AP Calculus BC FRQ Room

Applying Algebraic Techniques to Evaluate Limits

Examine the limit $$\lim_{x\to4} \frac{\sqrt{x+5}-3}{x-4}$$. Answer the following: (a) Evaluate the

Bacterial Growth Experiment

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

Comparing Methods for Limit Evaluation

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

Continuity in a Parametric Function Context

A particle moves such that its coordinates are given by the parametric equations: $$x(t)= t^2-4$$ an

Continuity in Piecewise-Defined Functions

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

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 $$f(0)=L$$.

Endpoint Behavior of a Continuous Function

Let $$m(x)=\sqrt{x+4}$$ be defined on the interval $$[-4,5]$$. Answer the following:

Epsilon-Delta Proof for a Polynomial Function

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

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 a Limit Involving a Radical and Trigonometric Component

Consider the function $$f(x)= \frac{\sqrt{1+x}-\sqrt{1-x}}{x}$$. Answer the following:

Implicitly Defined Function and Differentiation

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

Inflow Function with a Vertical Asymptote

A water reservoir is fed by an inflow given by $$R_{in}(t)=\frac{50\,t}{t-5}$$ liters per minute, de

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 Involving Infinity and Vertical Asymptotes

Consider the function $$f(x)=\frac{1}{x-3}$$. Answer the following:

Limits via Improper Integration Representation

Consider the function defined by the integral $$f(x)= \int_{1}^{x} \frac{1}{t^2} dt$$ for x > 1. Add

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

Mixed Function Inflow Limit Analysis

Consider the water inflow function defined by $$R(t)=10+\frac{\sqrt{t+4}-2}{t}$$ for \(t\neq0\). Det

One-Sided Limits and Discontinuities

Consider the function $$p(x)=\begin{cases} x^2+1, & x<2, \\ 4*x-3, & x\ge2. \end{cases}$$ Answer t

Parameterized Function Continuity and Differentiability

Let $$f(x)= \begin{cases} \frac{e^x - \ln(1+2x) - 1}{x} & x \neq 0 \\ k & x=0 \end{cases}.$$ Determi

Piecewise Function Continuity

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

Radioactive Material Decay with Intermittent Additions

A laboratory maintains a radioactive material by adding 10 units of the substance at the beginning o

Rational Function Analysis with Removable Discontinuities

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

Rational Function Limit and Continuity

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

Rational Functions and Limit at Infinity

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

Seasonal Temperature Curve Analysis

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

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 with an Oscillating Function

Let $$f(x)=x * \cos(\frac{1}{x})$$ for $$x \neq 0$$, and define $$f(0)=0$$. Answer the following:

Advanced Analysis of a Composite Piecewise Function

Consider the function $$g(x)= \begin{cases} \frac{2*x^2-8}{x-2} & x \neq 2 \\ 5 & x=2 \end{cases}$$

Analysis of a Piecewise Function's Differentiability

Consider the function $$f(x)= \begin{cases} x^2+2, & x<1 \\ 3*x-1, & x\ge 1 \end{cases}$$. Answer th

Analysis of a Quadratic Function

Consider the function $$f(x)=3*x^2 - 2*x + 5$$. Using the limit definition of the derivative, answer

Analyzing Motion Through Derivatives

A car’s position as a function of time is given by $$s(t) = 4*t^3 - 12*t^2 + 9*t + 5,$$ where $$s

Application of Derivative to Relative Rates in Related Variables

Water is being pumped into a conical tank, and the volume of water is given by $$V=\frac{1}{3}\pi*r^

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 $

Chain Rule in Biological Growth Models

A biologist models the growth of a bacterial population by the function $$P(t) = (5*t + 2)^4$$, wher

Chemical Reaction Rate

In a chemical reaction, the concentration of a reactant is modeled by $$C(t)=10 - 2*\ln(t+1)$$, wher

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

Differentiation in Polar Coordinates

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

Differentiation of a Trigonometric Function

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

Error Analysis in Approximating Derivatives

Consider the function $$f(x)= \ln(1+x)$$. (a) Write the Maclaurin series for \(f(x)\) up to and inc

Evaluation of Derivative at a Point Using the Limit Definition

Let $$f(x)=3*x^2-7$$. Use the limit definition of the derivative to evaluate $$f'(2)$$.

Fuel Storage Tank

A fuel storage tank receives oil at a rate of $$F_{in}(t)=40\sqrt{t+1}$$ liters per hour and loses o

Graphical Estimation of Tangent Slopes

Using the provided graph of a function g(t), analyze its rate of change at various points.

Implicit Differentiation in a Geometric Context

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

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

Optimization in a Chemical Reaction

The rate of a chemical reaction is modeled by the function $$R(x)=x*e^{-x}+\ln(x+2)$$, where $$x$$ r

Population Growth Rate

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

Related Rates: Two Moving Vehicles

A car is traveling east at 60 km/h and a truck is traveling north at 80 km/h. Let $$x$$ and $$y$$ be

Savings Account Growth: From Discrete Deposits to Continuous Derivatives

An individual deposits $$P$$ dollars at the beginning of each month into an account that earns a con

Secant Line Approximation in an Experimental Context

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

Sine Function Analysis

Let $$g(x)=3*\sin(x)+2$$, where $$x$$ is in radians. Analyze its rate of change.

Analyzing a Composite Function from a Changing Systems Model

The displacement of an object is given by $$s(t) = \sqrt{t^2 + 4*t + 5}$$ (in meters), where $$t$$ i

Analyzing the Rate of Change in an Economic Model

Suppose the profit function is given by $$P(x)=e^{x}-4*\ln(x+2)$$, where x represents the number of

Biological Growth Model Differentiation

In a biological model, the concentration of a chemical is modeled by $$C(t)=e^{-0.5*t}+\ln(2*t+3)$$.

Composite Differentiation with Nested Logarithmic Functions

Consider the function $$F(x)= \sqrt{\ln(3*x^2+1)}$$.

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))

Differentiation in a Logistic Population Model

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

Differentiation of a Logarithmic-Square Root Composite Function

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

Differentiation of an Arctan Composite Function

For the function $$f(x) = \arctan\left(\frac{3*x}{x+1}\right)$$, differentiate with respect to $$x$$

Differentiation of an Inverse Trigonometric Composite Function

Let $$f(x)= \arctan(e^{2*x})$$. 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 an Economic Model

A company’s production is modeled by the implicit relationship $$x*y^2 + \ln(x+y) = 10$$, where $$x$

Implicit Differentiation of a Product Equation

Consider the equation $$ x*y + x + y = 10 $$.

Implicit Differentiation of an Implicit Curve

Consider the curve defined by $$x*y + x^2 - y^2 = 5$$. Answer the following parts.

Implicit Differentiation with Logarithmic Equation

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

Inverse Analysis of an Exponential-Linear Function

Consider the function $$f(x)=e^{x}+x$$ defined for all real numbers. Analyze its inverse function.

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 of a Radical Function with Domain Restrictions

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

Nested Composite Function Differentiation

Consider the function $$ h(x)= \sqrt{\cos(3*x^2+1)} $$.

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

Revenue Model and Inverse Analysis

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

Shadow Length and Related Rates

A 1.8 m tall person walks away from a 4 m tall lamppost at a speed of 1.2 m/s. Let $$x$$ be the dist

Tangent Line to a Circle via Implicit Differentiation

Consider the circle defined by $$x^2 + y^2 = 25$$. At the point $$(3, -4)$$, determine the slope of

Water Tank Composite Rate Analysis

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

Air Pressure Change in a Sealed Container

The air pressure in a sealed container is modeled by $$P(t)=\frac{100}{1+e^{-0.5*(t-10)}}$$, where $

Analysis of Particle Motion

A particle’s velocity is given by $$v(t)= 4t^3 - 3t^2 + 2$$. Analyze the particle’s motion by invest

Analyzing Temperature Change of Coffee

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

Approximating Function Values Using Linearization

Consider the function $$f(x)=x^4$$. Use linearization at x = 4 to approximate the value of $$f(3.98)

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

Chemical Reactor Concentration Monitoring

A chemical reactor receives a coolant at an inflow rate $$I(t)=\frac{10}{1+t}$$ (liters per minute)

Comparing Rates: Temperature Change and Coffee Cooling

The temperature of a freshly brewed coffee is modeled by $$T(t)=95*e^{-0.05*t}+25$$ (in °F), where $

Cooling Analysis using Newton’s Law of Cooling

An object cools in a room according to Newton's Law of Cooling, given by $$T(t)=T_{env}+ (T(0)-T_{en

Cooling Coffee Temperature

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

Economic Rates: Marginal Profit Analysis

A manufacturer’s profit (in dollars) from producing $$x$$ items is modeled by $$P(x)=500*x-2*x^2$$.

Exponential Cooling Rate Analysis

The temperature of a cup of coffee is modeled by $$T(t)=70+50e^{-0.05t}$$ (in °F), where t is in min

Horizontal Tangents on Cubic Curve

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

Implicit Differentiation in Astronomy

The trajectory of a comet is given by the ellipse $$x^2 + 4*y^2 = 16$$, where \(x\) and \(y\) (in as

Inflating Balloon

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

Inflating Spherical Balloon

A spherical balloon is being inflated so that its volume increases at a constant rate of $$\frac{dV}

Ladder Sliding Down a Wall

A 10-meter ladder leans against a vertical wall and begins to slide. The bottom slides away from the

Logarithmic Function Series Analysis

The function $$L(x)= \sum_{n=1}^{\infty} \frac{(-1)^{n+1}(x-1)^n}{n}$$ represents $$\ln(x)$$ centere

Maclaurin Series for ln(1+x)

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

Marginal Cost Analysis

A company’s cost function is given by $$C(x)=100+25*x+4*x^2$$, where $$x$$ represents the number of

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

Optimization in Design: Maximizing Inscribed Rectangle Area

A rectangle is inscribed in a semicircle of radius $$R$$ (with the rectangle's base along the diamet

Particle Motion Analysis

A particle's position along a line is given by $$s(t) = t^3 - 6*t^2 + 9*t + 2$$, where $$t$$ is meas

Polar Coordinates: Arc Length of a Spiral

Consider the polar curve defined by $$r(\theta)=1+\frac{\theta}{2}$$ for $$0 \le \theta \le \pi$$.

Pollution Decay and Inversion

A model for pollution decay is given by the function $$f(t)=\frac{100}{1+t}$$ where $$t\ge0$$ repres

Rational Function Inversion

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

Revenue Concavity Analysis

A company’s revenue from sales is modeled by the function $$R(x)= 300*x - 2*x^2$$, where \(x\) repre

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 with Center Shift

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

Series Differentiation and Approximation of Arctan

Consider the function $$F(x)= \sum_{n=0}^{\infty} \frac{(-1)^n x^{2*n+1}}{2*n+1}$$, which represents

Solids of Revolution: Washer vs Shell Methods

Consider the region enclosed by $$y = \sin(x)$$ and $$y = \cos(x)$$ for $$0 \le x \le \frac{\pi}{4}$

Temperature Conversion Model Inversion

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

Trigonometric Implicit Relation

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

Varying Acceleration and Particle Motion

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

Analysis of an Exponential-Linear Function

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

Analysis of Critical Points for Increasing/Decreasing Intervals

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

Car Motion: Velocity and Total Distance

A car’s position along a straight road is modeled by $$s(t) = t^3 - 6*t^2 + 9*t + 15$$ (in meters),

Combining Series and Integration to Analyze a Population Model

A population's growth rate is approximated by the series $$P'(t)=\sum_{n=0}^\infty \frac{t^n}{(n+1)!

Concavity and Inflection Points in Particle Motion

Consider the position function of a particle $$s(x)=x^3-6*x^2+9*x+2$$.

Convergence and Differentiation of a Series with Polynomial Coefficients

The function $$P(x)=\sum_{n=0}^\infty \frac{n^2 * (x-1)^n}{3^n}$$ is used to model stress in a mater

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

Curve Sketching with Second Derivative

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

Determining Absolute Extrema for a Trigonometric-Polynomial Function

Consider the function $$f(x)= x+\cos(x)$$ defined on the closed interval $$[0, 2\pi]$$. Determine th

Error Approximation using Linearization

Consider the function $$f(x) = \sqrt{4*x + 1}$$.

Graph Analysis of Experimental Data

A set of experimental measurements was recorded over time. Analyze the following data regarding the

Interpreting a Velocity-Time Graph

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

Intervals of Increase and Decrease in Vehicle Motion

A vehicle’s position along a straight road is given by the function $$s(t) = t^3 - 6*t^2 + 9*t + 10$

Investigation of a Series with Factorials and Its Operational Calculus

Consider the series $$F(x)=\sum_{n=0}^\infty \frac{x^n}{n!}$$, which represents an exponential funct

Investment Portfolio Dividends

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

Maximizing the Area of an Inscribed Rectangle

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

Mean Value Theorem in Motion

A car travels along a straight highway with its position given by $$s(t)=4*t^3 - 12*t^2 + 9*t + 5$$

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 $$t$$

Optimizing Material for a Container

An open-top rectangular container with a square base must have a fixed volume of $$32$$ cubic feet.

Profit Maximization in Business

A company’s profit function is given by $$P(x) = -2*x^3 + 30*x^2 - 100*x + 150$$, where x represents

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

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

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

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

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

Water Tank Volume Analysis

Water is being added to a tank at a varying rate given by $$r(t) = 3*t^2 - 12*t + 15$$ (in liters/mi

Arc Length of a Power Function

Find the arc length of the curve $$y=\frac{1}{3}*x^{3/2}$$ on the interval $$[0,9]$$.

Arc Length of an Architectural Arch

An architectural arch is described by the curve $$y=4 - 0.5*(x-2)^2$$ for $$0 \le x \le 4$$. The len

Area Under a Piecewise-Defined Curve with a Jump Discontinuity

Consider the function $$ g(x)= \begin{cases} 2x+1 & \text{if } 0 \le x < 2, \\ 3x-2 & \text{if } 2 \

Bacterial Growth with Logarithmic Integration

A bacterial culture grows at a rate given by $$P'(t)=100/(t+2)$$ (in bacteria per hour). Given that

Cyclist's Displacement from Variable Acceleration

A cyclist's acceleration is given by $$a(t)= 7 - 3*t$$ (in m/s²). At time $$t=0$$, the cyclist has a

Displacement and Distance from a Velocity Function

A particle moves along a straight line with its velocity given by $$v(t)=3\sin(t)$$ (in m/s) for $$t

Evaluating an Integral via U-Substitution

Evaluate the integral $$\int_{1}^{5} (x-4)^{10}\,dx$$ using u-substitution.

Integration by Substitution and Inverse Functions

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

Integration Involving Inverse Trigonometric Functions

Consider the function $$f(x)= \tan^{-1}(x)$$. Answer the following questions regarding its inverse a

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

Inverse Functions in Economic Models

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

Marginal Cost and Total Cost in Production

A company's marginal cost function is given by $$MC(q)=12+2*q$$ (in dollars per unit) for $$q$$ in t

Mechanical Systems: Work Done and Inverse Length Function

Let $$f(x)= \sqrt{x+4}$$ for $$x \ge -4$$, representing displacement in a mechanical system. Answer

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

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 Sum Estimation from Tabular Data

The following table lists values of a function $$f(x)$$ at selected points, which are used to approx

Tank Filling Problem

Water flows into a tank at a rate given by $$R(t)=8e^{-0.5*t}+2$$ (in liters per minute) for $$t\geq

Temperature Change in a Material

A laser heats a material such that its temperature changes at a rate given by $$\frac{dT}{dt} = 2*\s

Trapezoidal Approximation for a Curved Function

Consider the function $$f(x)=x^2+2$$ on the interval [1, 5]. Answer the following:

Water Flow and the Trapezoidal Rule

Water flows into a reservoir at a rate given by $$R(t)$$ (in m³/hour) as provided in the table below

Work Done by a Variable Force

A force acting along a displacement is given by $$F(x)=5*x^2-2*x$$ (in Newtons), where x is measured

Work on a Nonlinear Spring

A nonlinear spring exerts a force given by $$F(x)=8 * e^(0.3 * x)$$ (in Newtons) as a function of di

Analysis of a Piecewise Function with Potential Discontinuities

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

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$

Compound Interest with Continuous Payment

An investment account grows with a continuous compound interest rate $$r$$ and also receives continu

Cooling Cup of Coffee

A cup of coffee at an initial temperature of $$95^\circ C$$ is placed in a room. For the first 5 min

Cooling Model Using Newton's Law

Newton's law of cooling states that the temperature T of an object changes at a rate proportional to

Differential Equations in Economic Modeling

An economist models the rate of change of a commodity price $$P(t)$$ with the differential equation

Direction Fields and Isoclines

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

Direction Fields and Stability Analysis

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

Disease Spread Model

In a simplified epidemiological model, the number of infected individuals \(I(t)\) evolves according

Economic Investment Growth Model with Regular Deposits

An investment account grows with continuously compounded interest at a rate $$r$$ and receives conti

Environmental Modeling Using Differential Equations

The concentration $$C(t)$$ of a pollutant in a lake is modeled by the differential equation $$\frac{

Existence and Uniqueness in an Implicit Differential Equation

Consider the implicit initial value problem given by $$y\,e^{y}+x=0$$ with the initial condition $$y

Exponential Population Growth and Doubling Time

A certain population is modeled by the differential equation $$\frac{dP}{dt} = k*P$$. This equation

Forced Oscillation in a Damped System

Consider the differential equation $$\frac{dx}{dt}=-0.2*x+\sin(t)$$ with initial condition $$x(0)=1$

FRQ 2: Separable Differential Equation with Initial Condition

Consider the differential equation $$\frac{dy}{dx} = \frac{4*x}{y}$$ with the initial condition $$y(

FRQ 11: Linear Differential Equation via Integrating Factor

Solve the differential equation $$\frac{dy}{dx}+\frac{1}{x}y=\sin(x)$$ with the initial condition $$

Growth and Decay with External Forcing Term

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

Implicit Solution of a Separable Differential Equation

Solve the differential equation $$\frac{dy}{dx}=\frac{y+1}{x}$$ with the initial condition $$y(1)=2$

Infectious Disease Spread Model

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

Integrating Factor for a Non-Exact Differential Equation

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

Integration Factor Method

Solve the linear differential equation $$\frac{dy}{dx}+ 2*y = \sin(x)$$ with the initial condition $

Investment Growth Model

An investment account grows continuously at a rate proportional to its current balance. The balance

Logistic Model in Population Dynamics

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

Logistic Model in Product Adoption

A company models the adoption rate of a new product using the logistic equation $$\frac{dP}{dt} = 0.

Modeling Ambient Temperature Change

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

Modeling Free Fall with Air Resistance

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

Motion along a Line with a Separable Differential Equation

A particle moves along a straight line according to the differential equation $$\frac{dy}{dx} = \fra

Pollutant Concentration in a Lake

A lake receives a pollutant at a constant rate of $$5$$ kg/day and the pollutant is removed at a rat

Population Growth with Harvesting

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

Population Growth with Logistic Differential Equation

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

Projectile Motion with Air Resistance

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

Radio Signal Strength Decay

A radio signal's strength $$S$$ decays with distance r according to the differential equation $$\fra

Radioactive Decay Data Analysis

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

Radioactive Decay with Constant Source

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

RC Circuit: Voltage Decay

In an RC circuit, the voltage across a capacitor satisfies $$\frac{dV}{dt} = -\frac{1}{R*C} * V$$. G

Separable Differential Equation with Parameter Identification

A chemical reaction is modeled by the differential equation $$\frac{dC}{dt} = -a*C$$, where $$C(t)$$

Sketching a Solution Curve from a Slope Field

A slope field for the differential equation $$\frac{dy}{dt}=y(1-y)$$ is provided. Use the slope fiel

Accumulated Interest in a Savings Account

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

Accumulated Rainfall

The rainfall intensity in a region is given by $$R(t)=0.2*t^2+1$$ (in cm/hour), where $$t$$ is measu

Advanced Parameter-Dependent Integration Problem

Consider the function $$g(x)=e^{-a*x}$$, where $$a>0$$ and $$x$$ lies within $$[0,b]$$. The average

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 in a Business Context

A company’s revenue and cost (in dollars) for producing items (in hundreds) are modeled by the funct

Area Between Nonlinear Curves

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

Area Under a Curve with a Discontinuity

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

Average Concentration of a Drug in Bloodstream

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

Average Power Consumption

A household's power consumption is modeled by the function $$P(t)=3+2*\sin\left(\frac{\pi}{12}*t\rig

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\

Averaging Chemical Concentration in a Reactor

In a chemical reactor, the concentration of a substance is given by $$C(t)=100*e^{-0.5*t}+20$$ (mg/L

Car Motion Analysis

A car's acceleration is given by $$a(t) = 4 - 2 * t$$ (in m/s²) for $$0 \le t \le 4$$ seconds. The c

Center of Mass of a Non-uniform Rod

A thin rod of length 10 m has a linear density given by $$\lambda(x)= 3 + 0.5*x$$ (in kg/m) for $$0

Center of Mass of a Plate

A lamina occupies the rectangular region defined by $$0 \le x \le 2$$ and $$0 \le y \le 3$$, with a

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}{

Determining the Length of a Curve

Find the arc length of the curve given by $$y=\sqrt{4*x}$$ for $$x\in[0,9]$$.

Electric Charge Distribution Along a Rod

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

Fluid Force on a Submerged Plate

A vertical plate submerged in water experiences a force due to fluid pressure given by $$F(y)=\rho*g

Movement Under Variable Acceleration

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

Position from Velocity Function

A particle moves along a horizontal line with a velocity function given by $$v(t)=4*\cos(t) - 1$$ fo

Projectile Motion Analysis

A projectile is launched vertically upward with an initial velocity of $$20$$ m/s. The only accelera

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 Cross-Section: Rotated Region

Consider the region bounded by the curves $$y=x+2$$ and $$y=x^2$$, with the intersection points form

Volume by the Washer Method: Between Curves

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

Volume of a Hollow Cylinder Using the Shell Method

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

Work Done in Pumping Water from a Tank

A cylindrical tank has a radius of $$3$$ meters and a height of $$10$$ meters. The tank is completel

Arc Length and Curvature Comparison

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

Arc Length of a Parametric Curve

The curve defined by $$x(t)=\ln(t+1)$$ and $$y(t)=\sqrt{t}$$ for $$t \in [0,4]$$ is given.

Area Between Polar Curves

In the polar coordinate plane, consider the region bounded by the curves $$r = 2 + \cos(\theta)$$ (t

Circular Motion: Speed and Acceleration Components

A car travels around a circle of radius 5, described by the parametric equations $$x(t)=5\cos(t)$$ a

Concavity and Inflection Points of a Parametric Curve

For the curve defined by $$x(t)=e^{t}-t$$ and $$y(t)=\ln(1+t^2)$$ for $$t \ge 0$$, answer the follow

Conversion and Analysis of Polar and Rectangular Forms

Consider the polar equation $$r=3e^{\theta}$$. Answer the following:

Designing a Race Track with Parametric Equations

An engineer designs a race track whose left and right boundaries are described by the curves $$C_1:

Distance Traveled in a Turning Curve

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

Error Analysis in Taylor Approximations

Consider the function $$f(x)=e^x$$.

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

Integration of Speed in a Parametric Motion

For the parametric curve defined by $$x(t)=\cos(t)$$ and $$y(t)=\sin(t)$$ for $$0 \le t \le 2\pi$$,

Intersection and Area Between Polar Curves

Two polar curves are given by $$r_1(\theta)=2\sin(\theta)$$ and $$r_2(\theta)=1+\cos(\theta)$$.

Kinematics in Polar Coordinates

An object moves so that its position in polar coordinates is given by $$r(t)= 4 - t$$ and $$\theta(t

Lissajous Figures and Their Properties

A Lissajous curve is defined by the parametric equations $$x(t)=5*\sin(3*t)$$ and $$y(t)=5*\cos(2*t)

Maclaurin Series for Trigonometric Functions

Let $$f(x)=\sin(x)$$.

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

Motion Along a Parametric Curve

Consider the parametric equations $$x(t)= 2*t - 3$$ and $$y(t)= t^2$$, where $$t$$ represents time i

Motion Along an Elliptical Path

Consider a particle moving along the curve defined by $$ x(t)=2*\cos(t) $$ and $$ y(t)=3*\sin(t) $$

Parametric Curve Intersection

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

Parametric Egg Curve Analysis

An egg-shaped curve is modeled by the parametric equations $$x(t)= \cos(t)+0.5\cos(2t)$$ and $$y(t)=

Parametric Tangent Line and Curve Analysis

For the curve defined by the parametric equations $$x(t)=t^{2}$$ and $$y(t)=t^{3}-3t$$, answer the f

Particle Motion in Circular Motion

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

Polar and Parametric Form Conversion

A curve is given in polar form by $$r(\theta)=\frac{2}{1+\cos(\theta)}$$. This curve represents a co

Polar Coordinates and Dynamics

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

Polar Differentiation and Tangent Lines

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

Polar Equations and Slope Analysis

Given the polar curve $$r = 4\sin(\theta)$$, analyze its Cartesian form and tangent properties.

Polar Plots and Intersection Points in Design

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

Roller Coaster Design: Parametric Path

A roller coaster is modeled by the parametric equations $$x(t)=t-\cos(t)$$ and $$y(t)=t-\sin(t)$$ fo

Sensitivity Analysis and Linear Approximation using Implicit Differentiation

The variables $$x$$ and $$y$$ satisfy the equation $$xy+\ln(y)=5$$.

Tangent Line to a Polar Curve

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

Vector-Valued Functions: Tangent and Normal Components

A car’s motion on a plane is described by the vector-valued function $$\mathbf{r}(t)=\langle t^2, 4*