AP Calculus AB FRQ Room

Absolute Value Function and Discontinuity

Consider the function $$f(x)=\begin{cases} \frac{|x-5|}{x-5} & x\neq5 \\ 0 & x=5 \end{cases}$$. Answ

Algebraic Manipulation and Limit Evaluation

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

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

Applying the Squeeze Theorem to an Oscillatory Function

Let the function $$g(x)= \begin{cases} x \cos\left(\frac{1}{x}\right) & \text{if } x \neq 0, \\ 0 &

Area and Volume Setup with Bounded Regions

Consider the region R bounded by the curves $$y = x^2$$, $$y = 4$$, and $$x = 0$$. Though integratio

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 of Constant Functions

Consider the constant function $$f(x)=7$$ for all x. Answer the following parts.

Determining Horizontal Asymptotes for Rational Functions

Given the rational function $$R(x)= \frac{2*x^3+ x^2 - x}{x^3 - 4}$$, answer the following:

Determining Parameters for a Continuous Log-Exponential Function

Suppose a function is defined by $$ v(x)=\begin{cases} \frac{\ln(e^{p*x}+x)-q*x}{x} & \text{if } x \

Discontinuities in a Rational-Exponential Function

Consider the function $$ f(x) = \begin{cases} \frac{e^{x} - 1}{x}, & x \neq 0 \\ k, & x = 0. \en

Discontinuity in Acceleration Function and Integration

A particle’s acceleration is defined by the piecewise function $$a(t)= \begin{cases} \frac{1-t}{t-2}

Evaluating a Limit with Radical Expressions

Evaluate the limit $$\lim_{x \to 9} \frac{\sqrt{x} - 3}{x - 9}$$. Answer the following:

Factorization and Removable Discontinuity

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

Graph Reading: Left and Right Limits

A graph of a function f is provided below which shows a discontinuity at x = 2. Use the graph to det

Identifying Discontinuities in a Rational Function

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

Implicit Differentiation Involving Logarithms

Consider the curve defined implicitly by $$\ln(x) + \ln(y) = \ln(5)$$. Answer the following:

Intermediate Value Theorem and Continuity

Consider the function $$f(x)=\begin{cases} 2*x+1 & x<3 \\ x^2-4 & x\ge3 \end{cases}$$. Answer the f

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

Suppose a continuous function $$f(x)$$ is defined on the interval $$[1,5]$$, with $$f(1)=-3$$ and $$

Limit Involving an Exponential Function

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

Limits at Infinity and Horizontal Asymptotes

Consider the rational function $$R(x) = \frac{2x^2 - 3x + 4}{x^2 + 5}$$. Analyze its behavior as x a

Limits Involving Radicals and Algebra

Consider the function $$f(x)= \sqrt{x^2 + x} - x$$. Answer the following parts.

One-Sided Limits and Continuity of a Piecewise Function

Consider the piecewise function $$w(x)= \begin{cases} \frac{e^{x}-1}{x} & \text{if } x<0, \\ \frac{\

Oscillatory Behavior and Discontinuity

Consider the function $$f(x)=\begin{cases} x\cos(\frac{1}{x}) & x\neq0 \\ 2 & x=0 \end{cases}$$. Ans

Particle Motion with Vertical Asymptote in Velocity

A particle moves along a number line with velocity function $$v(t)= \frac{3*t}{t-1}$$ for $$t > 1$$.

Piecewise-Defined Function Continuity Analysis

Let $$f(x)$$ be defined as follows: For $$x < 2$$: $$f(x)= 3x - 1$$. For $$2 \leq x \leq 5$$: $$f(x

Rational Functions with Removable Discontinuities

Examine the function $$f(x)= \frac{x^2 - 5x + 6}{x - 2}$$. (a) Factor the numerator and simplify th

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 Limit Evaluation

Consider the function $$f(x) = \frac{(x + 3) * (x - 2)}{x + 3}$$ for $$x \neq -3$$. Answer the follo

Squeeze Theorem Application with Trigonometric Functions

Let $$f(x)= x^2 \sin(1/x)$$ for $$x\neq0$$, and define $$f(0)=0$$.

Squeeze Theorem for an Exponential Damped Function

A physical process is modeled by the function $$h(x)= x*e^{-1/(x*x)}$$ for $$x \neq 0$$ and is defin

Trigonometric Limits

Consider the functions $$g(x)=\frac{\sin(3*x)}{\sin(2*x)}$$ and $$h(x)=\frac{1-\cos(4*x)}{x^2}$$. An

Advanced Implicit Differentiation

Given the equation $$e^{x*y} + x^2 - y^2 = 5$$, answer the following:

Average and Instantaneous Rates of Change

A function $$f$$ is defined by $$f(x)=x^2+3*x+2$$, representing the height (in meters) of a projecti

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

Derivation of $$h(x)= \ln(2*x+3)$$ Using the Chain Rule

Let $$h(x)= \ln(2*x+3)$$, a composition of a logarithmic and a linear function.

Derivative from First Principles: The Function $$f(x)=\sqrt{x}$$

Consider the function $$f(x) = \sqrt{x}$$. Use the definition of the derivative to find an expressio

Derivative from the Limit Definition

Let $$f(x)=\sqrt{x}$$. Using the limit definition of the derivative, answer the following:

Derivative of the Square Root Function via Limit Definition

Let $$g(x)=\sqrt{x}$$. Using the limit definition of the derivative, answer the following parts.

Derivatives on an Ellipse

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

Economic Model: Revenue and Rate of Change

The revenue for a product is given by $$R(x)= \frac{x^2 + 10*x}{x+2}$$, where $$x$$ is in hundreds o

Exponential Decay Analysis

A radioactive substance decays according to the function $$N(t)=N_0 \cdot e^{-0.03t}$$, where t is m

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 $$h'(x)$$.

Graphical Interpretation of Rate of Change

Consider the graph of a function provided in the stimulus which shows a vehicle's displacement over

Highway Traffic Flow Analysis

Vehicles enter a highway ramp at a rate given by $$f(t)=60+4*t$$ (vehicles/min) and exit the highway

Instantaneous Acceleration from a Velocity Function

A runner's velocity is given by $$v(t)= 3*t^2 - 12*t + 9$$, where $$t$$ is in seconds. Analyze the r

Instantaneous Acceleration from a Velocity Function

An object's velocity is given by $$v(t)=3*t^2 - 12*t + 9$$, where $$t$$ is in seconds. Answer the fo

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 $$

Integrating Graph and Table Data for Revenue Analysis

A company’s revenue function $$R(x)$$ (in thousand dollars) appears to be linear. Data from a recent

Inverse Function Analysis: Cosine and Linear Combination

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

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

Inverse Function Analysis: Restricted Rational Function

Consider the function $$f(x)=\frac{2*x}{x^2+1}$$ defined on the interval $$0\leq x\leq 1$$.

Inverse Function Analysis: Sum with Reciprocal

Consider the function $$f(x)=x+\frac{1}{x}$$ defined for $$x\geq1$$.

Meteor Trajectory Analysis

A meteor’s altitude is modeled by $$y(x)=-0.5x^2+3x+20$$, where x is in kilometers. Answer the follo

Optimizing Car Speed: Rate of Change Analysis

A car’s speed in km/h is modeled by the function $$s(t)=50+2*t^2-0.1*t^3$$ for $$0 \leq t \leq 10$$

Real-World Cooling Process

In an experiment, the temperature (in °C) of a substance as it cools is modeled by $$T(t)= 30*e^{-0.

Related Rates: Expanding Ripple Circle

Water droplets create circular ripples on a surface. The area of a ripple is given by $$A = \pi * r^

Related Rates: Shadow Length Change

A person 1.8 m tall is walking away from a streetlight that is 5 m high. Let x represent the distanc

Riemann Sums and Derivative Estimation

A car’s position $$s(t)$$ in meters is recorded in the table below at various times $$t$$ in seconds

Tangent Line Approximation

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

Tangent Line to a Cubic Function

The function $$f(x) = x^3 - 6x^2 + 9x + 1$$ models the height (in meters) of a roller coaster at pos

Tangent to an Implicit Curve

Consider the curve defined implicitly by \(x^2 + y^2 = 25\). Answer the following parts.

Advanced Implicit Differentiation: Second Derivative Analysis

Consider the curve defined implicitly by the equation $$x^2*y+\sin(y)= x^3$$.

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 Population Modeling

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

Chain Rule in Temperature Variation

A metal rod's temperature along its length is given by the function $$T(x)= \cos((4*x+2)^2)$$, where

Chain Rule with Logarithms

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

Chain Rule with Trigonometric Function

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

Composite Function Analysis in Temperature Change

A chemical reaction has its temperature modeled by the function $$T(t)= \sqrt{3*t^2+1}$$. Analyze th

Composite Function and Inverse Analysis via Graph

Consider the function $$f(x)= \sqrt{4*x-1}$$, defined for $$x \geq \frac{1}{4}$$. Analyze the functi

Composite Function with Nested Exponential and Trigonometric Terms

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

Composite Functions with Multiple Layers

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

Derivative of an Inverse Function: Quadratic Case

Let $$f(x)=x^2+2$$ for $$x \ge 0$$ and let $$g = f^{-1}$$ be its inverse function.

Differentiation of a Log-Exponential-Trigonometric Composite

Consider the function $$f(x)= \ln\left(e^(\cos(x)) + x^2\right)$$. Solve the following:

Differentiation of Inverse Trigonometric Composite Function

Given the function $$y = \arctan(\sqrt{x})$$, answer the following parts.

Economic Equilibrium: Composite and Inverse Functions

In an economic model, the demand function is given by the composite function $$D(p)= f(g(p))$$, wher

Estimating Derivatives Using a Table

An experiment measures a one-to-one function $$f$$ and its inverse $$g$$, yielding the following dat

Expanding Spherical Balloon

A spherical balloon has its volume given by $$V=\frac{4}{3}\pi r^3$$. The radius of the balloon incr

Exponential Form and Chain Rule Complexity

Define $$Q(x)=(\cos(x))^{\sin(x)}$$. Hint: Express Q(x) as an exponential function.

Implicit Differentiation in an Economic Demand-Supply Model

In an economic model, the relationship between supply (\(S\)) and demand (\(D\)) is given by the equ

Implicit Differentiation Involving a Logarithm

Consider the equation $$x*\ln(y) + y^2 = x^2$$. Answer the following parts.

Implicit Differentiation of a Circle

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

Implicit Differentiation with Mixed Functions

Consider the relation $$x\cos(y)+y^3=4*x+2*y$$.

Implicit Differentiation with Mixed Trigonometric and Polynomial Terms

Consider the equation $$x*\cos(y) + y^2 = x^2$$, which mixes trigonometric and polynomial expression

Implicit Differentiation with Product Rule

Consider the equation $$x*e^{y} + y*\ln(x)=5$$. Answer the following:

Implicit Trigonometric Equation Analysis

Consider the equation $$x \sin(y) + \cos(y) = x$$. Answer the following parts.

Inverse Function Differentiation

Let $$f(x)=x^3+x$$ which is one-to-one on its domain. Its inverse function is denoted by $$g(x)$$.

Inverse Function Differentiation in a Biological Growth Model

In a bacterial growth experiment, the population $$P$$ (in colony-forming units) at time $$t$$ (in h

Inverse Function Differentiation in a Piecewise Scenario

Consider the piecewise function $$f(x)=\begin{cases} x^2+1, & x \geq 0 \\ -x+1, & x<0 \end{cases}$$

Inverse Function Differentiation in an Exponential Context

Let $$f(x)= e^(3*x) - 2$$ and let $$g(x)$$ be the inverse function of f. Answer the following:

Inverse Function Differentiation in an Exponential Model

Let $$f(x) = e^{2*x} + x$$, and let g be its inverse function. Answer the following parts.

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 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

Multilayer Composite Function Differentiation

Let $$y=\cos(\sqrt{5*x+3})$$. Answer the following:

Projectile Motion and Composite Function Analysis

A projectile is launched and its height $$h(t)$$ (in meters) is recorded at various times t (in seco

Related Rates via Chain Rule

A spherical balloon is being inflated so that its volume increases at a rate of $$\frac{dV}{dt}=150\

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

Analysis of Wheel Rotation

Consider a wheel whose angular position is given by $$\theta(t) = 2t^2 + 3t$$, in radians, where $$t

Balloon Inflation Analysis

A spherical balloon inflates such that its volume increases at a constant rate of 10 cubic inches pe

Complex Piecewise Function Analysis

Consider the function $$f(x)=\begin{cases}\frac{\sin(x)}{x} & x<\pi \\ 2 & x=\pi \\ 1+\cos(x-\pi) &

Cost Efficiency in Production

A firm's cost function for producing $$x$$ items is given by $$C(x)=0.1*x^2 - 5*x + 200$$. Analyze t

Cost Function Optimization

A company’s cost is modeled by the function $$C(x)=0.5x^3-6x^2+20x+100$$, where x (in hundreds of un

Falling Object Analysis

An object is dropped from a height and its position is modeled by $$s(t)=100-4.9t^2$$ (in meters), w

FRQ 4: Revenue and Cost Implicit Relationship

A company’s revenue (R) and cost (C) are related by the equation $$R^2 + 3*R*C + C^2 = 1000$$. Treat

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

Implicit Differentiation and Related Rates in Conic Sections

A point moves along the ellipse defined by $$\frac{x^2}{9} + \frac{y^2}{16} = 1$$. At a certain inst

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

Linearization and Differential Approximations

Let $$f(x)=x^4$$. Use linearization to approximate $$f(3.98)$$ near $$x=4$$.

Linearization and Differentials Approximation

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

Linearization of a Nonlinear Function

Suppose $$f(x)=\ln(x)$$. Use linearization about $$x=4$$ to approximate $$\ln(4.1)$$. Answer the fol

Maximization of Profit

A company's revenue and cost functions are given by $$R(x)=-2x^2+120x$$ and $$C(x)=50+30x$$, respect

Maximizing the Area of an Enclosure with Limited Fencing

A farmer has 240 meters of fencing available to enclose a rectangular field that borders a river (th

Motion along a Straight Line: Changing Direction

A runner's position is modeled by $$s(t)= t^4 - 8*t^2 + 16$$, where $$s(t)$$ is in meters and $$t$$

Open-top Box Optimization

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

Population Growth Model and Asymptotic Limits

A population is modeled by $$P(t) = \frac{5000e^{0.1t}}{1 + e^{0.1t}}$$, where $$P(t)$$ is the popul

Projectile Motion Analysis

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

Rate of Change in a Freefall Problem

An object is dropped from a height. Its height (in meters) after t seconds is modeled by $$h(t)= 100

Reaction Rate and Temperature

The rate of a chemical reaction is modeled by $$r(T)= 0.5*e^{-0.05*T}$$, where $$T$$ is the temperat

Related Rates in a Conical Tank

Water is being poured into a conical tank at a rate of $$\frac{dV}{dt}=10$$ cubic meters per minute.

Related Rates in Expanding Circular Oil Spill

An oil spill forms a circular patch. Its area is given by $$A= \pi*r^2$$. If the area is increasing

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

Rocket Thrust: Analyzing Exponential Acceleration

A rocket’s velocity is modeled by $$v(t) = 100(1 - e^{-0.05t})$$, where $$t$$ is in seconds and $$v(

Route Optimization for a Rescue Boat

A rescue boat must travel from a point on the shore to an accident site located 2 km along the shore

Airport Runway Deicing Fluid Analysis

An airport runway is being de-iced. The fluid is applied at a rate $$F(t)=12*\sin(\frac{\pi*t}{4})+1

Analysis of a Trigonometric Piecewise Function

Consider the function $$ f(x) = \begin{cases} \frac{\sin(x)}{x}, & x \neq 0, \\ 2, & x = 0. \end{ca

Analyzing a Piecewise Function and Differentiability

Let $$f(x)$$ be defined piecewise by $$f(x)= x^2$$ for $$x \le 2$$ and $$f(x) = 4*x - 4$$ for $$x >

Analyzing Critical Points in a Piecewise Function

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

Applying the Mean Value Theorem and Analyzing Discontinuities

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

Biological Growth and the Mean Value Theorem

In a bacterial culture, the population is modeled by $$P(t)= 4*t^2 + 3*t + 7$$ for $$t$$ in hours on

Continuous Compound Interest

An investment account is governed by the formula $$A(t)= A_0 * e^{r*t}$$, where $$r$$ is the continu

Cost Function and the Mean Value Theorem in Economics

An economic model gives the cost function as $$C(x)= 100 + 20*x - 0.5*x^2$$, where x represents the

Determining Intervals of Concavity for a Logarithmic Function

Consider the function $$f(x)= \ln(x) - x$$ defined on the interval \([1, e]\). Answer the following:

Determining Intervals of Increase and Decrease with a Rational Function

Consider the function $$f(x) = \frac{x^2}{x+2}$$ defined on the interval $$[0, 4]$$. Answer the foll

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

Exponential Bacterial Growth

A bacterial culture grows according to $$P(t)= P_0 * e^{k*t}$$, where $$t$$ is in hours. The culture

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

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 Linear Function

Consider the function $$f(x)=3*x+2$$. Analyze its inverse function by answering all parts below.

Inverse Analysis of a Logarithm-Exponential Hybrid Function

Consider the function $$f(x)=\ln(x+2)+e^(x)$$ defined for $$x>-2$$. Address the following regarding

Investment with Continuous Compounding and Variable Rates

An investment grows continuously with a variable rate given by $$r(t)= 0.05+0.01e^{-0.5*t}$$. Its va

Local Linear Approximation of a Trigonometric Function

Consider the function $$f(x)= \cos(x)$$ and its behavior near $$x=0$$.

Modeling Disease Spread with an Exponential Model

In an epidemic, the number of infected individuals is modeled by $$I(t)= I_0 * e^{r*t}$$, where $$t$

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

Optimizing a Cylindrical Water Tank

A cylindrical water tank without a top is to be built with a fixed surface area of 100 m². Let $$r$$

Optimizing an Open-Top Box from a Metal Sheet

A rectangular sheet of metal with dimensions 24 cm by 18 cm is used to create an open-top box by cut

Pharmaceutical Drug Delivery

A drug is administered intravenously using an infusion device. The drug infusion rate is given by $$

Population Growth Analysis via the Mean Value Theorem

A country's population data over a period of years is given in the table below. Use the data to anal

Production Cost Optimization and the Extreme Value Theorem

A company monitored its production cost as a function of units produced. The following table gives e

Profit Analysis and Inflection Points

A company's profit is modeled by $$P(x)= -x^3 + 9*x^2 - 24*x + 10$$, where $$x$$ represents thousand

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

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

Antiderivative of a Transcendental Function

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

Antiderivatives and Initial Value Problems

Given that $$f'(x)=\frac{2}{\sqrt{x}}$$ for $$x>0$$ and $$f(4)=3$$, find the function $$f(x)$$.

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 $$^\circ C$$

Cooling of a Liquid Mixture

In a tank, the cooling rate is given by $$C(t)=20e^{-0.3t}$$ J/min while an external heater adds a c

Definite Integral and the Fundamental Theorem of Calculus

Consider the function $$f(x)= 3*x^2 - 2*x + 1$$ defined on the interval $$[1,4]$$. Use the Fundament

Definite Integral Evaluation via U-Substitution

Consider the integral $$\int_{2}^{6} 3*(x-2)^4\,dx$$ which arises in a physical experiment. Evaluate

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 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

Evaluating a Definite Integral Using U-Substitution

Compute the integral $$\int_{0}^{3} (2*t+1)^5\,dt$$ using u-substitution.

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

FRQ10: Inverse Analysis of a Production Accumulation Function

A company's production output (in thousands of units) over time (in days) is modeled by $$ P(t)=\int

FRQ18: Inverse Analysis of a Square Root Accumulation Function

Consider the function $$ R(x)=\int_{1}^{x} \sqrt{t+1}\,dt $$. Answer the following parts.

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

Integration by Parts: Evaluating $$\int_1^e \ln(x)\,dx$$

Evaluate the integral $$\int_1^e \ln(x)\,dx$$ using integration by parts.

Integration of a Piecewise Function

A function $$f(x)$$ is defined piecewise as follows: $$f(x)= \begin{cases} x^2, & x < 3, \\ 2*x+1,

Marginal Cost and Total Cost

In a production process, the marginal cost (in dollars per unit) for producing x units is given by $

Modeling Savings with a Geometric Sequence

A person makes annual deposits into a savings account such that the first deposit is $100 and each s

Motion Under Variable Acceleration

A particle moves along the x-axis with acceleration $$a(t) = 6 - 4*t$$ (in m/s²) for $$0 \le t \le 3

Net Change Calculation

The net change in a quantity $$Q$$ is modeled by the rate function $$\frac{dQ}{dt}=e^{t}-1$$ for $$0

Optimizing Fencing Cost for a Garden Adjacent to a River

A farmer plans to fence a rectangular garden adjacent to a river, so that no fence is required along

Particle Motion with Variable Acceleration

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

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 Change in a Wildlife Reserve

In a wildlife reserve, animals immigrate at a rate of $$I(t)= 10\cos(t) + 20$$ per month, while emig

Sand Pile Dynamics

A sand pile is being formed by delivering sand at a rate of $$r_{in}(t) = 3t$$ kg/min while erosion

Seismic Data Analysis: Ground Acceleration

A seismograph records ground acceleration (in m/s²) during an earthquake. Use the data in the table

Ski Lift Passengers: Boarding and Alighting Rates

On a ski lift, passengers board at a rate $$B(t)= 3e^{-t/2}$$ persons/min and alight at a constant r

Temperature Change in a Reactor

In a chemical reactor, the internal heating is modeled by $$H(t)= 10+2\cos(t)$$ °C/min and cooling o

Trapezoidal Rule in Estimating Accumulated Change

A rising balloon has its height measured at various times. A portion of the recorded data is given i

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

Bacterial Growth with Constant Removal

A bacterial colony (in thousands) grows according to the differential equation $$\frac{dP}{dt}=0.4P-

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

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

Chemical Reaction Rate

In a chemical reaction, the concentration $$C$$ (in mol/L) of a reactant is recorded over time as sh

CO2 Absorption in a Lake

A lake absorbs CO2 from the atmosphere. The concentration $$C(t)$$ of dissolved CO2 (in mol/m³) in t

Cooling of a Cup of Coffee

Newton's Law of Cooling states that the rate of change of temperature of an object is proportional t

Cooling of Electronic Components

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

Epidemic Spread with Limited Capacity

In a closed community, the number of infected individuals $$I(t)$$ (in people) is modeled by the log

Exponential Growth: Separable Equation

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

Heating and Cooling in an Electrical Component

An electronic component experiences heating and cooling according to the differential equation $$\fr

Implicit Differentiation Involving a Logarithmic Function

Consider the function defined implicitly by $$\ln(y) + x^2y = 7$$. Answer the following:

Implicit Differentiation of a Circle

Consider the circle defined by $$x^2+ y^2= 25$$. Answer the following:

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

Investment Growth with Withdrawals

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

Logistic Growth in a Population

A population $$P(t)$$ grows according to the logistic differential equation $$\frac{dP}{dt}=0.5P\lef

Logistic Growth Model

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

Logistic Population Growth

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

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 in a Tank

A tank initially contains 200 L of water with 10 kg of dissolved salt. Brine with a salt concentrati

Nonlinear Differential Equation

Consider the differential equation $$\frac{dy}{dx}=(1-y^2)\cos(x)$$ with the initial condition $$y(0

Particle Motion in the Plane

A particle moving in the plane has a constant x-component velocity of $$v_x(t)=2$$ m/s, and its y-co

Population Growth with Harvesting

A fish population in a lake grows according to $$\frac{dP}{dt}=0.08*P-50$$, where $$P(t)$$ represent

Population Growth with Harvesting

A fish population in a lake grows at a rate proportional to its current size, but fishermen harvest

Population Growth with Logistic Equation

A population grows according to the logistic differential equation $$\frac{dy}{dx} = 0.5*y\left(1-\f

Radioactive Decay

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

Related Rates: Expanding Balloon

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

Sand Pile Dynamics

Sand is added to a pile at a constant rate of $$15$$ kg/min while some sand is simultaneously lost d

Seasonal Temperature Variation

The temperature $$T(t)$$ in a region is modeled by the differential equation $$\frac{dT}{dt} = -0.2\

Separable Equation with Trigonometric Functions

Solve the differential equation $$\frac{dy}{dx} = \frac{\tan(x)}{1+y^2}$$ given that $$y(0)=0$$.

Slope Field and General Solution

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

Slope Field and Integrating Factor Analysis

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

Slope Field and Solution Curve Analysis

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

Vehicle Deceleration

A vehicle undergoing braking has its speed $$v$$ (in m/s) recorded over time (in seconds) as shown.

Analysis of an Inverse Function

Consider the function $$f(x)=(x-1)^3+2$$, defined for all real numbers. Analyze its inverse function

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 Curves: Revenue and Cost Analysis

A company’s revenue and cost are modeled by the functions $$f(x)=10-x^2$$ and $$g(x)=2*x$$, where $$

Area Between Two Curves from Tabulated Data

Consider two functions, $$f(x)$$ and $$g(x)$$, whose values are recorded in the table below over the

Average Density of a Rod

A rod of length $$10$$ cm has a linear density given by $$\rho(x)= 4 + x$$ (in g/cm) for $$0 \le x \

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 Temperature Analysis

A weather scientist models the temperature during a day by the function $$f(t)=5+2*t-0.1*t^2$$ where

Average Temperature of a Cooling Liquid

The temperature of a cooling liquid is modeled by $$T(t)=50*e^{-0.1*t}+20$$ (in $$^\circ C$$) for $$

Bloodstream Drug Concentration

A drug enters the bloodstream at a rate given by $$R(t)= 5*e^{-0.5*t}$$ mg/min for $$t \ge 0$$. Simu

Calculation of Consumer Surplus

The demand function for a product is given by $$p(x)=20-0.5*x$$, where $$p$$ is the price (in dollar

Cost Optimization for a Cylindrical Container

A manufacturer wishes to design a closed cylindrical container with a fixed volume $$V_0$$. The cost

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 Analysis of Consumer Surplus

A market demand function is given by $$P(x)=50 - 10*\ln(x+1)$$, where $$x$$ represents quantity dema

Electric Charge Accumulation

The current flowing into a capacitor is defined by $$I(t)=\frac{10}{1+e^{-2*(t-3)}}$$ (in amperes) f

Graduated Rent Increase

An apartment’s yearly rent increases by a fixed amount. The initial annual rent is $$1200$$ dollars

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 with Exponential Acceleration

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

Population Dynamics in a Wildlife Reserve

A wildlife reserve monitors the change in the number of a particular species. The rate of change of

Position and Velocity Relationship in Car Motion

A car's position along a highway is modeled by $$s(t)=t^3-6*t^2+9*t+2$$ (in kilometers) with time $$

Projectile Motion: Position, Velocity, and Maximum Height

A projectile is launched vertically upward with an initial velocity of $$20\,m/s$$ from a height of

Projectile Motion: Time of Maximum Height

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

Resource Consumption in an Ecosystem

The rate of consumption of a resource in an ecosystem is given by $$C(t)=50*\ln(1+t)$$ (in units per

River Current Analysis

The velocity of a river is given by $$v(x)=2+x-0.1*x^2$$ (in m/s) for 0 ≤ x ≤ 10, where x measures t

Technology Adoption Growth

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

Viral Video Views

A viral video’s daily views form a geometric sequence. On day 1, the video is viewed 1000 times, and

Volume by Rotation using the Disc Method

Consider the region bounded by $$y=\sqrt{x}$$, the $$x$$-axis, and the vertical lines $$x=0$$ and $$

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 Washer Method

Consider the region R bounded by $$y= x$$ and $$y= x^2$$ on the interval $$x \in [0,1]$$. This regio

Volume of a Solid of Revolution Using the Disk Method

Consider the region bounded by the graph of $$f(x)=\sqrt{x}$$, the x-axis, and the vertical line $$x

Volume of a Solid of Revolution: Disc Method

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

Volume of a Solid with Semicircular Cross Sections

A solid has a base in the xy-plane given by the region bounded by $$y=4-x^2$$ and the x-axis for $$0

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

Water Tank Filling with Graduated Inflow

A water tank is filled daily by adding a certain amount of water that increases by a fixed amount ea