AP Calculus BC FRQ Room

Analyzing a Composite Function Involving a Limit

Let $$f(x)=\sin(x)$$ and define the function $$g(x)=\frac{f(x)}{x}$$ for $$x\neq0$$, with the conven

Analyzing Continuity on a Closed Interval

Suppose a function $$f(x)$$ is continuous on the closed interval $$[0,5]$$ and differentiable on the

Analyzing Limits of a Composite Function

Let $$f(x)=\frac{\sin(\sqrt{4+x}-2)}{x}$$ for $$x \neq 0$$. Answer the following:

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

Car Braking Distance and Continuity

A car decelerates to a stop, and its velocity $$v(t)$$ in m/s is recorded in the following table, wh

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

Continuity and Asymptotes of a Log‐Exponential Function

Examine the function $$f(x)= \ln(e^x + e^{-x})$$.

Direct Substitution in Polynomial Functions

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

End Behavior and Horizontal Asymptote Analysis

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

Exponential Function Limits at Infinity

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

Graph Analysis of Discontinuities

A function $$q(x)$$ is defined piecewise as follows: $$q(x)=\begin{cases} x+2, & x<1, \\ 4, & x=1,

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

Intermediate Value Theorem in Water Tank Levels

The water volume \(V(t)\) in a tank is a continuous function on the interval \([0,10]\) minutes. It

Jump Discontinuity Analysis using Table Data

A function f is defined by experimental measurements near $$x=2$$. Use the table provided to answer

L'Hôpital's Rule for Indeterminate Forms

Evaluate the limit $$h(x)=\frac{e^{2*x}-1}{\sin(3*x)}$$ as x approaches 0.

Limits and Asymptotic Behavior of Rational Functions

Let $$k(x)=\frac{5*x^2-2*x+7}{x^2+4}.$$ Answer the following:

Limits Involving Exponential Functions

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

Limits Involving Radicals

Consider the function $$f(x)=\frac{\sqrt{x+4}-2}{x}$$ defined for $$x \neq 0$$. Answer the following

Oscillatory Behavior and Squeeze Theorem

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

Pollution Level Analysis and Removable Discontinuity

A function $$f(x)$$ represents the concentration of a pollutant (in mg/L) in a river as a function o

Radioactive Material Decay with Intermittent Additions

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

Rate of Change in a Chemical Reaction (Implicit Differentiation)

In a chemical reaction the concentration C (in M) of a reactant is related to time t (in minutes) by

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

Acceleration and Jerk in Motion

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

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

Chain Rule Verification with a Power Function

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

Cooling Model Rate Analysis

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

Cubic Rate of Change Analysis

Consider the function $$f(x) = x^3 - 4*x^2 + x + 6$$. This function models a certain process. Answer

Derivative of Inverse Functions

Let $$f(x)=3*x+\sin(x)$$, which is assumed to be one-to-one with an inverse function $$f^{-1}(x)$$.

Differentiation of Parametric Equations

A curve is defined by the following parametric equations: $$x(t)= t^2+1, \quad y(t)= 2*t^3-3*t+1.$$

Epidemiological Rate Change Analysis

In epidemiology, the spread of a disease is sometimes modeled by a combination of logarithmic and ex

Exploration of Derivative Notation and Higher Order Derivatives

Given the function $$f(x)=x^2*e^x$$, analyze its derivatives.

Implicit Differentiation with Exponential and Trigonometric Functions

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

Implicit Differentiation: Inverse Trigonometric Equation

Consider the function defined implicitly by $$\arctan(y) + y = x$$.

Implicit Differentiation: Square Root Equation

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

Instantaneous Velocity from a Displacement Function

A particle moves along a straight line with its position at time $$t$$ (in seconds) given by $$s(t)

Instantaneous Versus Average Rates: A Comparative Study

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

Irrigation Reservoir Analysis

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

Logarithmic Differentiation in Temperature Modeling

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

Logarithmic Differentiation Simplification

Consider the function $$h(x)=\ln\left( \frac{(x^2+1)^{3}*e^{2*x}}{\sqrt{x+2}} \right)$$.

Optimization Using Derivatives

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

Population Dynamics: Derivative and Series Analysis

A town's population is modeled by the continuous function $$P(t)= 1000e^{0.04t}$$, where t is in yea

Radioactive Decay and Derivative

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

Rainwater Harvesting System

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

Reconstructing Position from a Velocity Graph

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

Related Rates: Sweeping Spotlight

A spotlight located at the origin rotates at a constant rate of $$2 \text{ rad/s}$$. A wall is posit

Revenue Change Analysis via the Product Rule

A company’s revenue (in thousands of dollars) is modeled by $$R(x) = (2*x + 3)*(x^2 - x + 4)$$, wher

Secants and Tangents in Profit Function

A firm’s profit is modeled by the quadratic function $$f(x)=-x^2+6*x-8$$, where $$x$$ (in thousands)

Second Derivative of a Composite Function

Consider the function $$f(x)=\cos(3*x^2)$$. Answer the following:

Taylor Series Expansion of ln(x) About x = 2

For a financial model, the function $$f(x)=\ln(x)$$ is expanded about $$x=2$$. Use this expansion to

Taylor Series for sin(x) Approximation

A researcher studying oscillatory phenomena wishes to approximate the function $$f(x)=\sin(x)$$ for

Temperature Function Analysis

Suppose the temperature over time is modeled by $$T(t)=e^(2*t)*\sin(t)$$, where $$t$$ is measured in

Traffic Flow Analysis

A highway on-ramp has vehicles entering at a rate of $$V_{in}(t)=30+2*t$$ vehicles per minute and ve

Traffic Flow and Instantaneous Rate

The number of cars passing through an intersection per minute is modeled by $$F(t)= 3t^2 + 2t + 10$$

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

Warehouse Inventory Management

A warehouse receives shipments at a rate of $$I(t)=100e^{-0.05*t}$$ items per day and ships items ou

Chain Rule and Higher-Order Derivatives

Given the function $$f(x)= \ln(\sqrt{1 + e^{3*x}})$$, answer the following parts:

Chain Rule Combined with Inverse Trigonometric Differentiation

Let $$h(x)= \arccos((2*x-1)^2)$$. Answer the following:

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 Implicit Differentiation Involving Trigonometric and Polynomial Terms

Consider the relation $$\sin(x*y) + y^3 = x$$.

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 an Arctan Composite Function

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

Drug Concentration in the Bloodstream

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

Graphical Analysis of a Composite Function

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

Implicit Differentiation for a Spiral Equation

Consider the curve given by the equation $$x^2 + y^2 = 4*x*y$$. Analyze its derivative using implici

Implicit Differentiation in a Conical Sand Pile Problem

A conical sand pile has a constant ratio between its radius and height given by $$r= \frac{1}{2}*h$$

Implicit Differentiation in a Non-Standard Function

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

Implicit Differentiation in a Pressure-Temperature Experiment

In a chemistry experiment, the pressure $$P$$ (in atm) and temperature $$T$$ (in °C) of a system sat

Implicit Differentiation of a Product Equation

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

Inverse Function Differentiation in Economics

In an economic model, the price function $$f(x)$$ is differentiable and one-to-one, mapping the quan

Inverse Trigonometric Differentiation

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

Investment Growth and Rate of Change

An investor makes monthly deposits that increase according to an arithmetic sequence. The deposit am

Logarithmic and Exponential Composite Function with Transformation

Let $$g(x)=\ln((3*x+1)^2)-e^{x}$$. Answer the following questions.

Modeling with Composite Functions: Pollution Concentration

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

Parameter Dependent Composite Function

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

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

Rate of Change in a Biochemical Process Modeled by Composite Functions

The concentration of a biochemical in a cell is modeled by the function $$C(t) = \sin(0.2*t) + 1$$,

Revenue Model and Inverse Analysis

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

Analyzing Concavity through the Second Derivative

A particle’s position is given by $$x(t)=\ln(t^2+1)$$, where $$t$$ is in seconds.

Bacterial Culture Dynamics

In a bioreactor, bacteria are introduced at a rate given by $$I(t)=200e^{-0.1t}$$ (cells per minute)

Biology: Logistic Population Growth Analysis

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

Car Motion with Changing Acceleration

A car's velocity is given by $$v(t) = 3*t^2 - 4*t + 2$$, where $$t$$ is in seconds. Answer the follo

Conical Tank Filling

A conical water tank has a height of $$10$$ m and a top radius of $$4$$ m. The water in the tank for

Conical Tank Filling - Related Rates

A conical water tank has its volume given by $$V= \frac{1}{3}\pi*r^2*h$$, where \(r\) is the radius

Cooling Temperature Model

The temperature of a heated object cooling in a room is modeled by $$T(t)= 80 + 120*e^{-0.25*t}$$, w

Cost Function Analysis in Production

A company's cost for producing $$x$$ items is given by $$C(x)=0.5*x^3-4*x^2+10*x+500$$ dollars.

Cubic Curve Linearization

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

Data Table Inversion

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

Draining Hemispherical Tank

A hemispherical tank of radius $$5$$ m is draining. The volume of water in the tank is given by $$V

Financial Model Inversion

Consider the function $$f(x)=\ln(x+2)+x$$ which models a certain financial indicator. Although an ex

Fuel Consumption in a Truck

A truck’s fuel tank is refilled at a constant rate of $$I(t)=10$$ (gallons per minute) while fuel is

Graphical Analysis of an Inverse Function

Consider a continuously differentiable and strictly increasing function $$f(x)$$ as depicted in the

Implicit Differentiation: Tangent to a Circle

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

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: 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

Instantaneous vs. Average Speed in a Race

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

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

Ladder Sliding Problem

A 10-meter ladder is leaning against a vertical wall. The bottom of the ladder is pulled away from t

Limit Evaluation via L'Hopital's Rule

Evaluate the limit: $$L=\lim_{x\to 0}\frac{e^{2x}-1}{\ln(1+3x)}$$. Answer the following:

Logarithmic Function Series Analysis

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

Motion with Non-Uniform Acceleration

A particle moves along a straight line and its position is given by $$s(t)= 2*t^3 - 9*t^2 + 12*t + 3

Optimizing a Cylindrical Can Design

A manufacturer wants to design a cylindrical can with a fixed surface area of $$600\pi$$ cm² in orde

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

Pollution Accumulation in a Lake

A lake is subject to pollution with pollutants entering at a rate of $$I(t)=3e^{0.1t}$$ (kg per day)

Population Decline Modeled by Exponential Decay

A bacteria population is modeled by $$P(t)=200e^{-0.3t}$$, where t is measured in hours. Answer the

Related Rates in Conical Tank Draining

Water is draining from a conical tank. The volume of water is given by $$V=\frac{1}{3}\pi*r^2*h$$, a

Revenue and Marginal Analysis

A company’s revenue function is given by $$R(p)= p*(1000 - 5*p)$$, where $$p$$ is the price per unit

Security Camera Angle Change

A security camera is mounted on a 15 m tall tower. Let $$x$$ denote the horizontal distance from the

Tangent Lines in Motion Analysis

A particle's position is given by $$s(t)=t^3 - 6t^2 + 9t + 5$$. Analyze the tangent lines to the gra

Volume Measurement Inversion

The volume of a sphere is given by $$f(x)=\frac{4}{3}*\pi*x^3$$, where $$x$$ is the radius. Analyze

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)

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:

Application of the Mean Value Theorem

Let $$f(x)=\frac{x}{x^2+1}$$ be defined on the interval $$[0,2]$$. Answer the following questions us

Asymptotic Behavior and Limits of a Logarithmic Model

Examine the function $$f(x)= \ln(1+e^{-x})$$ and its long-term behavior.

Bacterial Culture with Periodic Removal

A laboratory experiment involves a bacterial culture that, at the beginning of an hour, has 200 bact

Concavity and Inflection Points Analysis

Consider the function \( f(x)=\ln(x) - x \) where \( x > 0 \). Answer the following parts:

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

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

Determining Convergence and Error Analysis in a Logarithmic Series

Investigate the series $$L(x)=\sum_{n=1}^\infty (-1)^{n+1} * \frac{(x-1)^n}{n}$$, which represents a

Economic Optimization: Maximizing Profit

The profit function for a product is given by $$P(x) = -2*x^3 + 27*x^2 - 108*x + 150$$, where \(x\)

Extremum Analysis Using the Extreme Value Theorem

Given the function $$f(x)= \cos(x)-x$$ defined on the interval $$\left[0, \frac{\pi}{2}\right]$$, an

Finding Local Extrema for an Exponential-Logarithmic Function

The function $$g(x)= e^x\ln(x)$$ is defined for $$x > 0$$. Analyze the function as follows:

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 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

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

Mean Value Theorem Application

Let $$f(x)=\ln(x)$$ be defined on the interval $$[1, e^2]$$. Answer the following parts using the Me

Optimization in Production Costs

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

Radius of Convergence and Series Manipulation in Substitution

Let $$f(x)=\sum_{n=0}^\infty c_n * (x-2)^n$$ be a power series with radius of convergence $$R = 4$$.

Related Rates: Draining Conical Tank

Water is draining from a conical tank with a height of \(10\,m\) and a top diameter of \(8\,m\). Wat

Series Convergence and Differentiation in Thermodynamics

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

Sign Chart Construction from the Derivative

Consider the function $$ f(x)=x^4-4x^3+6x^2.$$ Answer the following parts:

Square Root Function Inverse Analysis

Consider the function $$f(x)= \sqrt{3*x+4}$$ defined for $$x \ge -\frac{4}{3}$$. Answer the followin

Tangent Line to a Parametric Curve

A curve is defined by the parametric equations $$x(t) = \cos(t)$$ and $$y(t) = \sin(t) + \frac{t}{2}

Taylor Polynomial for $$\ln(x)$$ about $$x=1$$

For the function $$f(x)=\ln(x)$$, find the third degree Taylor polynomial centered at $$x=1$$. Then,

Taylor Series for $$\ln(1+3*x)$$

Let $$f(x)=\ln(1+3*x)$$. Develop its Maclaurin series up to the 3rd degree, determine the radius of

Vector Analysis of Particle Motion

A particle moves in the plane with its position given by the vector function $$\mathbf{r}(t) = \lang

Volume by Cross Sections Using Squares

A region in the xy-plane is bounded by $$y=x$$, $$y=0$$, and $$x=3$$. Perpendicular to the x-axis, c

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

Analyzing and Integrating a Function with a Removable Discontinuity

Consider the function $$ f(x)= \begin{cases} \frac{x^2-9}{x-3} & \text{if } x \neq 3,\\ 4 & \text{if

Arc Length of $$y=x^{3/2}$$ on $$[0,4]$$

The curve defined by $$y=x^{3/2}$$ is given for $$x\in[0,4]$$. The arc length of a curve is determin

Area Between Two Curves

Given the functions $$f(x)= x^2$$ and $$g(x)= 4*x$$, determine the area of the region bounded by the

Chemical Reactor Concentration

In a chemical reactor, a reactant enters at a rate of $$C_{in}(t)=5+t$$ grams per minute and is simu

Cooling of a Hot Object

An object cools in a room with ambient temperature 20°C according to Newton's Law of Cooling, modele

Determining Constant in a Height Function

A ball is thrown upward with a constant acceleration of $$a(t)= -9.8$$ m/s² and an initial velocity

Filling a Tank: Antiderivative with Initial Value

Water is entering a tank at a rate given by $$r(t)= \frac{2}{t+1}$$ liters per minute. The initial

Finding the Area Between Curves

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

Graphical Analysis of Riemann Sums

A graph titled 'Graph of Experimental Data' shows a curve representing the height function $$h(t)$$

Heat Energy Accumulation

The rate of heat transfer into a container is given by $$H(t)= 5\sin(t)$$ kJ/min for $$t \in [0,\pi]

Investment Growth Analysis with Exponentials

An investment grows according to the function $$f(t)= 100*e^{0.05*t}$$ for $$t \ge 0$$ (in years). A

Midpoint Riemann Sum Estimation

The function $$f(x)$$ is sampled at the following (possibly non-uniform) x-values provided in the ta

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

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

Net Change in Drug Concentration

The rate of change of a drug's concentration in the bloodstream is given by $$R(t)=8*e^{-0.5*t}$$ (i

Parameter-Dependent Integral Function Analysis

Define the function $$F(x)=\int_(1)^(x) \frac{\ln(t)}{t} dt$$ for x > 1. This function accumulates t

Particle Displacement and Total Distance

A particle moves along a straight line with a velocity function $$v(t)=3*t^2-2*t+1$$ for $$0\le t\le

Particle Motion in the Plane

A particle moves in the plane with its acceleration components given by $$a_x(t)=4-2*t$$ and $$a_y(t

Recovering Position from Velocity

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

Revenue Estimation Using the Trapezoidal Rule

A company recorded its daily revenue (in thousands of dollars) over four days. Use the data in the t

Riemann Sum from a Table: Plant Growth Data

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

Vehicle Distance Estimation from Velocity Data

A vehicle's velocity over time is recorded in the table provided. Use Riemann sums to estimate the v

Volume by Disk Method of a Rotated Region

Consider the region bounded by the curve $$y=\sqrt{x}$$, the x-axis, and the vertical line x = 4. T

Volume of a Solid: Cross-Sectional Area

A solid has cross-sectional area perpendicular to the x-axis given by $$A(x)= (4-x)^2$$ for $$0 \le

Work Done by a Variable Force

A variable force given by $$F(x)=4*x^2$$ (in Newtons) is applied along a straight line over the disp

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$

Bacteria Growth with Antibiotic Treatment

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

Bacterial Growth with Time-Dependent Growth Rate

A bacterial culture grows according to the differential equation $$\frac{dP}{dt}=(r_0+r_1*t)P$$, whe

Basic Separation of Variables: Solving $$\frac{dy}{dx}=\frac{x}{y}$$

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

Differential Equation Involving Logarithms

Consider the differential equation $$\frac{dy}{dx} = (y-1)*\ln|y-1|$$ with the initial condition $$y

Disease Spread Model

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

Estimating Instantaneous Rate from a Table

A function $$f(x)$$ is defined by the following table of values:

Euler's Method and Differential Equations

Consider the initial value problem $$\frac{dy}{dx} = x*y$$ with the initial condition $$y(0)=1$$. Eu

FRQ 6: Radioactive Decay

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

Logistic Model in Product Adoption

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

Maclaurin Series Solution for a Differential Equation

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

Mixing Problem in a Tank

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

Mixing Problem with Time-Dependent Inflow

A tank initially contains $$100$$ L of salt water with a salt concentration of $$0.5$$ kg/L. Pure wa

Mixing Tank with Variable Inflow

A tank initially contains 50 L of water with 5 kg of salt dissolved in it. A brine solution with a s

Modeling Ambient Temperature Change

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

Newton's Law of Cooling

An object with an initial temperature of $$80^\circ C$$ is placed in a room at a constant temperatur

Picard Iteration for Approximate Solutions

Consider the initial value problem $$\frac{dy}{dt}=y+t, \quad y(0)=1$$. Use one iteration of the Pic

Population Growth with Logistic Differential Equation

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

Radioactive Decay

A radioactive substance decays according to the law $$\frac{dN}{dt} = -k*N$$. The half-life of the s

Separable DE with Trigonometric Component

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

Series Convergence and Error Analysis

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

Vibration of a Suspension Bridge

A suspension bridge’s vertical displacement is modeled by the differential equation $$\frac{d^2y}{dt

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

Arc Length of a Logarithmic Curve

Determine the arc length of the curve $$f(x)= \ln(x)$$ on the interval $$[1,e]$$.

Area Between Curves in a Physical Context

The heights of two particles moving along parallel tracks are given by $$h_1(t)=t^2$$ and $$h_2(t)=4

Area Between Curves: Park Design

A park layout is bounded by two curves: $$f(x)=10-x^2$$ and $$g(x)=2*x+2$$. Answer the following par

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 Parabola and Tangent

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

Average Daily Temperature

The temperature during a day is modeled by $$T(t)=10+5*\sin((\pi/12)*t)$$ (in °C), where $$t$$ is th

Average Temperature Analysis

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

Average Temperature Over a Day

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

Average Value of a Trigonometric Function

A function representing sound intensity is given by $$I(t)= 4*\cos(2*t) + 10$$ over the time interva

Center of Mass of a Lamina

A thin lamina occupies the region under the curve $$y=\sqrt{x}$$ on the interval $$[0,4]$$ and has a

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

Complex Integral Evaluation with Exponential Function

Evaluate the integral $$I=\int_1^e \frac{2*\ln(t)}{t}dt$$ by applying a suitable substitution.

Cost Analysis of a Water Channel

A water channel has a cross-sectional shape defined by the region bounded by $$y=\sqrt{x}$$ and $$y=

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 and Distance from a Variable Velocity Function

A particle moves along a straight line with velocity function $$v(t)= \sin(t) - 0.5$$ for $$t \in [0

Fluid Flow Rate and Total Volume

A river has a flow rate given by $$Q(t)=50+10*\cos(t)$$ (in cubic meters per second) for $$t\in[0,\p

Implicit Function Differentiation

Consider the implicitly defined function $$\sin(x * y) + x^2 = \ln(y)$$. Answer the following:

Particle Motion: Position, Velocity, and Acceleration

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

Piecewise Velocity Analysis

A particle moves along a straight line with velocity given by the following piecewise function: $$v

Position from Velocity Function

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

Series and Integration Combined: Error Bound in Integration

Consider the integral $$\int_{0}^{0.5} \frac{1}{1+x^2} dx$$. Use the Taylor series expansion of the

Surface Area of a Solid of Revolution

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

Volume by Shell Method: Rotating a Region

Consider the region bounded by the curves $$y=x$$ and $$y=x^2$$. This region is rotated about the y-

Volume of a Rotated Region via Washer Method

Consider the region bounded by the curves $$y=x$$ and $$y=\sqrt{x}$$ along with the vertical line $$

Volume of a 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 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 via Shell Method

Consider the region bounded by $$y=x^2$$ and $$y=4$$ in the first quadrant. This region is revolved

Volume of a Solid with Variable Cross Sections

A solid has a cross-sectional area perpendicular to the x-axis given by $$A(x)=4-x^2$$ for $$x\in[-2

Volume with Equilateral Triangle Cross Sections

The region bounded by $$y=4-x^2$$ and $$y=0$$ for $$x\in[-2,2]$$ serves as the base of a solid. Cros

Work Done by a Variable Force

A variable force acting along the x-axis is given by $$F(x) = 2 * x + 3$$ (in Newtons). An object mo

Work Done in Pumping Water from a Parabolic Tank

A water tank has a parabolic cross-section described by $$y=x^2$$ (with y in meters, x in meters). T

Acceleration in Polar Coordinates

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

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 and Speed from Parametric Equations

Consider the curve defined by $$x(t)=e^t$$ and $$y(t)=e^{-t}$$ for $$-1 \le t \le 1$$. Analyze the a

Arc Length of a Parametric Curve

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

Arc Length of a Polar Curve

Consider the polar curve given by $$r(θ)= 1+\sin(θ)$$ for $$0 \le θ \le \pi$$. Answer the following:

Arc Length of a Polar Curve

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

Area Enclosed by a Polar Curve

Consider the polar curve given by $$r = 2*\sin(\theta)$$.

Area Enclosed by a Polar Curve: Lemniscate

The lemniscate is defined by the polar equation $$r^2=8\cos(2\theta)$$.

Catching a Thief: A Parametric Pursuit Problem

A police car and a thief are moving along a straight road. Initially, both are on the same road with

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

Computing the Area Between Two Polar Curves

Consider the polar curves given by $$R(\theta)=3+2*\cos(\theta)$$ (outer curve) and $$r(\theta)=1+\c

Concavity and Inflection in Parametric Curves

A curve is defined by the parametric functions $$x(t)=t^3-3*t$$ and $$y(t)=t^2$$ for \(-2\le t\le2\)

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-Logarithmic Particle Motion

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

Intersection of Parametric Curves

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

Intersection Points of Polar Curves

Two polar curves are given by \(r=2\sin(\theta)\) and \(r=1\). Answer the following:

Kinematics on a Circular Path

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

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)

Logarithmic Exponential Transformations in Polar Graphs

Consider the polar equation $$r=2\ln(3+\cos(\theta))$$. Answer the following:

Motion Analysis of a Cycloid

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

Parametric Equations and Intersection Points

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

Parametric Intersection and Enclosed Area

Consider the curves C₁ given by $$x=\cos(t)$$, $$y=\sin(t)$$ for $$0 \le t \le 2\pi$$, and C₂ given

Parametric Motion Analysis

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

Parametric Plotting and Cusps

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

Parametric Spiral Curve Analysis

The curve defined by $$x(t)=t\cos(t)$$ and $$y(t)=t\sin(t)$$ for $$t \in [0,4\pi]$$ represents a spi

Polar Coordinates: Area Between Curves

Consider two polar curves: the outer curve given by $$R(\theta)=4$$ and the inner curve by $$r(\thet

Polar Equations and Slope Analysis

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

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/

Vector-Valued Function and Particle Motion

Consider the vector-valued function $$\vec{r}(t)= \langle e^t, \sin(t), \cos(t) \rangle$$ representi

Vector-Valued Function Integration

A particle moves along a straight line with constant acceleration given by $$ a(t)=\langle 6,\;-4 \r

Vector-Valued Motion: Acceleration and Maximum Speed

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