral Intersection on the X-Axis

Consider the spiral defined by the parametric equations $$x(t) = e^{-t}\cos(2*t)$$ and $$y(t)= e^{-t

Vector-Valued Functions in 3D

A space curve is described by the vector function $$\mathbf{r}(t)=\langle e^t,\cos(t),\ln(1+t) \rang