AP Calculus AB FRQ Room

Absolute Value Function Limits

Consider the function $$f(x)=\frac{|x-3|}{x-3}$$.

Advanced Analysis of a Piecewise Function

Consider the function $$f(x)=\begin{cases} x^2*\sin\left(\frac{1}{x}\right) & x\neq 0 \\ 0 & x=0 \en

Algebraic Simplification and Limit Evaluation of a Log-Exponential Function

Consider the function $$z(x)=\ln\left(\frac{e^{3*x}+e^{2*x}}{e^{3*x}-e^{2*x}}\right)$$ for $$x \neq

Analysis of Three Functions

The table below lists the values of three functions f, g, and h at selected x-values. Use the table

Analyzing Asymptotic Behavior in a Rational Function

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

Application of the Squeeze Theorem in Trigonometric Limits

Consider the function $$f(x) = x^2 * \sin(1/x)$$ for $$x \neq 0$$ with $$f(0)=0$$. Answer the follow

Arithmetic Sequence in Temperature Data and Continuity Correction

A temperature sensor records the temperature every minute and the readings follow an arithmetic sequ

Continuity Analysis with a Piecewise-defined Function

A particle’s displacement is described by the piecewise function $$s(t)= \begin{cases} t^2+1, & t <

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 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 &

Continuity of a Radical Function

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

Ensuring Continuity for a Piecewise-Defined Function

Consider the piecewise function $$p(x)= \begin{cases} ax + 3 & \text{if } x < 2, \\ x^2 + b & \text{

Evaluating a Compound Limit Involving Rational and Trigonometric Functions

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

Evaluating Trigonometric Limits Without a Calculator

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

Graphical Interpretation of Limits and Continuity

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

Implicit Differentiation with Rational Exponents

Consider the curve defined by $$x^{2/3} + y^{2/3} = 4$$. Answer the following:

Intermediate Value Theorem and Root Existence

Consider the function $$f(x)= x^3 - 6*x + 1$$ on the interval [1, 3].

Intermediate Value Theorem Application

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

Intermediate Value Theorem Application

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

Intermediate Value Theorem Application

Let $$p(x)=x^3-4x-5$$. Use the Intermediate Value Theorem (IVT) to show that the equation \(p(x)=0\)

Jump Discontinuity in a Piecewise Function

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

Limit Involving an Exponential Function

Evaluate the limit $$\lim_{x \to 0} \frac{e^{2*x} - 1}{x}$$.

Logarithmic Limit Evaluation

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

One-Sided Limits and Vertical Asymptotes

Consider the function $$ f(x)= \frac{1}{x-4} $$.

Oscillatory Function and the Squeeze Theorem

Consider the function $$f(x)= x*\sin(1/x)$$ for $$x \neq 0$$, with $$f(0)=0$$. Answer the following:

Oscillatory Function and the Squeeze Theorem

Consider the function $$f(x)=x*\sin(1/x)$$ for x ≠ 0, with f(0)=0.

Squeeze Theorem with Bounded Function

Suppose that for all x in some interval around 0, the function $$f(x)$$ satisfies $$-x^2 \le f(x) \l

Table Analysis for Estimating a Limit

The table below shows values of the function $$g(x)$$ for x near 4. Use this data to answer the foll

Trigonometric Limit Computation

Consider the function $$f(x)= \frac{\sin(5*(x-\pi/4))}{x-\pi/4}$$.

Trigonometric Limit Evaluation

Examine the function $$ f(x)= \frac{\sin(3*x)}{x} $$ for $$x\ne0$$.

Air Quality and Pollution Removal

A city experiences pollutant inflow at a rate of $$f(t)=30+2*t$$ (micrograms/m³·hr) and pollutant re

Analyzing a Function with a Removable Discontinuity

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

Analyzing a Projectile's Motion

A projectile is launched vertically, and its height (in feet) at time $$t$$ seconds is given by $$s(

Analyzing Function Behavior Using Its Derivative

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

Analyzing Rates Without a Calculator: Average vs Instantaneous Rates

Consider the function $$f(x)= x^2$$.

Approximating Derivative using Secant Lines

Consider the function $$f(x)= \ln(x)$$. A student records the following data: | x | f(x) | |---

Bacterial Culture Growth with Washout

In a bioreactor, bacteria grow at a rate of $$f(t)=50*e^{0.05*t}$$ (cells/min) while simultaneous wa

Critical Points of a Log-Quotient Function

Let $$f(x)= \frac{\ln(x)}{x}$$ for $$x > 0$$. This function is used to analyze efficiency in logarit

Derivative of an Exponential Decay Function

Consider the function $$f(t)=e^{-0.5*t}$$, which may represent the decay of a substance over time. A

Derivatives of Trigonometric Functions

Let $$f(x)=\sin(x)+\cos(x)$$, where $$x$$ is measured in radians. This function may represent a comb

Differentiability and Continuity

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

Differentiation of Exponential Functions

Consider the function $$f(x)=e^{2*x}-3*e^{x}$$.

Differentiation Using the Product Rule

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

DIY Rainwater Harvesting System

A household's rainwater harvesting system collects rain at a rate of $$f(t)=12-0.5*t$$ (liters/min)

Electricity Consumption with Renewable Generation

A household has solar panels that generate power at a rate of $$f(t)=50*\sin\left(\frac{\pi*t}{12}\r

Evaluating Limits and Discontinuities in a Piecewise Function

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

Exponential Growth Rate

Consider the function $$f(x)= 3*e^{2*x}$$ which models a quantity growing exponentially over time.

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.

Implicit Differentiation in Motion

A particle’s motion is given by the implicit equation $$y^2 + x*y = 10$$, where x represents time (i

Inverse Function Analysis: Exponential Transformation

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

Inverse Function Analysis: Rational Function

Consider the function $$f(x)=\frac{2*x+1}{x+3}$$ defined for all x except $$x=-3$$.

Inverse Function Analysis: Rational Function 2

Consider the function $$f(x)=\frac{x+4}{x+2}$$ defined for $$x\neq -2$$, with the additional restric

Marginal Profit Calculation

A company’s profit (in thousands of dollars) is given by $$P(x)= -2*x^2 + 50*x - 100$$, where $$x$$

Physical Motion with Variable Speed

A car's velocity is given by $$v(t)= 2*t^2 - 3*t + 1$$, where $$t$$ is in seconds.

Population Growth and Instantaneous Rate of Change

A town's population is modeled by $$P(t)= 2000*e^{0.05*t}$$, where $$t$$ is in years. Analyze the ch

Product and Chain Rule Combined

Let \(f(x)=(3*x+1)^2 * \cos(x)\). Answer the following parts.

Quotient Rule Challenge

For the function $$f(x)= \frac{3*x^2 + 2}{5*x - 7}$$, find the derivative.

Rainfall-Runoff Model

A reservoir receives water from rainfall at a rate modeled by $$R_{in}(t)=10*\sin\left(\frac{\pi*t}{

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

Using the Limit Definition to Derive the Derivative

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

Using the Quotient Rule for a Rational Function

Let $$f(x) = \frac{3*x+5}{x-2}$$. Differentiate $$f(x)$$ using the quotient rule.

Water Treatment Plant's Chemical Dosing

A water treatment plant adds a chemical at a rate of $$f(t)=5+0.2*t$$ (liters/min) while the chemica

Chain Rule in an Implicitly Defined Function

Consider the equation $$\tan(x+y)=x^2-y^2$$. Answer the following:

Chain Rule in Population Modeling

A biologist models the population of a species with the function $$P(t)= f(g(t))$$, where $$g(t)=25*

Chain Rule with Trigonometric and Exponential Functions

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

Chemical Reaction Rate: Exponential and Logarithmic Model

The concentration of a chemical reaction is modeled by $$C(t)= \ln\left(3*e^(2*t) + 7\right)$$, wher

Composite Function and Tangent Line

Consider the function $$f(x)=\sqrt{3*x^2+2*x+1}$$. (Note: All derivatives are to be computed without

Composite Function Differentiation with Logarithms

A function is given by $$h(x)=\ln((5*x+1)^2)$$. Use the chain rule to differentiate $$h(x)$$.

Composite Function Modeling in Finance

A bank models the growth of a savings account by the function $$B(t)= f(g(t))$$, where $$g(t)= \ln(t

Composite Function with Nested Chain Rule

Let $$h(x)=\sqrt{\ln(4*x^2+1)}$$. Answer the following:

Composite Functions with Multiple Layers

Let $$f(x)=\sqrt{\ln(5*x^2+1)}$$. Answer the following:

Composite Log-Exponential Function Analysis

A function is defined by $$f(x)=\ln\left(e^(2*x^2) + 5\right)$$. This function is composed of an exp

Composite Temperature Model

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

Differentiating an Inverse Trigonometric Function

Let $$y = \arcsin\left(\frac{2*x}{1+x^2}\right)$$.

Differentiation Involving Exponentials and Inverse Trigonometry

Consider the function $$M(x)=e^{\arctan(x)}\cdot\cos(x)$$.

Differentiation Involving Inverse Sine and Exponentials

Let $$f(x)= \arcsin(e^(-x))$$, where the domain is chosen so that $$e^(-x)$$ is within [-1, 1]. Solv

Implicit Differentiation Involving Logarithms

Consider the equation $$\ln(x) + x*y = \ln(y) + x$$ which relates $$x$$ and $$y$$. Use implicit diff

Implicit Differentiation with Exponential and Trigonometric Functions

Consider the equation $$e^{x*y}+\sin(y)= x$$, which relates \(x\) and \(y\). This equation may repre

Inverse Function Derivative

Suppose that $$f$$ is a differentiable and one-to-one function. Given that $$f(4)=10$$ and $$f'(4)=2

Inverse Function Derivative with Composite Functions

Consider the function $$f(x)=x^3+2*x+1$$, which is one-to-one on its domain. Given that $$f(1)=4$$,

Inverse Function Differentiation

Let $$f(x)=x^3+x+1$$, a one-to-one function, and let $$g$$ be the inverse of $$f$$. Use inverse func

Inverse Function Differentiation in Temperature Conversion

In a temperature conversion model, the function $$f(T)=\frac{9}{5}*T+32$$ converts Celsius temperatu

Inverse Function Differentiation Involving a Polynomial

Let $$f(x)= x^3 + 2*x + 1$$. Analyze its invertibility and the derivative of its inverse function.

Inverse Function Differentiation: Composite Inversion

Let $$f(x) = \frac{x}{1-x}$$ for x < 1, and let g denote its inverse function. Answer the following

Logarithmic Differentiation of a Composite Function

For the function $$y= (x^2+1)^(\tan(x))$$, use logarithmic differentiation to address the following

Manufacturing Optimization via Implicit Differentiation

A manufacturing cost relationship is given implicitly by $$x^2*y + x*y^2 = 1000$$, where $$x$$ repre

Temperature Change Model Using Composite Functions

The temperature of an object is modeled by the function $$T(t)=e^{-\sqrt{t+2}}$$, where $$t$$ is tim

Accelerating Car Motion Analysis

A car's velocity is modeled by $$v(t)=4t^2-16t+12$$ in m/s for $$t\ge0$$. Analyze the car's motion.

Airplane Altitude Change

An airplane's altitude (in meters) as a function of time is modeled by $$A(t)= 500*t - 4.9*t^2 + 100

Analyzing a Nonlinear Rate of Revenue Change

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

Coffee Cooling Analysis Revisited

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

Cooling Hot Beverage

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

Critical Points and Concavity Analysis

Consider the polynomial function $$f(x)= x^4 - 4*x^3 + 6*x^2 - 4*x + 1$$ modeling the position of an

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 $$

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

FRQ 11: Shadow Length Change

A 2‑m tall person walks away from a 10‑m tall lamp post. Let x be the distance from the lamp post to

FRQ 17: Water Heater Temperature Change

The temperature of water in a heater is modeled by $$T(t) = 20 + 80e^{-0.05*t}$$, where t is in minu

Implicit Differentiation in Related Rates

A 5-foot ladder leans against a wall such that its bottom slides away from the wall. The relationshi

L’Hôpital’s Rule in Limit Evaluation

Evaluate the limit $$\lim_{x\to\infty}\frac{5*x^3 - 4*x + 1}{2*x^3 + 7*x^2 - x + 6}$$ using L’Hôpita

Linear Approximation in Estimating Function Values

Let $$f(x)= \ln(x)$$. Analyze its linear approximation.

Linear Approximation of Natural Logarithm

Estimate $$\ln(1.05)$$ using linear approximation for the function $$f(x)=\ln(x)$$ at $$a=1$$.

Marginal Analysis in Economics

A company’s cost function is given by $$C(x)=0.5*x^3 - 3*x^2 + 5*x + 8$$, where $$x$$ represents the

Medicine Dosage: Instantaneous Rate of Change

The concentration of a medicine in the bloodstream is given by $$C(t) = 25e^{-0.2t}+5$$, where $$t$$

Modeling Coffee Cooling

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

Optimization for Minimizing Time in Road Construction

A new road connecting two towns involves two segments with different construction speeds. The travel

Optimization: Minimizing Surface Area of a Box

An open-top box with a square base is to have a volume of 500 cubic inches. The surface area (materi

Optimizing Crop Yield

The yield per acre of a crop is modeled by the function $$Y(p) = 100\,p\,e^{-0.1p}$$, where $$p$$ is

Projectile Motion Analysis

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

Projectile Motion Analysis

The height of a projectile is modeled by the function $$h(t) = -4.9t^2 + 20t + 2$$, where $$t$$ is i

Projectile Motion with Velocity Components

A projectile is launched from the ground with a constant horizontal velocity of 15 m/s and a vertica

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

Radioactive Decay: Rate of Change and Half-life

A radioactive substance decays according to the formula $$N(t)=N_0e^{-kt}$$, where $$N(t)$$ is the a

Rate of Change in a Population Model

A population model is given by $$P(t)=30e^{0.02t}$$, where $$P(t)$$ is the population in thousands a

Rate of Change in Pool Volume

The volume $$V(t)$$ (in gallons) of water in a swimming pool is given by $$V(t)=10t^2-40t+20$$, wher

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 Oil Spill

An oil spill on calm water forms a perfect circle. The area of the spill is increasing at a constant

Revenue Function and Marginal Revenue Analysis

A company's revenue is modeled by $$R(x)= -0.5*x^3 + 20*x^2 + 15*x$$, where $$x$$ represents the num

Savings Account Growth Modeled by a Geometric Sequence

A savings account has an initial balance of $$B_0=1000$$ dollars. The account earns compound interes

Temperature Change Analysis

The temperature of a chemical solution is recorded over time. Use the table below, where $$T(t)$$ (i

Temperature Change in a Cooling Process

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

Vehicle Deceleration Analysis

A vehicle’s position is given by $$s(t)=100t-5t^2$$ where $$s(t)$$ is in meters and $$t$$ in seconds

Water Tank Volume Change

A water tank is being filled and its volume is given by $$V(t)= 4*t^3 - 9*t^2 + 5*t + 100$$ (in gall

Application of Rolle's Theorem

Let $$f(x)$$ be a function that is continuous on $$[0,5]$$ and differentiable on $$(0,5)$$ with $$f(

Application of Rolle's Theorem to a Trigonometric Function

Consider the function $$f(x) = \sin(x)$$ defined on the interval $$[0, \pi]$$. Answer the following:

Chemical Reaction Rate and Exponential Decay

In a chemical reaction, the concentration of a reactant declines according to $$C(t)= C_0* e^{-k*t}$

Derivative and Concavity of f(x)= e^(x) - ln(x)

Consider the function $$f(x)= e^{x}-\ln(x)$$ for $$x>0$$. Answer the following:

Discontinuity and Derivative in a Hybrid Piecewise Function

Consider the function $$ f(x) = \begin{cases} x^2, & x \le 1, \\ 3x - 2, & x > 1. \end{cases} $$ A

FRQ 7: Maximizing Revenue in Production

A company’s revenue function is modeled by $$R(x)= -2*x^2 + 40*x$$ (in thousands of dollars), where

FRQ 10: First Derivative Test for a Cubic Profit Function

A company’s profit function is given by $$P(x)= x^3 - 9*x^2 + 24*x + 1$$, where $$x$$ represents the

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

Function Behavior Analysis Using Derivatives

Examine the function $$f(x) = \ln(x) + x$$, where $$x > 0$$.

Graphical Analysis and Derivatives

A function \( f(x) \) is represented by the graph provided below. Answer the following based on the

Inflection Points and Concavity in a Real-World Cost Function

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

Inverse Analysis in a Modeling Context: Population Growth

A population is modeled by the function $$f(t)=\frac{500}{1+50*e^{-0.1*t}}$$, where t represents tim

Inverse Analysis of a Composite Trigonometric-Linear Function

Consider the function $$f(x)=2*\sin(x)+x$$ defined on the interval $$\left[-\frac{\pi}{2}, \frac{\pi

Liquid Cooling System Flow Analysis

A specialized liquid cooling system operates with non-linear flow rates. The inflow rate is given by

Logistic Growth Model and Derivative Interpretation

Consider the logistic growth model given by $$f(t)= \frac{5}{1+ e^{-t}}$$, where $$t$$ represents ti

Mean Value Theorem Analysis

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

Mean Value Theorem Applied to Car Position Data

A car’s position (in meters) is recorded at various times during a journey. Use the information prov

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

Motion Analysis with Acceleration Function

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

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

Optimizing a Box with a Square Base

A company is designing an open-top box with a square base of side length $$x$$ and height $$h$$. The

Pharmacokinetics: Drug Concentration Decay

A drug in the bloodstream decays according to $$D(t)= D_0*e^{-k*t}$$, where $$t$$ is in hours. Answe

Piecewise Function and the Mean Value Theorem

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

Reservoir Evaporation and Rainfall

A reservoir gains water through rainfall and loses water by evaporation. Rainfall occurs at a rate g

Solving a Log-Exponential Equation

Solve the equation $$\ln(x)+x=0$$ for $$x>0$$. Answer the following:

Temperature Analysis Over a Day

The temperature $$f(x)$$ (in $$^\circ C$$) at time $$x$$ (in hours) during the day is modeled by $$f

Temperature Regulation in a Greenhouse

A greenhouse is regulated by an inflow of warm air and an outflow of cooler air. The inflow temperat

Traffic Flow Modeling

A highway segment experiences varying traffic flows. Cars enter at a rate $$I(t)=50+10*\sin(\frac{\p

Trigonometric Function Behavior

Consider the function $$f(x)= \sin(x) + \cos(2*x)$$ defined on the interval $$[0,2\pi]$$. Analyze it

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

Accumulated Change Function Evaluation

Let $$F(x)=\int_{1}^{x} (2*t+3)\,dt$$ for $$x \ge 1$$. This function represents the accumulated chan

Accumulated Chemical Concentration

A scientist observes that the rate of change of chemical concentration in a solution is given by $$r

Accumulation and Flow Rate in a Tank

Water flows into a tank at a rate given by $$R(t)=3*t^2-2*t+1$$ (in m³/hr) for $$0\le t\le2$$. The t

Accumulation and Total Change in a Population Model

A population grows at a rate given by $$r(t)=0.2*t^2 - t + 5$$ (in thousands per year), where t is i

Antiderivative of a Transcendental Function

Consider the function $$f(x)=\frac{2}{x}$$. Answer the following parts:

Area Between Curves

Consider the curves defined by $$f(x)=x^2$$ and $$g(x)=2*x$$. The region enclosed by these curves is

Comparing Riemann Sum Methods for a Complex Function

Consider the function $$f(x)=\frac{1}{1+x^2}$$ on the interval [0,1]. Answer the following:

Elevation Profile Analysis on a Hike

A hiker records the elevation (in meters) along a trail at various distances. Use this data to analy

Estimating Accumulated Water Inflow Using Riemann Sums

A water tank fills at varying rates. The table below shows the inflow rate in liters per second at d

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 an Integral with a Piecewise Function

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

Evaluating the Accumulated Drug Concentration

In a pharmacokinetics study, a drug is infused into a patient's bloodstream at a rate given by $$R(t

Experimental Data Analysis using Trapezoidal Sums

A chemical reaction is monitored over time, and the reaction rate $$f(t)$$ (in moles per minute) is

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

FRQ2: Inverse Analysis of an Antiderivative Function

Consider the function $$ G(x)=\int_{0}^{x} (t^2+1)\,dt $$ for all real x. Answer the following parts

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

FRQ20: Inverse Analysis of a Function with a Piecewise Continuous Integrand

Consider the function $$ I(x)= \begin{cases} \int_{0}^{x}\cos(t)\,dt, & 0 \le x \le \pi/2 \\ \int_{0

Fuel Consumption: Approximating Total Fuel Use

A car's fuel consumption rate (in liters per hour) is modeled by $$f(t)=0.05*t^2 - 0.3*t + 2$$, wher

Fundamentals of Accumulation: Displacement and Total Distance

A cyclist's velocity is modeled by $$v(t)= 4 - |t-2|$$ (in m/s) for $$t$$ in the interval $$[0,4]$$.

Graphical Analysis of an Accumulation Function

Let $$f(t)$$ represent the rate of water flow (in $$m^3/hr$$) into a reservoir, and suppose the grap

Motion Along a Line: Changing Velocity

A particle moves along a line with a velocity given by $$v(t)=12-2*t$$ (in m/s) for $$0\le t\le8$$,

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

Pollutant Concentration in a River

Pollutants are introduced into a river at a rate $$D(t)= 8\sin(t)+10$$ kg/hr while a treatment plant

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

Roller Coaster Work Calculation

An amusement park engineer recorded the force applied by a roller coaster engine (in Newtons) at var

Temperature Change in a Chemical Reaction

During an exothermic chemical reaction, the temperature (in °C) is recorded over a 15-minute period.

Trapezoidal Approximation for a Changing Rate

The following table represents the flow rate (in L/min) of water entering a tank at various times:

Water Flow in a Tank

Water flows into a tank at a rate given by $$R(t)=3*t+2$$ (in liters per minute) for $$0 \le t \le 6

Work Done by a Variable Force

A variable force acting along a straight line is given by $$F(x)=3*x^{2} - 2*x + 1$$ (in newtons) fo

Area Under a Differential Equation Curve

Consider the differential equation $$\frac{dy}{dx} = y(1-y)$$, with a particular solution given by $

Bacterial Population with Time-Dependent Growth Rate

A bacterial population grows according to the differential equation $$\frac{dP}{dt}=\frac{k}{1+t^2}P

Balloon Inflation with Leak

A balloon is being inflated at a rate of $$5$$ liters/min, but it is also leaking air at a rate prop

Charging of a Capacitor

The voltage $$V$$ (in volts) across a capacitor being charged in an RC circuit is recorded over time

Cooling of Electronic Components

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

Cooling with Variable Ambient Temperature

An object cools in an environment where the ambient temperature varies with time. Its temperature $$

Economic Growth with Investment Outflow

A company’s investment fund grows continuously at an annual rate of $$5\%$$, but expenses lead to a

Falling Object with Air Resistance

A falling object experiences air resistance proportional to its velocity. Its motion is modeled by t

Heating a Liquid in a Tank

A liquid in a tank is being heated by mixing with an incoming fluid whose temperature oscillates ove

Implicit Differentiation and Slope Analysis

Consider the function defined implicitly by $$y^2+ x*y = 8$$. Answer the following:

Investigating a Piecewise Function's Discontinuity

Consider the function $$ f(x)=\begin{cases} \frac{x^2-9}{x-3}, & x\neq 3\\ 5, & x=3 \end{cases} $$

Investment Growth with Withdrawals

An investment account grows at a rate proportional to its current balance, but a constant amount is

Logistic Growth Model

A population $$P(t)$$ grows according to the logistic model $$\frac{dP}{dt} = rP\left(1-\frac{P}{K}\

Logistic Growth Model for Population Dynamics

A population $$P$$ is modeled by the logistic differential equation $$\frac{dP}{dt}=0.5*P\left(1-\fr

Mixing Problem in a Salt Solution Tank

A 100-liter tank initially contains a solution with 10 kg of salt. Brine with a salt concentration o

Mixing Problem: Salt in a Tank

A 100-liter tank initially contains 50 grams of salt. Brine with a salt concentration of $$0.5$$ gra

Mixing with Variable Inflow Rate

A 50-liter tank initially contains water with 1 kg of dissolved salt. Water containing 0.2 kg of sal

Motion Under Gravity with Air Resistance

An object is falling vertically under the influence of gravity and air resistance. Its velocity $$v(

Population Dynamics with Harvesting

A wildlife population (in thousands) is monitored over time and is subject to harvesting. The popula

Radioactive Decay and Half-Life

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

Radioactive Decay Model

A radioactive substance decays according to the differential equation $$\frac{dN}{dt} = -kN$$. At ti

Radioactive Material with Continuous Input

A radioactive substance decays at a rate proportional to its amount while being produced continuousl

RC Circuit Discharge

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

Related Rates: Expanding Balloon

A spherical balloon is inflated such that its radius increases at a constant rate of $$\frac{dr}{dt}

Separable Differential Equation with Trigonometric Component

Solve the differential equation $$\frac{dy}{dx}=\frac{3x^2}{1+\sin(y)}$$ with the initial condition

Separable Differential Equation: y and x

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

Separable Differential Equation: y' = (2*x)/y

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

Substitution to Linearize

The differential equation $$\frac{dy}{dx} = \frac{x + y}{1 - x*y}$$ appears non-linear. With the sub

Tank Draining Differential Equation

Water drains from a tank at a rate that depends on the square root of the volume, according to $$\fr

Accumulated Electrical Charge from a Current Function

An electrical device charges according to the current function $$I(t)= 10*e^{-0.3*t}$$ amperes, wher

Accumulated Nutrient Intake from a Drip

A medical nutrient drip administers a nutrient at a variable rate given by $$N(t)=-0.03*t^2+1.5*t+20

Accumulated Rainfall Calculation

During a 12-hour storm, the rainfall rate is modeled by $$r(t)=3+\sin(t)$$ (in mm/hr) for $$0 \le t

Area Between Curves: Exponential vs. Linear

Consider the functions $$f(x)=e^{-x}$$ and $$g(x)=1-x$$. A table of approximate values is provided b

Average Concentration in a Chemical Reaction

A chemical reaction in a laboratory setting is monitored by recording the concentration (in moles pe

Average Flow Rate in a River

The flow rate of a river (in $$m^3/s$$) is measured over a 12-hour period. Use the data provided in

Average Force Calculation

An object experiences a variable force given by $$F(x)=3*x^2-2*x+1$$ (in Newtons) for $$0 \le x \le

Average Growth Rate in a Biological Process

In a biological study, the instantaneous growth rate of a bacterial colony is modeled by $$k(t)=0.5*

Average Speed Over a Journey

A car travels along a straight road and its speed (in m/s) is modeled by the function $$v(x)=2*x^2-3

Average Value of a Deposition Rate Function

During a sediment deposition experiment, the deposition rate (in mm/hr) was recorded over a 10-hour

Average Velocity from Position Data

The position of a vehicle moving along a straight road is given in the table below. Use these data t

Boat Navigation Across a River with Current

A boat aims to cross a river that is 100 m wide. The boat moves due north at a constant speed of 5 m

Comparing Sales Projections

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

Cost Analysis: Area Between Quadratic Cost Functions

Two cost functions for production are given by $$C_1(x)=0.5*x^2+3*x+10$$ and $$C_2(x)=0.3*x^2+4*x+5$

Displacement from a Velocity Graph

A moving object has its velocity given as a function of time. A velocity versus time graph is provid

Distance Traveled by a Jogger

A jogger increases her daily running distance by a fixed amount. On the first day she runs $$2$$ km,

Environmental Impact: Average Pollutant Concentration

The pollutant concentration in a river is modeled by $$h(x)=0.01*x^3-0.5*x^2+5*x$$ (in mg/L) over a

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

Modeling Bacterial Growth

A bacterial culture grows at a rate modeled by $$g(t)=a*e^{0.3*t}$$, where $$t$$ is time in hours an

Particle Motion on a Line

A particle moves along a straight line with a velocity function given by $$v(t)=3*t^2-12*t+9$$ (in m

Particle Motion on a Parametric Path

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

Retirement Savings Auto-Increase

A person contributes to a retirement fund such that the monthly contributions form an arithmetic seq

Solid of Revolution: Water Tank

A water tank is formed by rotating the region bounded by the curve $$y=\sqrt{x}$$, the x-axis, and t

Temperature Average Calculation

A scientist records the temperature in a lab using a continuous function $$T(t)=3*t^2 - 4*t + 5$$, w

Volume by the Cylindrical Shells Method

A region bounded by $$y=\ln(x)$$, $$y=0$$, and the vertical line $$x=e$$ is rotated about the y-axis

Volume by the Washer Method: Solid of Revolution

Consider the region bounded by the curves $$y=x$$ and $$y=x^2$$ for $$0 \le x \le 1$$. This region i

Volume of a Solid of Revolution Rotated about a Line

Consider the region bounded by $$y=x^2$$ and $$y=x$$ for $$x\in [0,1]$$. This region is rotated abou

Volume of a Solid of Revolution: Curve Raised to a Power

Consider the region under the curve $$f(x)=x^{1/3}$$ for $$x\in [0,8]$$. This region is revolved abo

Volume of a Solid with a Hole Using the Washer Method

Consider the region bounded by $$y=x^2$$ and $$y=4$$. This region is revolved about the $$x$$-axis t

Volume with Square Cross-Sections

Consider the region bounded by the curve $$y=x^2$$ and the line $$y=4$$ for $$0 \le x \le 2$$. Squar

Water Flow in a River: Average Velocity and Flow Rate

A river has a width of 10 meters. The velocity of the water at a distance $$x$$ (in meters) from one

Work in Pumping Water

A water pump is used to empty a reservoir. The force required to pump water out at a depth $$y$$ (in

Work in Pumping Water from a Conical Tank

A water tank is in the shape of an inverted right circular cone with height $$10\,m$$ and top radius