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

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 for a Logarithmic Transform Function

Consider the function $$f(x)= \ln\Bigl(\frac{e^{3x}-1}{x}\Bigr)$$ for $$x \neq 0$$ and define $$f(0)

Continuity in Piecewise Functions with Parameters

A function is defined piecewise by $$f(x)=\begin{cases}a*x^2+3,& x<1,\\ b*x+1,& x\ge 1.\end{cases}$$

Continuity in Piecewise-Defined Functions

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

Establishing Continuity in a Piecewise Function

Consider the piecewise-defined function $$p(x)= \begin{cases} \frac{x^2-4}{x-2} & x \neq 2, \\ k & x

Evaluating Limits via Rationalizing Techniques

Let $$f(x)=\frac{\sqrt{2*x+9}-3}{x}.$$ Answer the following parts.

Exponential Function Limit and Continuity

Consider the function $$f(x)=\frac{e^{2*x}-1}{x}$$. Answer the following:

Horizontal Asymptote of a Rational Function

Consider the rational function $$f(x)= \frac{2*x^3+5*x^2-3}{x^3-4*x+1}$$. Answer the questions regar

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 in Temperature Analysis

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

Limit and Continuity with Parameterized Functions

Let $$ f(x)= \frac{e^{3x} - 1 - 3x}{\ln(1+4x) - 4x}, $$ for $$x \neq 0$$ and define \(f(0)=L\) for c

Limits Involving Trigonometric Functions

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

Limits Involving Trigonometric Functions and the Squeeze Theorem

Examine the following trigonometric limits: (a) Evaluate $$\lim_{x\to0} \frac{\sin(4*x)}{x}$$. (b) E

Limits with a Parameter in a Trigonometric Function

Consider the function $$f(x)= \begin{cases} \frac{\sin(a*x)}{x} & x \neq 0 \\ b & x=0 \end{cases}$$,

Limits with Infinite Discontinuities

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

Manufacturing Cost Sequence

A company's per-unit manufacturing cost decreases by $$50$$ dollars each year due to economies of sc

Manufacturing Process Tolerances

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

Oscillatory Behavior and Squeeze Theorem

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

Pendulum Oscillations and Trigonometric Limits

A pendulum’s angular displacement from the vertical is given by $$\theta(t)= \frac{\sin(2*t)}{t}$$ f

Rational Function Limit and Continuity

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

Squeeze Theorem in Oscillatory Functions

Consider the function $$f(x)= x\sin(1/x)$$ for $$x \neq 0$$ and define $$f(0)=0$$.

Analysis of a Quadratic Function

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

Analysis of Higher-Order Derivatives

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

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

Biochemical Reaction Rates: Derivative from Experimental Data

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

Car 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

Composite Exponential-Log Function Analysis

Consider the function $$f(x)=e^{x}*\ln(x+3)$$, which arises in the study of compound reactions in ch

Differentiation of a Rational Function

Consider the function $$f(x) = \frac{2*x^2+3*x}{x-1}$$, which is defined on its domain. Analyze the

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

Implicit Differentiation with Exponential and Trigonometric Functions

Consider the curve defined implicitly by $$e^(y) + x*\cos(y) = x^2$$.

Implicit Differentiation: Mixed Exponential and Polynomial Equation

Consider the curve defined by $$x^3 + e^(y) = 3*x*y$$.

Limit Definition of Derivative for a Rational Function

For the function $$f(x)=\frac{1}{x+1}$$, use the limit definition of the derivative to answer the fo

Limit Definition of the Derivative for a Quadratic Function

Let $$f(x)= 5*x^2 - 4$$. Use the limit definition of the derivative to compute $$f'(x)$$.

Manufacturing Production Rates

A factory produces items at a rate given by $$P_{in}(t)=\frac{200}{1+e^{-0.3*(t-4)}}$$ items per hou

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

Parametric Analysis of a Curve

A particle moves along a path given by the parametric equations $$x(t)=t^2+1$$ and $$y(t)=t^3-3*t$$,

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

Polar Coordinates and Tangent Lines

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

Pollutant Levels in a Lake

A lake receives pollutants at a rate of $$P_{in}(t)=30-0.5*t$$ concentration units per day and a tre

Population Growth Rates

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

Population Model Rate Analysis

A city's population is modeled by $$P(x)=2000+500\ln(x)$$, where $$x$$ represents years since a base

Product and Quotient Rules in Economic Modeling

A company’s revenue (in thousands of dollars) is modeled by the function $$R(x)= (x+2)(x-1)$$ where

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

Rates of Change in Economics

A company’s demand function for a product is given by $$D(p) = 120 - 3*p^2,$$ where \(p\) is the

Rational Function Derivative Using Quotient Rule

Consider the function $$g(x)=\frac{5*x-7}{x+2}$$. Find its derivative and analyze its critical featu

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 and Tangent Approximations from a Graph

A function f(t) has been graphed from t = 0 to 10 seconds. Use the graph to estimate rates of change

Tangent Lines and Related Approximations

For the function $$f(x)= \sin(x) + x^2,$$ (a) Compute \(f'(0)\). (b) Write the equation of the t

Using Taylor Series to Approximate the Derivative of sin(x²)

A physicist is analyzing the function $$f(x)=\sin(x^2)$$ and requires an approximation for its deriv

Vector Function and Motion Analysis

A particle moves according to the vector function $$\vec{r}(t)=\langle 2*\cos(t), 2*\sin(t)\rangle$$

Biological Growth Model Differentiation

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

Chain Rule and Taylor/Maclaurin Series for an Exponential Function

Consider the function $$h(x) = e^{\sin(2*x)}$$, which is a composite of the exponential and sine fun

Chain, Product, and Implicit: A Motion Problem

A particle moves along a curve defined by the parametric equations $$x(t)=e^{-t}\cos(t)$$ and $$y(t)

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 and Implicit Differentiation with Trigonometric Functions

Consider the equation $$\sin(x*y)+x^2=y^2$$ which relates x and y. Answer the following parts:

Composite Functions in a Biological Growth Model

A biologist models the substrate concentration by the function $$ g(t)= \frac{1}{1+e^{-0.5*t}} $$ an

Composite Functions in Biological Growth

Let a model for bacteria growth be represented by $$f(t)=e^{2*t}$$, and let the effect of nutrient c

Differentiation of an Inverse Trigonometric Composite Function

Let $$y = \arcsin(\sqrt{x})$$. Answer the following:

Differentiation of the Inverse Function in a Mechanics Experiment

An object's displacement is described by a one-to-one differentiable function \(s(t)\). It is given

Financial Flow Analysis: Investment Rates

An investment fund experiences deposits at a rate modeled by the composite function $$D(t)=g(h(t))$$

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 for an Elliptical Path

An ellipse is given by the equation $$4*x^2 + y^2 = 16$$. Answer the following parts.

Implicit Differentiation in a Conic Section

Consider the curve defined by $$x^2 + x*y + y^2 = 9$$.

Implicit Differentiation in a Logarithmic Equation

Given the equation $$\ln(x*y) + x - y = 0$$, answer the following:

Implicit Differentiation in Exponential Equation

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

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 Function Differentiation for a Cubic Function

Let $$f(x)= x^3 + x$$ be an invertible function with inverse $$g(x)$$. Use the inverse function deri

Inverse Function Differentiation for Cubic Functions

Let $$f(x)= x^3 + 2*x$$, and let $$g(x)$$ be its inverse function. Answer the following:

Inverse Function Differentiation in a Radical Context

Let $$f(x)= \sqrt{1+ x^3}$$ and let $$g$$ be its inverse function. Answer the following parts:

Inverse Function Differentiation in a Science Experiment

In an experiment, the relationship between an input value $$x$$ and the output is given by $$f(x)= \

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 Shifted Logarithmic Function

Analyze the function $$f(x)=\ln(x-1)+2$$ defined for $$x>1$$ and its inverse.

Inverse Trigonometric Functions in Navigation

A ship navigates such that its angular position relative to a fixed reference is given by $$\theta =

Logarithmic Differentiation of a Variable Exponent Function

Consider the function $$y= (x^2+1)^{\sqrt{x}}$$.

Multilayer Composite Differentiation in a Climate Model

A climate model gives the temperature $$T(t)$$ (in °C) as a function of time $$t$$ (in years) by $$T

Pipeline Pressure and Oil Flow

In an oil pipeline, the driving pressure is modeled by the composite function $$P(t)=r(s(t))$$, wher

Population Growth Analysis Using Composite Functions

A population model is defined by $$P(t)= f(g(t))$$ where $$g(t)= e^{-t} + 3$$ and $$f(u)= 2*u^2$$. H

Related Rates: Ladder Sliding Down a Wall

A ladder of length $$10\, m$$ leans against a wall such that its position is governed by $$x^2 + y^2

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

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 $$s$$

Comparison of Series Convergence and Error Analysis

Consider the series $$S(x)= \sum_{n=0}^{\infty} \frac{(-1)^n x^n}{2^n}$$ and $$T(x)= \sum_{n=0}^{\in

Derivative of Concentration in a Chemical Reaction

The concentration of a reactant in a chemical reaction is modeled by $$C(t)=\frac{8e^{-0.5t}}{1+e^{-

Expanding Circular Pool

A circular pool is being designed such that water flows in uniformly, expanding its surface area. Th

Exponential Relation

Consider the equation $$e^{x*y} = x + y$$.

GDP Growth Analysis

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

Inflating Spherical Balloon

A spherical balloon is being inflated so that the volume increases at a constant rate of $$dV/dt = 1

Instantaneous vs. Average Rate of Change in Temperature

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

Integration Region: Exponential and Polynomial Functions

Let the region be bounded by the curves $$y = x^2$$ and $$y = e^x$$. Analyze the area of the region

L'Hôpital's Rule Application

Evaluate the limit: $$\lim_{t \to \infty} \frac{5*t^3 - 4*t^2 + 7}{7*t^3 + 2*t - 6}$$ using L'Hôpita

Linearization in Engineering Load Estimation

In an engineering project, the load on a beam is modeled by $$L(x)=100*x^2+50*x+200$$, where $$L(x)$

Logarithmic Differentiation and Asymptotic Behavior

Let $$f(x)=\frac{(x^2+1)^3}{e^{2x}(x-1)^2}$$ for x > 1. Answer the following:

Maclaurin Series for ln(1+x)

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

Particle Motion with Measured Positions

A particle moves along a straight line with its velocity given by $$v(t)=4*t-1$$ for $$t \ge 0$$ (in

Pollution Decay and Inversion

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

Series Representation of a CDF

A cumulative distribution function (CDF) is given by $$F(x)= \sum_{n=0}^{\infty} \frac{(-1)^n (x-2)^

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

Water Filtration Plant Analysis

A water filtration plant processes water entering at a rate of $$I(t)=60-2t$$ (liters per minute) an

Water Tank Flow Analysis

A water tank receives water from an inlet at a rate given by $$I(t)=4+\cos(t)$$ (liters per minute)

Analysis of a Logarithmic Function

Consider the function $$q(x)=\ln(x)-\frac{1}{2}*x$$ defined on the interval [1,8]. Answer the follow

Application of Rolle's Theorem

Consider the function $$g(x)=x^3-3x$$ on the interval $$[-\sqrt{3},\sqrt{3}]$$. Answer the following

Application of the Extreme Value Theorem in Economics

A company's revenue is modeled by $$R(x)= -2*x^2+40*x+100$$, where $$x$$ is the number of units sold

Application of the Mean Value Theorem

Consider the function $$f(t)=t^3-3*t^2+2*t+5$$ representing the position (in meters) of a car along

Area Enclosed by a Polar Curve

Consider the polar curve defined by $$r(\theta) = 2 + 2*\sin(\theta)$$. This curve represents a lima

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

Car Depreciation Analysis

A new car is purchased for $$30000$$ dollars. Its value depreciates by 15% each year. Analyze the de

Concavity Analysis in a Revenue Model

A company’s revenue (in thousands of dollars) is modeled by the function $$R(x) = -0.5*x^3 + 6*x^2 -

Construction Payment Milestones

A construction project is structured around milestone payments. The first payment is $$10000$$ dolla

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

Elasticity Analysis of a Demand Function

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

Extreme Value Analysis of a Logarithmic-Exponential-Polynomial Function

Examine the function $$f(x)= x^2\,e^{-x} + \ln(x)$$ defined for $$x > 0$$. Apply methods to find its

Finding and Interpreting Inflection Points in a Complex Function

Analyze the function $$f(x)= e^{-x}\,\ln(x)$$ for $$x > 0$$. Investigate the points of inflection an

Graph Interpretation of a Function's First Derivative

A graph of the derivative function is provided below. Use it to determine the behavior of the origin

Mean Value Theorem in a Temperature Model

The temperature over a day (in °C) is modeled by $$T(t)=10+8*\sin\left(\frac{\pi*t}{12}\right)$$ for

Modeling Exponential Population Growth

A population is modeled by the function $$P(t)=500*e^{0.2*t}$$, where \(t\) is measured in years.

Optimizing Fencing for a Rectangular Garden

A homeowner plans to build a rectangular garden adjacent to a river (so the side along the river nee

Parameter-Dependent Concavity Conditions

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

Planar Particle Motion with Time-Dependent Accelerations

A particle moves in the plane with its position given by $$\vec{s}(t)=\langle t^2-4*t+4,\; \ln(t+1)\

Accumulated Displacement from a Piecewise Velocity Function

A particle moves along a line with velocity given by $$ v(t)= \begin{cases} t^2-1, & 0 \le t < 2, \\

Accumulation Function in an Investment Model

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

Analyzing a Cumulative Distribution Function (CDF)

A chemical reaction has a time-to-completion modeled by the cumulative distribution function $$F(t)=

Arc Length of a Power Function

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

Area and Volume for an Exponential Function Region

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

Area Between a Curve and Its Tangent

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

Area Between Curves

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

Area Between the Curves: Linear and Quadratic Functions

Consider the curves $$y = 2*t$$ and $$y = t^2$$. Answer the following parts to find the area of th

Bacterial Growth Accumulation

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

Bacterial Population Growth from Accumulated Rate

A bacteria population grows according to the rate function $$r(t)=k*t*e^{-t}$$ (in cells/hour) for \

Convergence of an Improper Integral Representing Accumulation

Consider the improper integral $$\int_{1}^{\infty} \frac{1}{t^2}dt$$, which can be interpreted as th

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

Definite Integral Evaluation via the Fundamental Theorem of Calculus

Let the function be $$f(x)=3*x^2+2*x$$. Evaluate the definite integral from $$x=1$$ to $$x=4$$.

Definite Integral using U-Substitution

Evaluate the integral $$\int_{1}^{5} (2*x - 3)^4\,dx$$ using the method of u-substitution.

Definite Integral via the Fundamental Theorem of Calculus

Consider the linear function $$f(x)=2*x+3$$ defined on the interval $$[1,4]$$. A graph of the functi

Definite Integral via U-Substitution

Evaluate the definite integral $$\int_{1}^{3} (2*x-1)^6\,dx$$ using u-substitution.

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

Estimating Area Under a Curve via Riemann Sums

The following table shows values of a function f(t): | t | 0 | 2 | 4 | 6 | 8 | |---|---|---|---|---

Evaluating a Complex Integral

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

Evaluating an Integral Involving an Exponential Function

Evaluate the definite integral $$\int_{0}^{\ln(4)} e^{x}\,dx$$.

Fundamental Theorem and Total Accumulated Growth

A bacteria culture grows according to the logistic model $$\frac{dN}{dt}=N\left(1-\frac{N}{10000}\r

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 of a Rational Function via Partial Fractions

Evaluate the indefinite integral $$\int \frac{2*x+3}{x^2+x-2}\,dx$$ by using partial fractions.

Rate of Production in a Factory

A factory has a production rate given by $$R(t)=100+20*\cos\left(\frac{\pi*t}{4}\right)$$ units per

Sandpile Accumulation

At an industrial site, sand is continuously added to and removed from a pile. The addition rate is g

Series Convergence and Integration with Power Series

Consider the power series $$\sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{(2n+1)!}$$, which represents $$

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

Trapezoidal Approximation for a Curved Function

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

Variable Interest Rate and Continuous Growth

An investment grows continuously with a variable interest rate given by $$r(t)=0.05+0.01*t$$. The in

Volume Accumulation in a Reservoir

A reservoir is being filled at a rate given by $$R(t)= e^{2*t}$$ liters per minute. Determine the t

Water Accumulation in a Reservoir

A reservoir receives water at an inflow rate modeled by $$r(t)=\frac{20}{t+1}$$ (in cubic meters per

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

Analysis of a Nonlinear Differential Equation

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

Bacterial Growth with Predation

A bacterial culture grows according to the differential equation $$\frac{dB}{dt}= r*B - P$$, where $

Car Engine Temperature Dynamics

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

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 of a Smartphone Battery

A smartphone battery cools according to Newton’s law: $$\frac{dT}{dt} = -k*(T-T_{room})$$. Initially

Cooling of an Object Using Newton's Law of Cooling

An object cools in a room with constant ambient temperature. The cooling process is modeled by Newto

Differential Equation with Exponential Growth and Logistic Correction

Consider the modified differential equation $$\frac{dy}{dt} = a*y - b*y^2$$, where $$a$$ and $$b$$ a

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.

Drug Concentration in the Bloodstream

A drug is administered intravenously, and its concentration in the bloodstream is modeled by the dif

Electrical Circuit Analysis Using an RL Circuit

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

Exact Differential Equations and Integrating Factors

Consider the differential equation $$ (2*x*y + y^2)\,dx + (x^2 + 2*x*y - 1)\,dy = 0$$. Answer the fo

Exponential Growth and Decay

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

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 13: Cooling of a Planetary Atmosphere

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

FRQ 20: Epidemic Decay with Intervention

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

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 $$x$$

Investment Account Growth with Fees

An investment account with balance $$A(t)$$ grows at a continuously compounded annual rate of $$6\%$

Loan Balance with Continuous Interest and Payments

A loan has a balance $$L(t)$$ (in dollars) that accrues interest continuously at a rate of $$5\%$$ p

Logistic Equation with Harvesting

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

Logistic Growth in Population Dynamics

The population of a small town is modeled by the logistic differential equation $$\frac{dP}{dt}=rP\l

Logistic Model with Harvesting

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

Modeling Currency Exchange Rates with Differential Equations

Suppose the exchange rate $$E(t)$$ (domestic currency per foreign unit) evolves according to the dif

Modeling Disease Spread with Differential Equations

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

Modeling Temperature in a Biological Specimen

A biological specimen initially at $$37^\circ C$$ is cooling in an environment where the ideal ambie

Nonlinear Differential Equation with Implicit Solution

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

Population Growth with Harvesting

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

Power Series Solutions for a Differential Equation

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

Radioactive Decay with Constant Source

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

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

Second-Order Differential Equation: Oscillations

Consider the second-order differential equation $$\frac{d^2y}{dx^2}= -9*y$$ with initial conditions

Slope Field Analysis for $$\frac{dy}{dx}=x$$

Consider the differential equation $$\frac{dy}{dx}= x$$. A slope field for this differential equatio

Temperature Change with Variable Ambient Temperature

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

Viral Spread on Social Media

Let $$V(t)$$ denote the number of viral posts on a social media platform. Posts go viral at a consta

Accumulated Change in a Population Model

A population of insects grows at a rate given by $$P'(t)=10e^{-0.2*t}$$, where $$t$$ is in days and

Analysis of a Function with a Removable Discontinuity

Consider the function $$f(x)=\frac{x^2-4}{x-2}$$ for $$x\neq2$$, with an assigned value of $$f(2)=3$

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 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 Exponential Curves

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

Area between Parabola and Tangent

Consider the parabola defined by $$y^2 = 4 * x$$. Let $$P = (1, 2)$$ be a point on the parabola. Ans

Area Between Two Curves

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

Average Fuel Consumption and Optimization

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

Average Reaction Concentration in a Chemical Reaction

In a chemical reaction, the concentration of a reactant is modeled by $$C(t)=20*\exp(-0.5*t)$$ (in m

Average Value of a Polynomial Function

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

Average Value of a Trigonometric Function

Let $$f(x)=C+\cos(2*x)$$ be defined on the interval $$[0,\pi]$$. Answer the following:

Average Value of a Velocity Function

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

Center of Mass of a Nonuniform Rod

A thin rod extends from $$x=0$$ to $$x=3$$ and has a linear density given by $$\delta(x)=1+x$$ (in k

Center of Mass of a Rod with Variable Density

A thin rod of length 10 meters lies along the x-axis from $$x=0$$ to $$x=10$$. Its density is given

Center of Mass of a Rod with Variable Density

A rod extending along the x-axis from $$x=0$$ to $$x=10$$ meters has a density given by $$\rho(x)=2+

Fluid Pressure on a Submerged Plate

A vertical rectangular plate with a width of 3 ft and a height of 10 ft is submerged in water so tha

Inflow Rate to a Reservoir

The inflow rate of water into a reservoir is given by $$R(t)=\frac{100*t}{5+t}$$ (in cubic meters pe

Inflow vs Outflow: Water Reservoir Capacity

A reservoir receives water with an inflow rate given by $$I(t)=20+5\sin(t)$$ (liters/min) and discha

Integration in Cost Analysis

In a manufacturing process, the cost per minute is modeled by $$C(t)=t^2 - 4*t + 7$$ (in dollars per

Inverse Function Analysis

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

Kinematics: Motion with Variable Acceleration

A particle moves along a straight line with acceleration $$a(t)=4-2*t$$ (in m/s²). The particle has

Optimization of Material Usage in a Container

A container's volume is given by $$V(h)=\int_0^h \pi*(3-0.5*\ln(1+x))^2dx$$, where $$h$$ is the heig

Particle Motion Analysis with Variable Acceleration

A particle moving along a straight line has an acceleration given by $$a(t)=4*e^{-t}-\sin(t)$$ (in m

Particle Motion with Variable Acceleration

A particle's acceleration is given by $$a(t)=4*e^{-t} - 2$$ for $$t$$ in seconds over the interval $

Projectile Motion under Gravity

An object is projected vertically upward with an initial velocity of $$20$$ m/s and from an initial

Rainfall Accumulation Analysis

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

River Cross Section Area

The cross-sectional boundaries of a river are modeled by the curves $$y = 5 * x - x^2$$ and $$y = x$

Volume of a Region via Washer Method

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

Volume of a Solid by the Disc Method

Consider the region bounded by the curve $$y=\sqrt{x}$$ and the x-axis for $$x\in[0,4]$$. This regio

Volume of a Solid Obtained by Rotation

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

Volume of a Solid of Revolution Between Curves

Consider the region bounded by the curves $$y=\sqrt{x}$$ and $$y=\frac{x}{2}$$ for $$x \in [0,4]$$.

Volume of a Solid Rotated about y = -1

The region bounded by $$y=\sqrt{x}$$ and $$y=0$$ for $$x\in[0,9]$$ is rotated about the line $$y=-1$

Volume of a Solid via the Disc Method

The region bounded by the curve $$y=\sqrt{x}$$ and the x-axis for $$x\in[0,4]$$ is revolved about th

Volume of a Solid with Equilateral Triangle Cross Sections

Consider the region bounded by $$y= \sqrt{x}$$ and $$y=0$$ for $$x \in [0,1]$$. A solid is formed by

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 of an Irregular Tank

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

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

Water Tank Dynamics: Inflow and Outflow

A water tank receives water through an inflow at a rate given by $$I(t)=10+2*t$$ (liters per minute)

Work Done by a Variable Force

A variable force is applied along a straight line and is given by $$F(x)=3*\ln(x+1)$$ (in Newtons),

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

Arc Length Calculation of a Cycloid

Consider a cycloid described by the parametric equations $$x(t)=r*(t-\sin(t))$$ and $$y(t)=r*(1-\cos

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 Parametrically Defined Curve

A curve is defined by the parametric equations $$x(t)=t^2$$ and $$y(t)=\frac{t^3}{3}$$ for $$0 \leq

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

Arc Length of a Quarter-Circle

Consider the circle defined parametrically by $$x(t)=\cos(t)$$ and $$y(t)=\sin(t)$$ for $$0 \le t \l

Circular Motion in Vector-Valued Form

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

Component-Wise Integration of a Vector-Valued Function

Given the acceleration vector $$\textbf{a}(t)= \langle 3\cos(t), -3\sin(t) \rangle$$, answer the fol

Curvature of a Space Curve

Let the space curve be defined by $$r(t)= \langle t, t^2, \ln(t+1) \rangle$$ for $$t > -1$$.

Dynamics in Polar Coordinates

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

Equivalence of Parametric and Polar Circle Representations

A circle is represented by the parametric equations $$x(t)=3\cos(t)$$ and $$y(t)=3\sin(t)$$ for $$0\

Intersection of Parametric Curves

Two curves are given by the parametric equations $$x_1(t)=t^2,\; y_1(t)=t^3$$ and $$x_2(s)=1-s^2,\;

Length of a Polar Spiral

For the polar spiral defined by $$r=\theta$$ for $$0 \le \theta \le 2\pi$$, answer the following:

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

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 Trajectory in Parametric Motion

A particle moves along a curve with parametric equations $$x(t)= t^2 - 4*t$$ and $$y(t)= t^3 - 3*t$$

Polar Coordinates and Area Computation

Examine the polar curve $$r = 2 + \sin(2\theta)$$ and determine the area of the region it encloses.

Polar Plots and Intersection Points in Design

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

Projectile Motion in Parametric Form

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

Projectile Motion Modeled by Vector-Valued Functions

A projectile is launched with an initial velocity vector $$\vec{v}_0=\langle 10, 20 \rangle$$ (in m/

Projectile Motion via Vector-Valued Functions

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

Spiral 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: Position, Velocity, and Acceleration

Let $$\textbf{r}(t)= \langle e^t, \ln(t+1) \rangle$$ represent the position of a particle in the pla

Vector-Valued Integrals in Motion

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

Velocity and Acceleration of a Particle

A particle’s position in three-dimensional space is given by the vector-valued function $$\mathbf{r}

Work Done Along a Path in a Force Field

A force field is given by \(\mathbf{F}(x,y)=\langle x,\,y \rangle\), and a particle moves along a pa