AP Calculus AB FRQ Room

Complex Rational Limit and Removable Discontinuity

Consider the function $$f(x)=\frac{x^3 - 27}{x^2 - 9}$$ for x \(\neq\) 3. Answer the following parts

Continuity and Asymptotic Behavior of a Rational Exponential Function

Consider the function $$q(x)= \frac{e^{2*x} - 4}{e^{x} - 2}$$. Notice that the function is not defin

Continuity in a Cost Function for a Manufactured Product

A company's cost function for producing $$n$$ items (with $$n > 0$$) is given by $$C(n)= \frac{50}{n

Continuity in a Piecewise Function with Square Root and Rational Expression

Consider the function $$f(x)=\begin{cases} \sqrt{x+6}-2 & x<-2 \\ \frac{(x+2)^2}{x+2} & x>-2 \\ 0 &

Discontinuity Analysis in Piecewise Functions

Consider the piecewise function $$f(x)= \begin{cases} \frac{x^2-4}{x-2} & x\neq2 \\ 5 & x=2 \end{cas

Epsilon-Delta Analysis of a Limit

Consider the linear function $$f(x) = 3*x + 1$$. For $$\epsilon = 0.5$$, answer the following:

Evaluating Limits Near Vertical Asymptotes

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

Evaluating Trigonometric Limits Without a Calculator

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

Graph Analysis of Discontinuities

Examine the provided graph of a function f(x) that displays both a removable discontinuity and a jum

Graph Analysis of Discontinuities

A graph of a function f(x) shows a jump discontinuity at x = 1 and a removable discontinuity (a hole

Graphical Interpretation of Limits and Continuity

The graph below represents a function $$f(x)$$ defined by two linear pieces with a potential discont

Horizontal Asymptote of a Rational Function

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

Intermediate Value Theorem Application

Consider the polynomial function $$f(x)=x^3-6*x^2+9*x+1$$ on the closed interval [0, 4].

Limit and Integration in Non-Polynomial Particle Motion

A particle moves along a line with velocity defined by $$v(t)= \frac{e^{2*t}-e^{4}}{t-2}$$ for \(t \

Limit Involving a Square Root and Removable Discontinuity

Consider the function $$h(x)=\frac{\sqrt{x+4}-2}{x}$$ for $$x\neq0$$ and $$h(0)=1$$. Answer the foll

Limits Involving Absolute Value

Consider the function $$f(x) = \frac{|x - 3|}{x - 3}$$. (a) Evaluate $$\lim_{x \to 3^-} f(x)$$ and

Logarithmic Limit Evaluation

Consider the function $$f(x)=\frac{\ln(x+1)}{x}$$.

Long-Term Behavior of Particle Motion: Horizontal Asymptotes

For a particle, the velocity function is given by $$v(t)= \frac{4*t^2-t+1}{t^2+2*t+3}$$. Answer the

Modeling Population Growth with a Limit

A population P(t) is modeled by the function $$P(t) = \frac{5000}{1 + 40e^{-0.5*t}}$$ for t ≥ 0. Ans

One-Sided Limits and Absolute Value Functions

Let $$f(x) = \frac{|x - 2|}{x - 2}$$. Analyze its behavior as x approaches 2.

Oscillatory Behavior in Damped Trigonometric Functions

Consider the function $$f(x) = x \cos(1/x)$$ for x ≠ 0 and define f(0) = 0. Answer the following:

Rational Function and Removable Discontinuity

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

Real-world Application: Economic Model of Inventory Growth

A company monitors its inventory \(I(t)\) (in units) over time (in months) using the rate function $

Redefining a Function for Continuity

A function is defined by $$f(x) = \frac{x^2 - 1}{x - 1}$$ for $$x \neq 1$$, while $$f(1)$$ is left u

Removable Discontinuity and Redefinition

Consider the function $$f(x)=\frac{x^2-4}{x-2}$$ for $$x\neq2$$. Note that f is undefined at $$x=2$$

Squeeze Theorem for an Oscillatory Function

Define the function $$f(x)= x \cos(\frac{1}{x})$$ for x ≠ 0, and let f(0)= 0.

Squeeze Theorem with an Oscillatory Term

Consider the function $$f(x) = x^2 \cdot \cos\left(\frac{1}{x^2}\right)$$ for $$x \neq 0$$, and defi

Analysis of Motion in the Plane

A particle moves in the plane with its position given by $$\mathbf{s}(t)=\langle t^2 - 4*t,\, 3*t +

Analyzing Function Behavior Using Its Derivative

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

Approximating Small Changes with Differentials

Let $$f(x)= x^3 - 5*x + 2$$. Use differentials to approximate small changes in the value of $$f(x)$$

Comparing Average vs. Instantaneous Rates

Consider the function $$f(x)= x^3 - 2*x + 1$$. Experimental data for the function is provided in the

Concavity and the Second Derivative

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

Curve Analysis – Increasing and Decreasing Intervals

Given the function $$f(x)= x^3 - 3*x^2 + 2$$, analyze its behavior.

Derivative from First Principles

Derive the derivative of the polynomial function $$f(x)=x^3+2*x$$ using the limit definition of the

Derivative using the Limit Definition for a Linear Function

For the linear function $$f(x)= 5*x - 3$$, perform an analysis of its derivative using the limit def

Derivatives on an Ellipse

The ellipse given by $$\frac{x^2}{9} + \frac{y^2}{4} = 1$$ represents a race track. Answer the follo

Differentiability and Continuity

A function is defined piecewise as follows: $$f(x)= \begin{cases} x^2 & \text{if } x \le 1, \\ 2*x +

Differentiation Using the Product Rule

Consider the function \(p(x)= (2*x+3)*(x^2-1)\). Answer the following parts.

Finding Derivatives of Composite Functions

Let $$f(x)= (3*x+1)^4$$.

Finding the Derivative using the Limit Definition

Let $$h(x)= 5*x^2 + 3*x - 7$$. Use the limit definition of the derivative to determine $$h'(x)$$.

Finding the Second Derivative

Given $$f(x)= x^4 - 4*x^2 + 7$$, compute its first and second derivatives.

Instantaneous and Average Velocity

A particle's position is given by $$s(t)=t^3-6*t^2+9*t+2$$, where $$s(t)$$ is in meters and $$t$$ is

Instantaneous Rate of Change in Motion

A particle’s position along a straight line is given by $$s(t)= 4*t^3 - 12*t^2 + 9*t + 5$$, where $$

Instantaneous Rate of Temperature Change in a Coffee Cup

The temperature of a cup of coffee is recorded at several time intervals as shown in the table below

Inverse Function Analysis: Hyperbolic-Type Function

Consider the function $$f(x)=\sqrt{x^2+1}$$ defined for $$x\geq0$$.

Investigating the Derivative of a Piecewise Function

The function $$f(x)$$ is defined piecewise by $$f(x)=\begin{cases} x^2 & \text{if } x \le 1, \\ 2*x

Particle Motion on a Straight Road

A particle moves along a straight road. Its position at time $$t$$ seconds is given by $$s(t) = t^3

Profit Function Analysis

A company's profit function is given by $$P(x)=-2x^2+12x-5$$, where x represents the production leve

Rates of Change from Experimental Data

A chemical experiment yielded the following measurements of a substance's concentration (in molarity

Rates of Change in Chemical Concentration

In a chemical reaction, the concentration $$C(t)$$ of a substance in a tank is modeled by $$C(t)=\fr

RC Circuit Voltage Decay

An RC circuit's capacitor voltage is modeled by $$V(t)= V_{0}*e^{-t/(R*C)}$$, where $$V_{0}$$ is the

Related Rates: Conical Tank Draining

A conical water tank drains so that its volume is given by $$V=\frac{1}{3}\pi r^2h$$. The radius r o

River Crossover: Inflow vs. Damming

A river receives water from two tributaries at rates $$f_1(t)=7+0.5*t$$ and $$f_2(t)=9-0.2*t$$ (lite

Sand Pile Growth with Erosion Dynamics

A sand pile is growing as sand is added at a rate of $$f(t)=8+0.3*t$$ (kg/min) and simultaneously lo

Secant and Tangent Lines Analysis

Consider the function $$g(t)=t^3-6*t^2+9*t+2$$ modeling the height (in meters) of a ball at time $$t

Tangent Line Approximation

Suppose a continuous function $$f(x)$$ is differentiable with $$f(2)=8$$ and $$f'(2)=5$$. Use this i

Tangent Line Approximation for a Cubic Function

Let $$f(x)=2*x^3 - 7*x + 1$$. At $$x=1$$, determine the equation of the tangent line and use it to a

Tangent Line to a Parabola

Consider the function $$f(x)=x^2 - 4*x + 3$$. A graph of this quadratic function is provided. Answer

Water Tank Inflow-Outflow Analysis

A water tank receives water at a rate given by $$f(t)=3*t+2$$ (liters/min) and loses water at a rate

Chain and Product Rules in a Rate of Reaction Process

In a chemical reaction, the rate is modeled by the function $$R(t)= \sqrt{t+3}*\cos(2*t)$$, where $$

Chain Rule in Temperature Model

A scientist models the temperature in a laboratory experiment by the function $$T(t)=\sqrt{3*t^2+2}$

Chain Rule with Exponential and Trigonometric Functions

A particle's position is given by $$s(t)=e^{3*t}\sin(2*t)$$. Use appropriate differentiation techniq

Chain Rule with Trigonometric and Exponential Functions

Let $$y = \sin(e^{3*x})$$. Answer the following:

Chain Rule with Trigonometric Function

Consider the function $$f(x)=\sin(5*x^2)$$. Answer the following:

Comparing the Rates between a Function and its Inverse

Let $$f(x)=x^5+2*x$$. Answer the following:

Composite Function Chain Reaction

A chemist models the concentration of a reacting solution at time $$t$$ (in seconds) with the compos

Composite Function Differentiation Involving Product and Chain Rules

Consider the function $$F(x)= (x^2 + 1)^3 * \ln(2*x+5)$$.

Composite Function Kinematics

A particle moves along a straight line with its position given by $$s(t) = (2*t+3)^4$$. Analyze the

Composite Function via Chain Rule in a Financial Context

A company’s profit (in dollars) based on production level (in thousands of units) is modeled by the

Composite Function with Inverse Trigonometric Components

Let $$f(x)= \sin^{-1}\left(\frac{2*x}{1+x^2}\right)$$. This function involves an inverse trigonometr

Composite Function with Nested Exponential and Trigonometric Terms

Let $$f(x)= e^{\sin(4*x)}$$. This function combines exponential and trigonometric elements.

Composite Temperature Model

Consider a temperature function given by $$T(t) = \sin(t^3 - 2*t)$$, where t is measured in seconds.

Designing a Tapered Tower

A tower has a circular cross-section where the relationship between the radius r (in meters) and the

Implicit Differentiation in Circular Motion

Consider the circle defined by the equation $$x^2+y^2=100$$, which could represent the track of an o

Implicit Differentiation Involving a Product

Consider the equation $$x^2*y + \sin(y) = x*y^2$$ which relates the variables $$x$$ and $$y$$ in a n

Implicit Differentiation with Trigonometric Components

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

Inverse Function Differentiation Combined with Chain Rule

Let $$f(x)=\sqrt{x-1}+x^2$$, and assume that it is one-to-one on its domain, with an inverse functio

Inverse Function Differentiation in a Biological Growth Curve

A biological measurement is modeled by the function $$f(t)= \frac{4*t}{t+2}$$, which is one-to-one o

Inverse Function Differentiation with an Exponential-Linear Function

Let $$f(x)=e^{2*x}+x$$ and assume it is invertible. Answer the following:

Inverse Function Differentiation with Exponentials and Trigonometry

Let $$f(x)=\sin(e^{x})+e^{x}$$ and assume that it is invertible. Answer the following:

Inverse Function in Currency Conversion

A function converting dollars to euros is given by $$f(d) = 0.9*d + 10\ln(d+1)$$ for $$d > 0$$. Let

Inverse Trigonometric Differentiation in Engineering Mechanics

In an engineering application, the angle of elevation $$\theta$$ is given by the function $$\theta=

Optimization in an Implicitly Defined Function

The curve defined by $$x^2y + \sin(y) = 10$$ implicitly defines $$y$$ as a function of $$x$$ near $$

Second Derivative via Implicit Differentiation

Consider the curve defined by $$x^2+x*y+y^2=7$$. Answer the following parts.

Tangent Lines on an Ellipse

Consider the ellipse given by $$\frac{x^2}{9} + \frac{y^2}{4} = 1$$. Use the graph provided to aid i

Water Tank Flow Analysis using Composite Functions

A water tank is equipped with an inflow system and an outflow system. At time $$t$$ (in minutes), wa

Analyzing a Nonlinear Rate of Revenue Change

A company's revenue in thousands of dollars is modeled by the function $$R(x)=100\ln(x+1) + 0.5x$$,

Chemical Reaction Rate

In a chemical reaction, the concentration of a reactant is given by $$C(t)=100e^{-0.05*t}$$ mg/L, wh

Cooling Coffee Temperature Analysis

A cup of coffee cools according to the function $$T(t)=80+20e^{-0.3t}$$ (in °F), where $$t$$ is meas

Cooling Coffee: Exponential Decay Model

A cup of coffee cools according to $$T(t) = 70 + 50e^{-0.1t}$$, where $$T(t)$$ (in °F) is the temper

Defect Rate Analysis in Manufacturing

The defect rate in a manufacturing process is modeled by $$D(t)=100e^{-0.05t}+5$$ defects per day, w

Drug Concentration in the Blood

A drug’s concentration in the bloodstream is modeled by $$C(t)= \frac{5}{1+e^{0.2(t-30)}}$$, where $

Economic Cost Analysis Using Derivatives

A company’s cost function for producing $$x$$ units is given by $$C(x)=0.05*x^3 - 2*x^2 + 40*x + 100

Economic Cost Function Linearization

A company's production cost is modeled by $$C(x)= 0.02*x^3 - 1.5*x^2 + 40*x + 200$$ dollars, where $

Elasticity of Demand Analysis

A product’s demand function is given by $$Q(p) = 150 - 10p + p^2$$, where $$p$$ is the price, and $$

Error Approximation in Engineering using Differentials

The cross-sectional area of a circular pipe is given by $$A=\pi r^2$$. If the radius is measured as

Expanding Balloon: Related Rates with a Sphere

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

Free Fall Motion Analysis

An object in free fall near Earth's surface has its position modeled by $$s(t)=-4.9t^2+20t+1$$ (in m

FRQ 5: Coffee Cooling Experiment

A cup of coffee cools according to the function $$T(t) = 70 + 50e^{-0.1*t}$$, where T is the tempera

FRQ 7: Conical Water Tank Filling

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

FRQ 8: Satellite Dish Design: Implicit Differentiation

A satellite dish’s cross‐section is modeled by the implicit equation $$y^2 + 4*x*y - 3*x^2 = 0$$. Th

Inflation of a Balloon: Surface Area Rate of Change

A spherical balloon is being inflated so that its volume increases at a rate of $$\frac{dV}{dt}=50$$

Inverse Function Analysis in a Real-World Model

Consider the function $$f(x)=x^3+1$$ which models a certain transformation of measurable quantities.

Inverse Trigonometric Analysis for Navigation

A navigation system relates the angle of elevation $$\theta$$ to a mountain with the horizontal dist

Linear Approximation for Function Values

Consider the function $$f(x) = x^3$$. Use linearization at $$x = 4$$ to approximate the value of $$f

Linear Approximations: Estimating Function Values

Let $$f(x)=x^4$$. Use linear approximation to estimate $$f(3.98)$$. Answer the following:

Linearization and Differentials

Given the function $$f(x)=x^4$$, use linear approximation to estimate the value of $$(3.98)^4$$.

Linearization and Differentials Approximation

A function $$f(x)= x^3$$ models the volume (in cubic centimeters) of liquid in a container as a func

Motion Analysis from Velocity Function

A particle moves along a straight line with a velocity given by $$v(t) = t^2 - 4t + 3$$ (in m/s). Th

Open-top Box Optimization

A manufacturer wants to design an open‐top rectangular box with a square base that has a fixed volum

Optimization: Minimizing Material for a Box

A company wants to design an open-top box with a square base that holds 32 cubic meters. Let the bas

Population Change Rate

The population of a town is modeled by $$P(t)= 50*e^{0.3*t}$$, where $$t$$ is in years and $$P(t)$$

Projectile Motion Analysis

A projectile is launched vertically, and its height (in meters) as a function of time is given by $$

Projectile Motion: Maximum Height

A ball is thrown upward and its height is modeled by $$h(t)=-5t^2+20t+2$$ (in meters). Analyze its m

Related Rates in Shadows: A Lamp and a Tree

A lamp post 5 m tall casts a shadow of a tree. At a certain moment, the tree is 3 m from the lamp an

Related Rates: Expanding Circle

A circular pool is being filled such that its surface area increases at a constant rate of $$10$$ sq

Related Rates: Shadow Length

A 1.8-meter tall person is walking away from a 4.5-meter tall streetlight at a constant speed of 1.2

Revenue and Cost Analysis

A company’s revenue is modeled by $$R(t)=200e^{0.05t}$$ and its cost by $$C(t)=10t^3-30t^2+50t+200$$

Shadow Length Problem

A person 1.80 m tall walks away from a 3.0 m tall lamppost at a rate of 1.2 m/s. Let $$x$$ be the di

Shadow Length: Related Rates

A 15-foot tall lamp post casts light on a 6-foot tall man walking away from the lamp at 4 ft/sec. Le

Studying a Bouncing Ball Model

A bouncing ball reaches a maximum height after each bounce modeled by $$h(n)= 100*(0.8)^n$$, where n

Tangent Line and Linearization Approximation

Let $$f(x)=\sqrt{x}$$. Use linearization at $$x=16$$ to approximate $$\sqrt{15.7}$$. Answer the foll

Temperature Cooling in a Cup of Coffee

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

Temperature Rate Change in Cooling Coffee

A cup of coffee cools following the model $$x(t)=70+50e^{-0.1t}$$, where x is in degrees Fahrenheit

Transcendental Function Temperature Change

A cooling object has its temperature modeled by $$T(t)= 100 + 50e^{-0.2*t}$$, where t is measured in

Water Flow Rate in a Tank

Water flows into a tank at a rate given by $$r(t)=\frac{2t+1}{t+4}$$ liters per minute, where $$t$$

Analyzing a Supply and Demand Model Using Derivatives

A product's price as a function of the number of units produced is given by $$P(q)= 50 - 3*q + 0.5*q

Analyzing Increasing/Decreasing Behavior of a Cubic Polynomial

Consider the function $$f(x)= x^3 - 6*x^2 + 9*x + 2$$. Analyze the function's behavior in terms of i

Area Growth of an Expanding Square

A square has a side length given by $$s(t)= t + 2$$ (in seconds), so its area is $$A(t)= (t+2)^2$$.

Asymptotic Behavior in an Exponential Decay Model

Consider the model $$f(t)= 100*e^{-0.3*t}$$ representing a decaying substance over time. Answer the

Car Speed Analysis via MVT

A car's position is given by $$f(t) = t^3 - 3*t^2 + 2*t$$ (in meters) for $$t$$ in seconds on the cl

Comprehensive Analysis of a Rational Function

Given the rational function $$f(x)= \frac{x^2-4}{x^2+1}$$, perform a comprehensive analysis includin

Continuity Analysis of a Rational Piecewise Function

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

Cost Minimization in Transportation

A transportation company recorded shipping costs (in thousands of dollars) for different numbers of

Drag Force and Rate of Change from Experimental Data

Drag force acting on an object was measured at various velocities. The table below presents the expe

Exploration of a Removable Discontinuity in a Rational Function

Consider the function $$ f(x) = \begin{cases} \frac{x^2 - 16}{x - 4}, & x \neq 4, \\ 7, & x = 4. \e

Extrema in a Cost Function

A company's cost function is given by $$C(x) = x^3 - 6*x^2 + 12*x + 7$$, where $$x$$ represents the

FRQ 1: Car's Motion and the Mean Value Theorem

A car’s position along a straight road is modeled by $$s(t) = t^3 - 6*t^2 + 9*t + 5$$ (in meters) fo

FRQ 11: Particle Motion with Non-Constant Acceleration

A particle moves along a straight line with acceleration given by $$a(t)= 12*t - 6$$ (in m/s²). If t

FRQ 15: Population Growth and the Mean Value Theorem

A town’s population (in thousands) is modeled by $$P(t)= t^3 - 3*t^2 + 2*t + 50$$, where $$t$$ repre

Inflection Points in a Population Growth Model

Population data from a species over several years is provided in the table below. Use this informati

Inverse Analysis of a Composite Function

Consider the function $$f(x)=e^(x)+x$$. Although its inverse cannot be written in closed form, answe

Inverse Analysis of a Function with Square Root and Linear Term

Consider the function $$f(x)=\sqrt{3*x+1}+x$$. Answer the following questions regarding its inverse.

Inverse Analysis of a Trigonometric Function on a Restricted Domain

Consider the function $$f(x)=\sin(x)$$ with the restricted domain $$\left[-\frac{\pi}{2},\frac{\pi}{

Investigating Limits and Discontinuities in a Rational Function with Complex Denominator

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

Limit Analysis of a Piecewise Function Involving a Rational Expression

Consider the function $$ f(x) = \begin{cases} \frac{2x^2-8}{x-2}, & x < 2, \\ x+2, & x \ge 2. \end{

Mean Value Theorem Applied to Exponential Functions

Consider the function $$f(x)= e^{x}$$ on the interval \([0, 1]\). Answer the following:

Mean Value Theorem for a Piecewise Function

Consider the function defined by $$ f(x)= \begin{cases} x^2, & x \le 2, \\ 4x - 4, & x > 2, \end

Minimizing Average Cost in Production

A company’s cost function is given by $$C(x)= 0.5*x^3 - 6*x^2 + 20*x + 100$$, where $$x$$ represents

Optimization of an Open-Top Box

A company is designing an open-top box with a square base. The volume of the box is modeled by the f

Rational Function Behavior and Extreme Values

Consider the function $$f(x)= \frac{2*x^2 - 3*x + 1}{x - 2}$$ defined for $$x \neq 2$$ on the interv

Water Reservoir Net Change

A water reservoir receives water from a river at a rate $$R_{in}(t)=5+0.5*t$$ and discharges water a

Application of the Fundamental Theorem of Calculus

Let $$f(x)=\ln(x)$$. Use the Fundamental Theorem of Calculus to evaluate the definite integral $$\in

Approximating the Area with Riemann Sums

Consider the linear function $$f(x) = 2*x + 1$$ on the interval $$[1,5]$$. Use Riemann sums to appro

Average Temperature Calculation over 12 Hours

In a city, the temperature over a 12-hour period is modeled by $$T(t) = -2*t + 20$$ (in $$^\circ C$$

Average Value of a Log Function

Let $$f(x)=\ln(1+x)$$ for $$x \ge 0$$. Find the average value of $$f(x)$$ on the interval [0,3].

Calculating Total Distance Traveled from a Changing Velocity Function

A particle moves with a velocity given by $$v(t)=t^2 - 4*t + 3$$ (in m/s) for $$0 \le t \le 5$$. Not

Car Fuel Consumption Analysis

A car engine’s fuel dynamics are modeled such that fuel is consumed at a rate of $$f(t)=0.1t^2$$ L/m

Coffee Brewing Dynamics

An advanced coffee machine drips water into the brewing chamber at a rate of $$W(t)=10+t$$ mL/s, whi

Comparing Riemann Sum and the Fundamental Theorem

Let $$f(x)=3*x^2$$ on the interval $$[1,4]$$.

Computing Accumulated Volume from a Filling Rate Function

A small pond is being filled at a rate given by $$r(t)=2*t + 3$$ (in $$m^3/hr$$), where $$t$$ is in

Economic Cost Function Analysis

A company's marginal cost (in dollars per unit) is recorded at various production levels. Use the da

Environmental Modeling: Pollution Accumulation

The pollutant enters a lake at a rate given by $$P(t)=5*e^{-0.3*t}$$ (in kg per day) for $$t$$ in da

Estimating Displacement with a Midpoint Riemann Sum

A vehicle’s velocity is modeled by the function $$v(t) = -t^{2} + 4*t$$ (in meters per second) over

Estimating Work Done Using Riemann Sums

In physics, the work done by a variable force is given by $$W=\int F(x)\,dx$$. A force sensor record

Evaluating Total Rainfall Using Integral Approximations

During a storm, the rainfall rate (in inches per hour) was recorded at several times. The table belo

Exploring the Fundamental Theorem of Calculus

Let the function $$F(x) = \int_{1}^{x} \frac{1}{t^2+1}\,dt$$ represent an accumulation function. Ans

FRQ4: Inverse Analysis of a Trigonometric Accumulation Function

Let $$ H(x)=\int_{0}^{x} (\sin(t)+2)\,dt $$ for $$ x \in [0,\pi] $$, representing a displacement fun

FRQ8: Inverse Analysis of a Piecewise-Defined Accumulation Function

Let $$ R(x)=\begin{cases} \int_{1}^{x} t\,dt, & 1 \le x \le 3 \\ \int_{1}^{x} (2*t-1)\,dt, & x > 3 \

FRQ9: Inverse Analysis of an Area Accumulation Function in a Meteorological Context

A region's accumulated rainfall over time (in inches) is given by $$ A(x)=\int_{0}^{x} (0.5*t+1)\,dt

FRQ12: Inverse Analysis of a Temperature Accumulation Function

The cumulative temperature above freezing over the morning is modeled by $$ T(t)=\int_{0}^{t} (0.8*t

FRQ13: Inverse Analysis of an Investment Growth Function

An investment's accumulated value is given by $$ G(t)=\int_{0}^{t} \frac{1}{1+u}\,du $$ for t ≥ 0. A

Mixed Method Approximation of an Integral

A function $$f(t)$$ that represents a biological rate is recorded over time. Use the table below to

Modeling Water Volume in a Tank via Integration

A tank is being filled with water at a rate given by $$R(t)= \frac{50}{t+2}$$ cubic meters per minut

Motion Analysis with Variable Acceleration

A particle moves along a straight line with acceleration given by $$a(t)=8-2*t$$ (in m/s²). The part

Net Change vs Total Accumulation in a Velocity Function

A particle moves with velocity $$v(t)=5-t^2$$ (in m/s) for t in [0,4]. Answer the following:

Net Surplus Calculation

A consumer's satisfaction is given by $$S(x)=100-4*x^2$$ and the marginal cost is given by $$C(x)=30

Oxygen Levels in a Bioreactor

In a bioreactor, oxygen is introduced at a rate $$O_{in}(t)= 7 - 0.5t$$ mg/min and is consumed at a

Particle Trajectory in the Plane

A particle moves in the plane with its velocity components given by $$v_x(t)=\cos(t)$$ and $$v_y(t)=

Population Growth and Accumulation

A rabbit population grows in an enclosed field at a rate given by the differential equation $$P'(t)=

Population Growth in a Bacterial Culture

A bacterial culture grows at a rate given by $$r(t)=0.5*e^{-0.3*t}$$ (in thousands of bacteria per h

Population Growth: Accumulation through Integration

A certain population grows at a rate modeled by $$R(t)= 0.5*t^2 - 3*t + 10$$ (individuals per year),

Rainfall Accumulation via Integration

A region experiences rain where the rate of rainfall (in inches per hour) is given by $$r(t)=0.5+0.2

Rainwater Collection in a Reservoir

Rainwater falls into a reservoir at a rate given by $$R(t)= 12e^{-0.5t}$$ L/min while evaporation re

River Flow Volume Calculation

A river has a flow rate given by $$Q(t)=4+\sin(t)$$ (in m³/s), where t is time in hours. Compute the

Total Distance from Velocity Data

A car’s velocity, in meters per second, is recorded over time as given in the table below: | Time (

Total Distance Traveled from Velocity Data

A particle moves along a line with velocity given by $$v(t)=t^3 - 6*t^2 + 9*t$$ (in m/s) for t in [0

Volume of a Solid of Revolution Using the Disk/Washer Method

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

Volume of a Solid with Square Cross Sections

A solid has a base in the xy-plane bounded by the curves $$y=x$$ and $$y=x^2$$. Cross sections perpe

Water Accumulation in a Tank

Water flows into a tank at a rate given by $$R(t)=2*\sqrt{t}$$ (in m³/min) for t in minutes. Answer

Work Done by a Variable Force

A force acting on an object is given by $$F(x)=3*x^2+2*x$$ where x is in meters and F in Newtons. Th

Analysis of an Autonomous Differential Equation

Consider the autonomous differential equation $$\frac{dy}{dx}=y(4-y)$$ with the initial condition $$

Applying the SIPPY Method to $$dy/dx = \frac{4x}{y}$$

Solve the differential equation $$\frac{dy}{dx}=\frac{4x}{y}$$ with the initial condition $$y(0)=5$$

Bank Account with Continuous Interest and Withdrawals

A bank account accrues interest continuously at an annual rate of $$6\%$$, while money is withdrawn

Bernoulli Differential Equation

Solve the Bernoulli differential equation $$\frac{dy}{dx}-\frac{1}{x}y=-x*y^2$$ for $$x>0$$ with the

Cooling of Electronic Components

After shutdown, the temperature $$T$$ (in °C) of an electronic component is recorded over time (in s

Drug Infusion and Elimination

The concentration of a drug in a patient's bloodstream is modeled by the differential equation $$\fr

Economic Decay Model

An asset depreciates in value according to the model $$\frac{dC}{dt}=-rC$$, where $$C$$ is the asset

Epidemic Model: Logistic Growth of Infected Individuals

In a closed population, the spread of an infection is modeled by the logistic differential equation

Epidemic Spread (Simplified Logistic Model)

In a simplified model of an epidemic, the number of infected individuals $$I(t)$$ (in thousands) is

Homogeneous Differential Equation

Consider the differential equation $$\frac{dy}{dx} = \frac{x+y}{x-y}$$ with the initial condition $$

Implicit Differential Equation and Asymptotic Analysis

Consider the differential equation $$\frac{dy}{dx}= \frac{y(1-y)}{x}$$ for $$x > 0$$ with the initia

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

Solve the differential equation $$\frac{dy}{dx} = \frac{x+2}{y+1}$$ with the initial condition $$y(0

Integrating Factor Initial Value Problem

Solve the initial value problem $$\frac{dy}{dx}+\frac{2}{x}y=x^2$$ for $$x>0$$ with $$y(1)=3$$.

Mixing a Salt Solution

A mixing tank experiment records the salt concentration $$C$$ (in g/L) at various times $$t$$ (in mi

Mixing Problem with Changing Volume

A tank initially contains 100 L of water with 5 kg of salt. Brine enters the tank at 3 L/min with a

Mixing Problem with Time-Dependent Inflow Concentration

A tank initially contains 100 liters of water with 8 kg of dissolved salt. Brine enters the tank at

Mixing Problem with Time-Dependent Inflow Rate

A tank initially holds 200 L of water with 10 kg of salt. Brine containing 0.2 kg/L of salt flows in

Mixing Problem with Variable Volume

A tank initially contains 200 liters of solution with 10 kg of solute. A solution with concentration

Modeling Cooling with Newton's Law of Cooling

A hot beverage cools according to Newton's Law of Cooling, modeled by the differential equation $$\f

Non-Separable to Linear DE

Consider the differential equation $$\frac{dy}{dx} = \frac{y}{x}+x^2$$ with the initial condition $$

Population Growth in a Bacterial Culture

A bacterial culture has its population measured (in thousands) at various times (in hours). The tabl

Radioactive Decay

A radioactive substance decays according to $$\frac{dy}{dt} = -0.05\,y$$ with an initial mass of $$y

Radioactive Material with Constant Influx

A laboratory receives radioactive waste material at a constant rate of $$3$$ g/day. Simultaneously,

RC Circuit Discharge

In an RC circuit, the voltage across a capacitor decays according to $$\frac{dV}{dt}=-\frac{1}{RC}V$

Separable Differential Equation involving $$y^{1/3}$$

Consider the differential equation $$\frac{dy}{dx} = y^{1/3}$$ with the initial condition $$y(8)=27$

Separable Differential Equation with Initial Condition

Consider the differential equation $$\frac{dy}{dx}= \frac{x^2}{2*y}$$ with the initial condition $$y

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

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

Slope Field and General Solution

Consider the differential equation $$\frac{dy}{dx}=x$$. The attached slope field shows the slopes at

Slope Field Exploration

Consider the differential equation $$\frac{dy}{dx} = \sin(x)$$. The provided slope field (see stimul

Tumor Growth with Allee Effect

The growth of a tumor is modeled by the differential equation $$\frac{dP}{dt}=rP\left(1-\frac{P}{K}\

Analysis of a Rational Function's Average Value

Consider the rational function $$f(x)=\frac{2*x}{x^2+1}$$ defined on the interval $$[-1,1]$$. Analyz

Area Between Curves in an Ecological Study

In an ecological study, the population densities of two species are modeled by the functions $$P_1(x

Area Between Curves: Complex Polynomial vs. Quadratic

Consider the functions $$f(x)= x^3 - 6*x^2 + 9*x+1$$ and $$g(x)= x^2 - 4*x+5$$. These curves interse

Area Between Transcendental Functions

Consider the curves $$f(x)=\cos(x)$$ and $$g(x)=\sin(x)$$ on the interval $$[0,\frac{\pi}{4}]$$.

Area Between Two Curves

Consider the curves $$f(x)= 5*x - x^2$$ and $$g(x)= x$$. Determine the area of the region enclosed b

Arithmetic Savings Account

A person makes monthly deposits into a savings account such that the amount deposited each month for

Average Force and Work Done on a Spring

A spring is compressed according to Hooke's Law, where the force required to compress the spring is

Average of a Logarithmic Function

Let $$f(x)=\ln(x+2)$$ represent a measured quantity over the interval $$[0,6]$$.

Comparing Sales Projections

A company’s projected sales (in thousands of dollars) are modeled by the function $$f(x)=5*x-x^2$$ w

Consumer Surplus Calculation

The demand function for a certain product is given by $$D(p)=100-5*p$$ and the supply function by $$

Determining Velocity and Position from Acceleration

A particle moves along a line with acceleration given by $$a(t)=4-2*t$$ (in $$m/s^2$$). At time $$t=

Distance Traveled Analysis from a Velocity Graph

An object’s motion is recorded, and its velocity is shown on the graph provided for $$t \in [0,10]$$

Economic Profit Analysis via Area Between Curves

A company's revenue and cost are modeled by the linear functions $$R(x)=50*x$$ and $$C(x)=20*x+1000$

Estimating Instantaneous Velocity from Position Data

A car's position along a straight road is recorded over a 10-second interval as shown in the table b

Exponential Decay Function Analysis

A lab experiment models the decay of a chemical concentration with the function $$f(t)=8*e^{-0.5*t}$

Funnel Design: Volume by Cross Sections

A funnel is designed by rotating the region bounded by the curve $$y=4-x^2$$ for -2 ≤ x ≤ 2 about th

Kinematics with Variable Acceleration

A particle is moving along a straight path with an acceleration given by $$a(t)=10-6*t$$ (in m/s²) f

Manufacturing Profit with Variable Rates

A manufacturer’s profit rate as a function of time (in hours) is given by $$P(t)=100\left(1-e^{-0.2*

Particle Motion with Exponential Acceleration

A particle moves along a straight line with acceleration given by $$a(t)=2*e^{-t} - 1$$ (in m/s²) fo

Position Analysis of a Particle with Piecewise Acceleration

A particle moving along a straight line experiences a piecewise constant acceleration given by $$a(

Projectile Motion: Time of Maximum Height

A projectile is launched vertically upward with an initial velocity of $$50\,m/s$$ and an accelerati

Related Rates: Shadow Length Change

A 2-meter tall lamp post casts a shadow of a moving 1.7-meter tall person. Let $$x$$ be the distance

River Discharge Analysis

The flow rate of a river is modeled by $$Q(t)=20+5*\sin\left(\frac{\pi*t}{12}\right)$$ (in cubic met

Technology Adoption Growth

A new technology is being adopted in a community such that the number of new users each day forms a

Volume by the Washer Method

A region in the xy-plane is bounded by $$y=x$$ and $$y=x^2$$ for $$0 \le x \le 1$$. This region is r

Volume of a Solid Using the Disc Method

Consider the region in the xy-plane bounded by $$y = \sqrt{x}$$ and $$y=0$$ for $$0 \le x \le 9$$. T

Volume of a Solid with Rectangular Cross Sections

A solid has a base on the x-axis from $$x=0$$ to $$x=3$$. The cross-sectional areas (in m²) perpendi

Volume with Semicircular Cross-Sections

A solid has a base on the interval $$[0,3]$$ along the x-axis, and its cross-sectional slices perpen

Volume with Semicircular Cross‐Sections

A region in the first quadrant is bounded by the curve $$y=x^2$$ and the x-axis for $$0 \le x \le 3$