AP Calculus AB FRQ Room

Application of the Intermediate Value Theorem in a Logistic Model

Let $$ f(x)=\frac{1}{1+e^{-x}} $$, a logistic function that is continuous for all x. Analyze its beh

Composite Function and Continuity Analysis

Define \(f(x)=\sqrt{1-\ln(x)}\) for \(x>0\) and consider the composite function \(g(x)=f(e^x)\). Ans

Composite Limit Problem Involving Absolute Value

Consider the function $$f(x)=\frac{|x-3|}{x-3}$$ for \(x\neq3\). Answer the following:

Compound Interest and Geometric Series

A bank account accrues interest compounded annually at an annual rate of 10%. The balance after $$n$

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 at Zero for a Trigonometric Function

Consider the function $$f(x)= x*\sin\left(\frac{1}{x}\right)$$ for x $$\neq 0$$ and $$f(0)=0$$. Answ

Continuous Extension and Removable Discontinuity

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

Determining Asymptotes and Holes in a Rational Function

Consider the function $$f(x) = \frac{2x^2 - 3x + 1}{x^2 - 4}$$. This function may exhibit vertical a

Determining Parameters for Continuity in a Piecewise Function

Let the function be defined as $$ f(x)=\begin{cases}ax+3, & x<2,\\ x^2+bx+1, & x\ge2.\end{cases} $$

Evaluating Limits Involving Square Roots

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

Exponential and Logarithmic Limits

Consider the functions $$f(x)=\frac{e^{2*x}-1}{x}$$ and $$g(x)=\frac{\ln(1+x)}{x}$$. Evaluate the li

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.

Intermediate Value Theorem in Context

Let $$f(x) = x^3 - 6x^2 + 9x + 2$$, which is continuous on the interval [0, 4]. Answer the following

Limit Evaluation with a Parameter in a Log-Exponential Function

Consider the function $$r(x)=\frac{e^{a*x} - e^{b*x}}{\ln(1+x)}$$ defined for $$x \neq 0$$, where $$

Limits at Infinity for Non-Rational Functions

Consider the function $$ h(x)=\frac{2*x+3}{\sqrt{4*x^2+7}} $$.

Numerical Estimation of a Limit Using a Table

A student investigates the function \(f(x)=\frac{x^2-1}{x-1}\) for \(x\neq1\) by creating a table of

One-Sided Limits and Discontinuity Analysis

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

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:

Oscillatory Function and the Squeeze Theorem

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

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: Expanding Circular Ripple

A circular ripple forms at the center of a pond and expands over time. The radius $$r$$ (in meters)

Removable Discontinuity and Limit Evaluation

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

Removable Discontinuity in a Cubic Function

Consider the function $$f(x)=\begin{cases} \frac{x^3-27}{x-3} & x\neq3 \\ 10 & x=3 \end{cases}$$. An

Squeeze Theorem for an Oscillatory Function

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

Vertical Asymptote Analysis

Consider the function $$f(x)=\frac{1}{x-3}$$. Answer the following parts.

Acceleration Through Successive Differentiation

A particle’s position is given by $$s(t)=t^3-6*t^2+9*t+4$$ (with s in meters and t in seconds). Answ

Approximating Derivative using Secant Lines

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

Approximating Tangent Line Slopes

A curve is given by the function $$f(x)= \ln(x) + e^{-x}$$, modeling a physical measurement obtained

Average vs. Instantaneous Rate of Change

Consider the function $$f(x)=2*x^2-3*x+1$$ defined for all real numbers. Answer the following parts

Comparative Analysis of Secant and Tangent Slopes

A function $$f(x)$$ is represented by the data in the following table: | x | f(x) | |---|------| |

Derivative from the Limit Definition

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

Derivative of a Logarithmic Function

Let $$g(x)= \ln(x)$$, a fundamental function in many natural logarithm applications.

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

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

Exploring the Difference Quotient for a Trigonometric Function

Consider the trigonometric function $$f(x)= \sin(x)$$, where $$x$$ is measured in radians. Use the d

Exponential Rate of Change

A population growth model is given by $$P(t)=e^{2*t}$$, where $$t$$ is in years.

Finding the Second Derivative

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

Finding the Tangent Line Using the Product Rule

For the function $$f(x)=(3*x^2-2)*(x+5)$$, which models a physical quantity's behavior over time (in

Graph vs. Derivative Graph

A graph of a function $$f(x)$$ and a separate graph of its derivative $$f'(x)$$ are provided in the

Implicit Differentiation in Motion

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

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 $$

Interpreting Derivative Graphs and Tangent Lines

A graph of the function $$f(x)=x^2 - 2*x + 1$$ along with its tangent line at $$x=2$$ is provided. A

Inverse Function Analysis: Cubic Function

Consider the function $$f(x)=x^3+2*x+1$$ defined for all real numbers.

Inverse Function Analysis: Exponential Transformation

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

Inverse Function Analysis: Quadratic Function

Consider the function $$f(x)=x^2$$ restricted to $$x\geq0$$.

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 $$x\geq1$$.

Kinematics and Position Function Analysis

A particle’s position is modeled by $$s(t)=4*t^3-12*t^2+5*t+2$$, where $$s(t)$$ is in meters and $$t

Limit Definition for a Quadratic Function

For the function $$h(x)=4*x^2 + 2*x - 7$$, answer the following parts using the limit definition of

Marginal Cost Analysis

A company's total cost function is given by $$C(x)=5*x^2+20*x+100$$, where $$x$$ represents the numb

Medication Infusion with Clearance

A patient receives medication via an IV at a rate of $$f(t)=5*e^{-0.1*t}$$ mg/min, while the body cl

Optimization in Revenue Models

A company's revenue function is given by $$R(x)= x*(50 - 2*x)$$, where $$x$$ represents the number o

Particle Motion on a Straight Road

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

Product and Quotient Rule Combination

Given $$u(x)=3*x^2+2$$ and $$v(x)=x-4$$, consider the function $$F(x)=\frac{u(x)*v(x)}{x+1}$$. Answe

Projectile Motion Analysis

A projectile is launched with its height (in meters) modeled by the function $$f(t)= -5*t^2 + 20*t +

Rate of Water Flow in a Rational Function Model

The water flow from a reservoir is modeled by $$F(t)= \frac{3*t}{t+2}$$, where $$t$$ is time in hour

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.

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

River Crossover: Inflow vs. Damming

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

Secant and Tangent Lines to a Curve

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

Social Media Followers Dynamics

A social media account gains followers at a rate $$f(t)=150-10*t$$ (followers/hour) and loses follow

Tangent Line and Instantaneous Rate at a Point with a Radical Function

Consider the function $$f(x)= (x+4)^{1/2}$$, which represents a physical measurement (with the domai

Tangent Line Equation for an Exponential Function

Consider the function $$f(x)= e^{x}$$ and its graph.

Temperature Change Analysis

A weather station models the temperature (in °C) with the function $$T(t)=15+2*t-0.5*t^2$$, where $$

Advanced Implicit and Inverse Function Differentiation on Polar Curves

Consider the curve defined implicitly by $$x^2+y^2= \sin(x*y)$$. Although not a typical polar curve,

Advanced Implicit Differentiation: Second Derivative Analysis

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

Chain Rule with Logarithmic and Radical Functions

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

Chain Rule with Logarithmic Differentiation

A measurement device produces an output given by $$y=\ln(\sin(3*t^2+2))$$. This function involves mu

Chain Rule with Nested Trigonometric Functions

Consider the function $$f(x)= \sin(\cos(3*x))$$. This function involves nested trigonometric functio

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 in Finance

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

Composite Function Kinematics

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

Composite Function with Logarithm and Trigonometry

Let $$h(x)=\ln(\sin(2*x))$$.

Composite Trigonometric Differentiation in Sound Waves

The sound intensity in a room is modeled by the function $$I(t)= \cos(3*t^2+\sin(t))$$, where $$t$$

Composite, Implicit, and Inverse Combined Challenge

Consider a dynamic system defined by the equation $$\sin(y)+\sqrt{x+y}=x$$, which implicitly defines

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 Composite Rational Function

Let $$f(x)=\frac{(2*x+1)^3}{\sqrt{5*x-2}}$$. Use the chain rule and the quotient (or product) rule t

Differentiation of Inverse Trigonometric Composite Function

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

Implicit and Inverse Function Analysis

Consider the function defined implicitly by $$e^{x*y}+x^2=5$$. Answer the following parts.

Implicit Differentiation for an Ellipse

Consider the ellipse defined by the equation $$\frac{x^2}{9} + \frac{y^2}{4} = 1$$. This equation re

Implicit Differentiation in a Circle

Consider the circle $$x^2 + y^2 = 25$$. Answer the following parts.

Implicit Differentiation in a Cubic Relationship

Examine the curve defined by $$x^3 + y^3 = 6*x*y$$, which represents a complex relationship between

Implicit Differentiation in Circular Motion

A runner is moving along a circular track described by the equation $$x^2+y^2=16$$, where $$x$$ and

Implicit Differentiation in Circular Motion

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

Implicit Differentiation in Elliptical Orbits

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

Implicit Differentiation Involving Sine

Consider the equation $$\sin(x*y)+x-y=0$$.

Implicit Differentiation of a Circle

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

Implicit Differentiation of a Logarithmic-Exponential Equation

Consider the equation $$\ln(x+y) + e^{x*y} = 7$$, which implicitly defines $$y$$ as a function of $$

Implicit Differentiation of a Trigonometric Composite Function

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

Implicit Differentiation of an Exponential-Product Equation

Consider the curve defined implicitly by $$e^(x*y) + x - y = 0.$$ Solve the following:

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

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 Trigonometric and Logarithmic Terms

Consider the equation $$\sin(x) + \ln(y) + x*y = 0.$$ Solve the following:

Implicit Differentiation: Combined Product and Chain Rules

Consider the equation $$x^2*y + \sin(x*y) = 0$$. Answer the following parts.

Intersection of Curves via Implicit Differentiation

Two curves are defined by the equations $$y^2= 4*x$$ and $$x^2+ y^2= 10$$. Consider their intersecti

Inverse Analysis in Exponential Decay

A radioactive substance decays according to $$N(t)= N_0*e^(-0.5*t)$$, where N(t) is the quantity at

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 Differentiation in a Biological Growth Curve

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

Related Rates and Composite Functions

A 10-foot ladder is leaning against a wall such that its bottom moves away from the wall according t

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 a Function Combining Polynomial and Exponential Terms

The concentration of a substance over time t (in hours) is modeled by $$C(t)= t^2 e^{-0.5*t} + 5$$.

Analysis of Experimental Data

The graph below shows the displacement of an object moving in a straight line. Analyze the object's

Chemical Reaction Rate Analysis

A chemical reaction follows the concentration model $$c(t)=\frac{100}{1+5e^{-0.3t}}$$, where c is in

Cost Analysis through a Rational Function

A company's average cost function is given by $$C(x)= \frac{2*x^3 + 5*x^2 - 20*x + 40}{x}$$, where $

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

Designing a Flower Bed: Optimal Shape

A landscape designer wants to create a rectangular flower bed with a fixed area of 200 square meters

Free Fall Motion Analysis

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

Friction and Motion: Finding Instantaneous Rates

A block slides down an inclined plane. The height of the plane at a horizontal distance $$x$$ is giv

FRQ 3: Ladder Sliding Problem

A 13­m ladder leans against a vertical wall. Its position satisfies the equation $$x^2 + y^2 = 169$$

Function with Vertical Asymptote

Consider the function $$f(x)=\frac{1}{x-3}+2$$. Analyze its behavior.

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

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 for Function Values

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

Linearization and Differentials

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

Local Linearization Approximation

Let $$f(x)=x^3.$$ We want to approximate $$f(4.02)$$ using linearization near $$x=4$$.

Logarithmic Differentiation in Exponential Functions

Let $$y = (2x^2 + 3)^{4x}$$. Use logarithmic differentiation to find $$y'$$.

Modeling a Bouncing Ball with a Geometric Sequence

A ball is dropped from a height of 10 m. Each time it bounces, it reaches 70% of the height of the p

Modeling Coffee Cooling

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

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

Open-top Box Optimization

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

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: Inflating Balloon

A spherical balloon is being inflated such that its volume increases at a rate of $$15\;cm^3/s$$. Th

Related Rates: Inflating Spherical Balloon

A spherical balloon is being inflated such that its volume increases at a constant rate of $$20$$ cu

Runner’s Speed Analysis

During a sprint, a runner's distance from the starting line is modeled by $$d(t)=-2t^2+12t$$, where

Studying a Bouncing Ball Model

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

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

Temperature Change in Cooling Coffee

A cup of coffee cools according to the model $$T(t) = 90e^{-0.05t} + 20$$, where $$t$$ is the time i

The Sliding Ladder

A 10 m long ladder is leaning against a vertical wall. The bottom of the ladder slides away from the

Vehicle Position and Acceleration

A vehicle's position along a straight road is modeled by $$s(t)=4\sqrt{t+1}$$ (in kilometers), where

Water Tank Dynamics

A water tank is subjected to an inflow and an outflow. The inflow rate is given by $$f(t)=10+2*t$$ m

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

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 a Supply and Demand Model Using Derivatives

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

Analyzing Critical Points in a Piecewise Function

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

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 the Mean Value Theorem on a Piecewise Function

Consider the function $$ f(x) = \begin{cases} x^2, & x < 2, \\ 4x - 4, & x \ge 2. \end{cases} $$ A

Behavior Analysis of a Logarithmic Function

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

Capacitor Discharge in an RC Circuit

The voltage across a capacitor during discharge is given by $$V(t)= V_0*e^{-t/(RC)}$$, where $$t$$ i

Chemical Reactor Temperature Optimization

In a chemical reactor, the temperature is controlled by the rate of coolant inflow. The coolant infl

Concavity Analysis of a Trigonometric Function

For the function $$f(x)= \sin(x) - \frac{1}{2}\cos(x)$$ defined on the interval $$[0,2\pi]$$, analyz

Concavity and Inflection Points in a Quartic Function

Analyze the concavity and determine any points of inflection for the function $$f(x)= x^4 - 4*x^3$$.

Continuity Analysis of a Rational Piecewise Function

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

Critical Numbers and Concavity in a Polynomial Function

Analyze the polynomial function $$f(x)= x^4 - 4*x^3 + 6*x^2 - 4*x + 1$$ by determining its critical

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

Economic Demand and Revenue Optimization

The demand for a product is modeled by $$D(p) = 100 - 2*p$$, where $$p$$ is the price in dollars. Th

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

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

FRQ 2: Daily Temperature Extremes and the Extreme Value Theorem

A function modeling the ambient temperature (in $$^\circ C$$) during the first 6 hours of a day is g

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 12: Optimization in Manufacturing: Minimizing Cost

A company’s cost function is given by $$C(x)= 0.5*x^2 - 10*x + 125$$ (in dollars), where $$x$$ repre

Graphical Analysis Using First and Second Derivatives

The graph provided represents the function $$f(x)= x^3 - 3*x^2 + 2*x$$. Analyze this function using

Investigating Limits and Discontinuities in a Rational Function with Complex Denominator

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

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

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

Oil Spill Cleanup

In an oil spill scenario, oil continues to enter an affected area while cleanup efforts remove oil.

Optimization of a Fenced Enclosure

A farmer wants to construct a rectangular garden using 120 meters of fencing along three sides, with

Piecewise Function with Trigonometric and Constant Segments

Consider the function $$ f(x) = \begin{cases} \cos(x), & x < \frac{\pi}{2}, \\ 0, & x = \frac{\pi}{

Relative Extrema in an Economic Demand Model

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

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 Altitude Change: Hiking Profile

During a hike, a climber's rate of change of altitude (in m/hr) is recorded as shown in the table be

Antiderivatives of Trigonometric Functions

Evaluate the integral $$\int \sin(2*t)\,dt$$, and then use your result to compute the definite integ

Antiderivatives with Initial Conditions: Temperature

The rate of temperature change in a chemical reaction is given by $$T'(t)=-0.2*t+3$$ (in °C/min), wi

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 with a Discontinuous Function

Consider the function $$h(x)= \begin{cases} x+2 & \text{if } 0 \le x < 3,\\ 7 & \text{if } x = 3,\\

Average Value of a Log Function

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

Bacterial Growth Modeling with Antibiotic Administration

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

Calculating Total Distance Traveled from a Changing Velocity Function

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

Coffee Brewing Dynamics

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

Comparing Riemann Sum and the Fundamental Theorem

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

Economic Cost Function Analysis

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

Economic Revenue Analysis from Marginal Revenue Data

A company's marginal revenue (in thousands of dollars per hour) is recorded over a 4-hour production

Electric Charge Accumulation

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

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

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

FRQ8: Inverse Analysis of a Piecewise-Defined Accumulation Function

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

FRQ11: Inverse Analysis of a Parameterized Function

For a positive constant a, consider the function $$ F(x)=\int_{a}^{x} \frac{1}{t+a}\,dt $$ for x > a

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.

FRQ19: Inverse Analysis with a Fractional Integrand

Let $$ M(x)=\int_{2}^{x} \frac{t}{t+2}\,dt $$. Answer the following parts.

Fuel Consumption Analysis

A truck's fuel consumption rate (in L/hr) is recorded at various times during a 12-hour drive. Use t

Implicit Differentiation of a Conic

Consider the relation $$x^2 + x*y + y^2 = 7.$$ Answer the following parts:

Marginal Cost and Total Cost

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

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

Modeling Water Volume in a Tank via Integration

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

Motion 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 $$0\le t\le8$$,

Particle Motion with Changing Direction

A particle moves along a line with its velocity given by $$v(t)=t*(t-4)$$ (in m/s) for $$0\le t\le6$

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

Population Accumulation in a Lake

A researcher is studying a fish population in a lake. The rate of change of the fish population is m

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

Riemann Sum Approximation for Sin(x)

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

Volume of a Solid with Square Cross Sections

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

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

Bernoulli Differential Equation

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

Boat Crossing a River with Current

A boat is attempting to cross a river flowing at a constant speed of $$2$$ m/s. The boat is directed

Charging of an RC Circuit

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

Chemical Reaction Rate

The concentration $$y$$ (in moles per liter) of a reactant in a chemical reaction is modeled by the

Chemical Reactor Temperature Profile

In a chemical reactor, the temperature $$T$$ (in °C) is recorded over time (in minutes) as shown. Th

Cooling of a Liquid

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

Environmental Pollution Model

Pollutant concentration in a lake is modeled by the differential equation $$\frac{dC}{dt}=\frac{R}{V

Implicit Differentiation and Tangent Lines of an Ellipse

Consider the ellipse defined by $$4x^2+ 9y^2= 36$$. Answer the following:

Implicit Differentiation of a Circle

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

Logistic Growth Model Analysis

A population $$y(t)$$ grows according to the logistic differential equation $$\frac{dy}{dt} = k * y

Logistic Population Growth

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

Logistic Population Model Analysis

A population $$P$$ grows according to the logistic equation $$\frac{dP}{dt}=0.4P\left(1-\frac{P}{100

Mixing of a Pollutant in a Lake

A lake with a constant volume of $$10^6$$ m\(^3\) receives polluted water from a river at a rate of

Modeling Continuous Compound Interest

An account accrues interest continuously according to the differential equation $$\frac{dA}{dt}=rA$$

Population Growth with Harvesting

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

Radioactive Decay and Half-Life

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

Radioactive Decay Differential Equation

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

Radioactive Decay with Production

A radioactive substance decays while being produced at a constant rate, and its mass $$M(t)$$ (in kg

RC Circuit Discharge

In an RC circuit, the voltage across a capacitor discharging through a resistor follows $$\frac{dV}{

Salt Mixing in a Tank

A tank initially contains 100 L of water with 5 kg of salt dissolved. Brine with a concentration of

Salt Mixing Problem

A tank initially contains $$100$$ kg of salt dissolved in $$1000$$ L of water. A salt solution with

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

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

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 Analysis for a Linear Differential Equation

Consider the linear differential equation $$\frac{dy}{dx}=\frac{1}{2}*x-y$$ with the initial conditi

Slope Field and Integrating Factor Analysis

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

Temperature Regulation in a Greenhouse

The temperature $$T$$ (in °F) inside a greenhouse is recorded over time (in hours) as shown. The war

Area Between \(\ln(x+1)\) and \(\sqrt{x}\)

Consider the functions $$f(x)=\ln(x+1)$$ and $$g(x)=\sqrt{x}$$ over the interval $$[0,3]$$.

Area Between a Function and Its Tangent

A function $$f(x)$$ and its tangent line at $$x=a$$, given by $$L(x)=m*x+b$$, are considered on the

Area Between a Parabola and a Line

Consider the functions $$f(x)=5*x-x^2$$ and $$g(x)=x$$. A set of experimental data provided the foll

Area Between Transcendental Functions

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

Area Between Two Curves

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

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 Temperature Analysis

A research facility recorded the temperature in a greenhouse over a period of 5 hours. The temperatu

Average Value of a Deposition Rate Function

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

Bacterial Colony Growth Analysis

A bacterial colony grows at a rate given by $$r(t)=20e^{0.1*t}$$ (in thousands per hour) over the ti

Cell Phone Battery Consumption

A cell phone’s battery life degrades over time such that the effective battery life each month forms

Consumer Surplus Calculation

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

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$

Cost Optimization for a Cylindrical Container

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

Designing an Open-Top Box

An open-top box with a square base is to be constructed with a fixed volume of $$5000\,cm^3$$. Let t

Determining Field Area from Intersection of Curves

A farmer's field is bounded by the curves $$y=0.5*x^2$$ and $$y=4*x$$. Find the area of the field wh

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$

Electrical Charge Calculation

The current in an electrical circuit is given by $$I(t)=6*e^{-0.5*t} - 3*e^{-t}$$ (in amperes) for $

Estimating Instantaneous Velocity from Position Data

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

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

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

Medication Dosage Increase

A patient receives a daily medication dose that increases by a fixed amount each day. The first day'

Net Change in Concentration of a Chemical Reaction

In a chemical reaction, the rate of production of a substance is given by $$r(t)$$ (in mol/min). The

Population Dynamics in a Wildlife Reserve

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

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 Increase in a Chemical Reaction

During a chemical reaction, the rate of temperature increase per minute follows an arithmetic sequen

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: 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

Washer Method with Logarithmic and Exponential Curves

Consider the region bounded by the curves $$f(x)=\ln(x+1)$$ and $$g(x)=e^{-x}$$ on the interval $$[0

Water Flow into a Reservoir

Water flows into a reservoir at a rate given by $$R(t)= 20 - 2*t$$ cubic meters per hour, where $$t$

Water Pumping from a Parabolic Tank

A water tank has ends shaped by the region bounded by $$y=x^2$$ and $$y=4$$, and the tank has a unif

Work Done by a Variable Force

A variable force is applied along a straight line such that $$F(x)=6-0.5*x$$ (in Newtons). The force