AP Calculus AB FRQ Room

Algebraic Manipulation in Limit Calculations

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

Analysis of Three Functions

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

Analysis of Vertical Asymptotes

Examine the function $$h(x)= \frac{x^2-9}{x^2-4*x+3}$$. Answer the following:

Analyzing Process Data for Continuity

A manufacturing process produces items whose lengths (in mm) are recorded as a function f(x) of time

Continuity Analysis of a Radical Function

Consider the function $$f(x) = \frac{\sqrt{x+4} - 2}{x}$$. (a) Evaluate $$\lim_{x \to 0} f(x)$$. (b

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 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

Determining Parameters for Continuity

Consider the function $$f(x)= \begin{cases} 2*x + k, & x < 1 \\ x^2, & x \geq 1 \end{cases}$$, where

Evaluating Limits Near Vertical Asymptotes

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

Factoring a Cubic Expression for Limit Evaluation

Consider the function $$f(x)= \frac{x^3 - 8}{x - 2}$$ for x \(\neq\) 2. Answer the following parts.

Graphical Analysis of Limit Behavior

The graph of f(x) is provided in the stimulus below. Analyze the behavior of f(x) around x = 2.

Horizontal Asymptote and End Behavior

Consider the rational function $$f(x)=\frac{3*x^2 + x - 5}{6*x^2 - 4*x + 7}$$. Answer the following

Implicit Differentiation in an Exponential Equation

Consider the function defined implicitly by $$e^{x*y} + x - y = 0$$. 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].

Limit Analysis in Population Modeling

A population is modeled by the function $$P(t)= \frac{1000*t}{t+5}$$ where $$t \geq 0$$ (in years).

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 \

Limits from Data in Chemical Reaction Rates

In a chemical reaction, the concentration of a reactant (in M) is monitored over time (in seconds).

Oscillatory Behavior and Continuity

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

Oscillatory Behavior and Non-Existence of Limit

Let \(f(x)=\sin(1/x)\) for \(x\neq0\). Answer the following:

Particle Motion with Squeeze Theorem Application

A particle moves along a line with velocity given by $$v(t)= t^2 \sin(1/t)$$ for $$t>0$$ and is defi

Related Rates: Shadow Length of a Moving Object

A 1.8 m tall person is walking away from a 3 m tall lamppost at a speed of 1.2 m/s. Let $$x$$ be the

Removable Discontinuity in a Rational Function

Consider the function $$f(x)=\begin{cases} \frac{x^2-16}{x-4} & x\neq4 \\ 3*x+1 & x=4 \end{cases}$$.

Removing Discontinuities

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

Return on Investment and Asymptotic Behavior

An investor’s portfolio is modeled by the function $$P(t)= \frac{0.02t^2 + 3t + 100}{t + 5}$$, where

Squeeze Theorem Application

Let $$f(x)=x^2\sin(1/x)$$ for \(x\neq 0\) and define \(f(0)=0\). Use the Squeeze Theorem to complete

Trigonometric Function Discontinuity

Consider the function $$f(x)=\begin{cases} \frac{\sin(2*x)}{x} & x\neq0 \\ 4 & x=0 \end{cases}$$. An

Trigonometric Limit Computation

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

Vertical Asymptotes and Horizontal Limits

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

Analyzing Differentiability of an Absolute Value Function

Consider the function $$f(x)= |x-2|$$.

Analyzing Function Behavior Using Its Derivative

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

Derivative from the Limit Definition

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

Derivative of an Absolute Value Function

Consider the function \(f(x)=|x|\). Answer the following parts, restricting your analysis to \(x\ne

Graphical Estimation of a Derivative

Consider the graph provided which plots the position $$s(t)$$ (in meters) of an object versus time $

Identifying Points of Non-Differentiability

Consider the function $$h(x)= |2*x - 5|$$.

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 Velocity from a Position Function

A ball is thrown upward, and its height in feet is modeled by $$s(t)= -16*t^2 + 64*t + 5$$, where $$

Inverse Function Analysis: Cubic Transformation

Consider the function $$f(x)=(x-1)^3$$ defined for all real numbers.

Inverse Function Analysis: Quadratic Transformation

Consider the function $$f(x)=x^2+2*x+2$$ with the domain restricted to $$x\geq -1$$ so that f is one

Inverse Function Analysis: Trigonometric Function with Linear Term

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

Linking Derivative to Kinematics: the Position Function

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

Profit Function Analysis

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

Proof of Scaling in Derivatives

Let $$f(x)$$ be a differentiable function and let $$k$$ be a constant. Consider $$g(x)= k*f(x)$$. Us

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

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 for a Cubic Function

Consider the function $$f(x)= x^3 - 4*x$$.

Secant and Tangent Lines for a Trigonometric Function

Let $$f(x)=\sin(x)+x^2$$. Use the definition of the derivative to find $$f'(x)$$ and evaluate it at

Using Derivative Rules on a Trigonometric Function

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

Advanced Composite Function Differentiation in Biological Growth

A biologist models bacterial growth by the function $$P(t)= e^{\sqrt{t+1}}$$, where $$t$$ is time in

Chain Rule Basics

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

Chain Rule in an Implicitly Defined Function

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

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 Logarithmic and Radical Functions

Let $$R(x)=\sqrt{\ln(1+x^2)}$$.

Chain Rule with Logarithms

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

Comparing the Rates between a Function and its Inverse

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

Composite and Product Rule Combination

The function $$F(x)= (3*x^2+2)^{4} * \cos(x^3)$$ arises in modeling a complex system. Answer the fol

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 in a Sand Pile Model

Sand is added to a pile at an inflow rate of $$A(t)= 4 + t^2$$ (kg/min) and removed at an outflow ra

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 in Finance

An account balance is modeled by the function $$B(t)=(2*t+1)^{3/2}$$ dollars, where $$t$$ represents

Differentiation Involving Exponentials and Inverse Trigonometry

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

Differentiation of Inverse Trigonometric Functions in Physics

In an optics experiment, the angle of refraction \(\theta\) is given by $$\theta= \arcsin\left(\frac

Differentiation of Nested Exponential Functions

Let $$F(x)=e^{\sin(x^2)}$$.

Finding Second Derivative via Implicit Differentiation

Given the curve defined by $$x^2+y^2+ x*y=7$$, answer the following:

Implicit Curve Analysis: Horizontal Tangents

Consider the curve defined implicitly by $$x^2+ e^(y)= 5$$. Answer the following:

Implicit Differentiation in Elliptical Orbits

Consider an elliptical orbit described by the equation $$\frac{x^2}{16}+\frac{y^2}{9}=1$$, where the

Implicit Differentiation with Exponential Terms

Consider the equation $$e^{x} + y = x + e^{y}$$ which relates $$x$$ and $$y$$ via exponential functi

Implicit Differentiation with Logarithmic and Trigonometric Combination

Consider the equation $$\ln(x+y)+\cos(x*y)=0$$, where $$y$$ is an implicit function of $$x$$. Find $

Implicit Differentiation with Product Rule

Consider the equation $$x*y+e^{y}=x^2$$. Answer the following:

Implicit Differentiation with Product Rule

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

Inverse Function Derivative for a Logarithmic Function

Let $$f(x)=\ln(x+1)-\sqrt{x}$$, which is one-to-one on its domain.

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 for a Log Function

Let $$f(x)=\ln(3*x+2)$$, and assume that $$f$$ is invertible with inverse function $$g$$. Find the d

Inverse Trigonometric Differentiation in Engineering Mechanics

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

Multiple Applications: Chain Rule, Implicit, and Inverse Differentiation

Consider the function \(f(x)= e^{x^2}\) and note that it has an inverse function \(g\). In addition,

Related Rates in a Circular Colony

A circular microorganism colony expands such that its radius at time $$t$$ (in seconds) is given by

Second Derivative via Implicit Differentiation

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

Temperature Profile and the Chain Rule

A metal rod has a temperature distribution given by $$T(x)=100*e^{-0.05*x^2}$$ (in °C), where x is t

Analysis of a Piecewise Function with Discontinuities

Consider the function $$f(x)=\begin{cases} \frac{x^2-4}{x-2} & \text{if } x<2 \\ x+1 & \text{if } x\

Analyzing a Nonlinear Rate of Revenue Change

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

Application of L'Hospital's Rule

Consider the limit $$\lim_{x\to\infty} \frac{5*x^3 - 2*x + 1}{10*x^3 + 3*x^2 - 4}.$$ Answer the f

Balloon Inflation Related Rates

A spherical balloon is being inflated, and its volume is increasing at a constant rate of $$12$$ cub

Biochemical Reaction Rate Analysis

A biochemical reaction proceeds with a rate modeled by $$R(t)=50t(1-t)^2$$ for $$0\le t\le1$$ (where

Chemical Reaction Rate

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

Chemistry Reaction Rate

The concentration of a chemical in a reaction is given by $$C(t)= \frac{100}{1+5*e^{-0.3*t}}$$ (in m

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

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

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 Oil Spill

The area of an oil spill is modeled by $$A(t)=\pi (2+t)^2$$ square kilometers, where $$t$$ is in hou

FRQ 1: Vessel Cross‐Section Analysis

A designer is analyzing the cross‐section of a vessel whose shape is given by the ellipse $$\frac{x^

FRQ 10: Chemical Kinetics Analysis

In a chemical reaction, the concentration of reactant A, denoted by [A], and time t (in minutes) are

FRQ 18: Chemical Reaction Concentration Changes

During a chemical reaction, the concentrations of reactants A and B are related by $$[A]^2 + 3*[A]*[

Graphing a Function via its Derivative

Consider the function $$f(x) = x^{1/3}$$ defined for all real numbers.

Hybrid Exponential-Logarithmic Convergence

Consider the function $$f(x)=e^{-x}\ln(1+2x)$$, which combines exponential decay with logarithmic gr

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 Trigonometric Analysis for Navigation

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

L'Hôpital's Rule in Analysis of Limits

Consider the limit $$L = \lim_{x\to \infty} \frac{5*x^3 - 4*x^2 + 1}{7*x^3 + 2*x - 6}$$. Use L'Hôpit

Linear Approximation for Function Values

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

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 for Approximating Powers

Let $$f(x) = x^3$$. Use linear approximation to estimate $$f(4.98)$$.

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

Medicine Dosage: Instantaneous Rate of Change

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

Minimizing Materials for a Cylindrical Can

A manufacturer aims to design a closed cylindrical can that holds exactly $$500$$ cubic centimeters

Motion Along a Curved Path

An object moves along the curve given by $$y=\ln(x)$$ for $$x\geq 1$$. Suppose the x-component of 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 $$

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

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

Reaction Rates in Chemistry

In a chemical reaction, the concentration of a reactant is modeled by $$C(t)=50e^{-0.3*t}+10$$, wher

Region Area and Volume by Rotation

Consider the region R bounded by the curves $$y=\sqrt{x}$$ and $$y=\frac{x}{2}$$ from $$x=0$$ to $$x

Related Rates in a Conical Tank

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

Shadow Length Problem

A 10-meter tall streetlight casts a shadow of a 1.8-meter tall person. If the person walks away from

Using L'Hospital's Rule to Evaluate a Limit

Consider the limit $$L=\lim_{x\to\infty}\frac{5x^3-4x^2+1}{7x^3+2x-6}$$. Answer the following:

Absolute Extrema for a Transcendental Function

Examine the function $$f(x)= e^{-x}*(x-2)$$ on the closed interval $$[0,3]$$ to determine its absolu

Analysis of an Exponential-Logarithmic Function

Consider the function $$f(x)= e^{x} - 3*\ln(x)$$ defined for $$x>0$$. Answer the following:

Analyzing Acceleration Functions Using Derivatives

For the position function $$s(t)= t^3 - 6*t^2 + 9*t + 1$$ (in meters), where \( t \) is in seconds,

Analyzing Differentiability of a Piecewise Function

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

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:

Behavior Analysis of a Logarithmic Function

Consider the function $$f(x)= \frac{\ln(x)}{x}$$ for $$x>0$$. Analyze the critical points and concav

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

Comprehensive Analysis of a Rational Function

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

Concavity and Inflection Points of a Cubic Function

Consider the cubic function $$f(x)=x^3-6*x^2+9*x+2$$. Answer the following questions regarding its d

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

Designing an Enclosure along a River

A farmer wants to build a rectangular enclosure adjacent to a river, using the river as one side of

Determining Absolute and Relative Extrema

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

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:

Discontinuity in a Rational Function Involving Square Roots

Consider the function $$ f(x) = \begin{cases} \frac{\sqrt{x+3} - 2}{x - 1}, & x \neq 1, \\ -1, & x

Evaluating Rate of Change in Electric Current Data

An electrical engineer recorded the current (in amperes) in a circuit over time. The table below sho

Exponential Bacterial Growth

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

Finding Local Extrema Using the First Derivative Test

Consider the function $$f(x) = x^3 - 6*x^2 + 9*x + 2$$. Answer the following:

FRQ 5: Concavity and Points of Inflection for a Cubic Function

For the function $$f(x) = x^3 - 6*x^2 + 9*x + 2$$, analyze its concavity.

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 19: Analysis of an Exponential-Polynomial Function

Consider the function $$f(x)= e^{-x}*x^2$$ defined for $$x \ge 0$$.

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

Inflection Points in a Population Growth Model

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

Instantaneous Velocity Analysis via the Mean Value Theorem

A particle moves along a straight line with its displacement given by $$s(t)= t^3 - 6*t^2 + 9*t + 3$

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 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 an Absolute Value Term

Consider the function $$f(x)=x+|x-2|$$ with the domain restricted to $$x\ge 2$$. Analyze the inverse

Inverse Analysis of a Logarithmic Function

Consider the function $$f(x)=\ln(x-1)$$ defined for $$x>1$$. Answer the following questions about it

Mean Value Theorem Applied to Exponential Functions

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

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

Motion Analysis: A Runner's Performance

A runner’s distance (in meters) is recorded at several time intervals during a race. Analyze the run

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 $$

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{

Relationship Between Integration and Differentiation

Let $$F(x)= \int_{0}^{x} (t^2 - t + 1)\,dt$$. Explore the relationship between the integral and its

Relative Extrema in an Economic Demand Model

An economic study recorded the quantity demanded of a product at different price points. Use the tab

Slope Analysis for Parametric Equations

A curve is defined parametrically by $$x(t)= t^2$$ and $$y(t)= t^3 - 3*t$$ for $$t$$ in the interval

Tangent Line to an Implicitly Defined Curve

The curve is defined by the equation $$x^2 + x*y + y^2 = 7$$.

Trigonometric Function Behavior

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

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

Antiderivatives and Initial Value Problems

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

Application of the Fundamental Theorem in a Discounted Cash Flow Model

A continuous cash flow is given by $$C(t)=500(1+0.05*t)$$ dollars per year. Using a continuous disco

Area Between Two Curves

Consider the functions $$f(x)=x^2$$ and $$g(x)=2*x+3$$. They intersect at two points. Using the grap

Area Under a Curve Using Riemann Sums

A function $$f(x)$$ is defined over the interval $$[1,7]$$ and its values are provided in the table

Area Under a Parabola

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

Area Under a Piecewise Function

Consider the piecewise function $$ f(x)= \begin{cases} x & \text{for } 0\le x<3,\\ 9-x & \text{for

Bacterial Growth Modeling with Antibiotic Administration

A bacterial culture is subject to both growth and treatment simultaneously. The bacterial growth rat

Chemical Production via Integration

The production rate of a chemical in a reactor is given by $$r(t)=5*(t-2)^3$$ (in kg/hr) for $$t\ge2

Chemical Reactor Conversion Process

In a chemical reactor, the instantaneous reaction rate is given by $$R(t)=4t^2-t+3$$ mol/min, while

Comparing Riemann Sum and the Fundamental Theorem

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

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:

Convergence of Riemann Sum Estimations

Consider the function $$f(x)=\sqrt{x}$$ on the interval $$[1,4]$$. Answer the following questions re

Definite Integral Evaluation via U-Substitution

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

Economic Analysis: Consumer Surplus

In a competitive market, the demand function is given by $$D(p)=100-2*p$$ and the supply function is

Electric Charge Accumulation

An electrical circuit records the current (in amperes) at various times during a brief experiment. U

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 Total Biomass in an Ecosystem

An ecologist measured the population density (in kg/km²) of a species along an 8 km transect. Use th

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 U-substitution

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

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

Evaluating Total Rainfall Using Integral Approximations

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

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

FRQ17: Inverse Analysis of a Biologically Modeled Accumulation Function

In a biological study, the net concentration of a chemical is modeled by $$ B(t)=\int_{0}^{t} (0.6*t

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 for a Rocket Launch

During a rocket launch, fuel is consumed at a rate $$F_{cons}(t)=50-3t$$ kg/s while additional fuel

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

Medication Infusion in Bloodstream

A patient receives medication through an IV at a rate $$I(t)= 5\sqrt{t+1}$$ mg/min, while the body m

Net Change Calculation

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

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:

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

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

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

U-Substitution in a Rate of Flow Model

A river's flow rate in cubic meters per second is modeled by the function $$Q(t)= (t-2)^3$$ for $$t

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

Analyzing Direction Fields for $$dy/dx = y-1$$

Consider the differential equation $$dy/dx = y - 1$$. A slope field for this equation is provided. A

Analyzing Slope Fields for $$dy/dx=x\sin(y)$$

Consider the differential equation $$\frac{dy}{dx}=x\sin(y)$$. A corresponding slope field is provid

Area Under a Differential Equation Curve

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

Bernoulli Differential Equation

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

Charging of an RC Circuit

An RC circuit is being charged with a battery of voltage $$12\,V$$. The voltage across the capacitor

Cooling of a Liquid

A liquid is cooling in a lab experiment. Its temperature $$T$$ (in °C) is recorded at several times

Disease Spread Modeling

The spread of an infection in a closed population is modeled by the differential equation $$\frac{dI

Exact Differential Equation

Consider the differential equation written in differential form: $$(2*x*y + y^2)\,dx + (x^2 + 2*x*y)

Exponential Growth and Doubling Time

A bacterial culture grows according to the differential equation $$\frac{dy}{dt} = k * y$$ where $$y

Implicit Differentiation Involving a Logarithmic Function

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

Inverse Function Analysis of a Differential Equation Solution

Consider the function $$f(x)=\sqrt{4*x+9}$$, which arises as a solution to a differential equation i

Investment Growth with Continuous Contributions

An investment account grows continuously with an annual interest rate of 5% while continuous deposit

Logistic Population Growth

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

Mixing a Salt Solution

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

Mixing Problem in a Tank

A tank initially contains 100 liters of brine with 10 kg of dissolved salt. Brine with a concentrati

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

Pollutant Concentration in a Lake

A lake receives a constant pollutant input so that the concentration $$C(t)$$ (in mg/L) satisfies th

Radioactive Decay

A radioactive substance decays according to $$\frac{dN}{dt} = -\lambda N$$. Initially, there are 500

Radioactive Decay

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

RC Circuit Discharge

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

Salt Tank Mixing Problem

A tank initially contains 100 liters of pure water. A salt solution with concentration 0.5 kg/L is p

Sand Erosion in a Beach Model

During a storm, a beach loses sand. Let $$S(t)$$ (in tons) be the amount of sand on a beach at time

Separable Differential Equation: $$dy/dx = x*y$$

Consider the differential equation $$dy/dx = x*y$$ with the initial condition $$y(0)=2$$. Solve the

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

A slope field for the differential equation $$\frac{dy}{dx}=y-x$$ is provided in the stimulus. Use t

Soot Particle Deposition

In an environmental study, the thickness $$P$$ (in micrometers) of soot deposited on a surface is me

Volumes from Cross Sections of a Bounded Region

The solution to a differential equation is given by $$y(x) = \ln(1+x)$$. This curve, combined with t

Water Tank Flow Analysis

A water tank receives an inflow of water at a rate $$Q_{in}(t)=50+10*\sin(t)$$ (liters/min) and an o

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 Cost Functions in a Business Analysis

A company analyzes its cost structure using two functions: the fixed-plus-variable cost function $$C

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

Average Concentration in a Chemical Reaction

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

Average Value and the Mean Value Theorem

For the function $$f(x)=\cos(x)$$ on the interval [0, $$\pi/2$$], compute the average value and find

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 Value of a Function in a Production Process

A factory machine's temperature (in $$^\circ C$$) during a production run is modeled by $$T(t)= 5*t

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

Car Braking Analysis

A car decelerates with acceleration given by $$a(t)=-4e^{-t/2}$$ (in m/s²) and has an initial veloci

Consumer Surplus Calculation

The demand and supply for a product are given by $$p_d(x)=20-0.5*x$$ and $$p_s(x)=10+0.2*x$$ respect

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$

Hollow Rotated Solid

Consider the region in the first quadrant bounded by the curves $$y=x$$ and $$y=x^2$$. This region i

Implicit Differentiation in Thermodynamics

In a thermodynamics experiment, the pressure $$P$$ and volume $$V$$ of a gas are related by the equa

Investment Compound Interest

An investment account starts with an initial deposit of $$1000$$ dollars and earns $$5\%$$ interest

Loaf Volume Calculation: Rotated Region

Consider the region bounded by $$y=\sqrt{x}$$ and $$y=\frac{x}{2}$$ for 0 ≤ x ≤ 4. This region is ro

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

Net Change in Biological Population

A species' population changes at a rate given by $$P'(t)=0.5e^{-0.2*t}-0.05$$ (in thousands per year

Piecewise Function Analysis

Consider a piecewise function defined by: $$ f(x)=\begin{cases} 3 & \text{for } 0 \le x < 2, \\ -x+5

Population Accumulation through Integration

A town’s rate of population growth is modeled by $$r(t)=500*e^{-0.2*t}$$ (people per year), where $$

Population Growth with Variable Growth Rate

A city's population changes with time according to a non-constant growth rate given in thousands per

Reconstructing Position from Acceleration Data

A particle traveling along a straight line has its acceleration given by the values in the table bel

Retirement Savings Auto-Increase

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

Volume Calculation via Cross-Sectional Areas

A solid has cross-sectional areas perpendicular to the x-axis that are circles with radius given by

Volume of a Solid Using the Washer Method

Consider the region bounded by $$y=x$$ and $$y=x^2$$ between $$x=0$$ and $$x=1$$. This region is rev

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 of a Solid with Square Cross Sections

A solid is formed over the region under the line $$f(x)=4-x$$ from $$x=0$$ to $$x=4$$ in the x-y pla

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

Water Tank Volume and Average Cross-Sectional Area

A water tank has a shape where the horizontal cross-sectional area at a depth $$x$$ (in feet) from t

Work to Pump Water from a Cylindrical Tank

A cylindrical tank with a radius of 3 m and a height of 10 m is completely filled with water (densit