AP Calculus AB FRQ Room

Analysis of a Removable Discontinuity in a Log-Exponential Function

Consider the function $$p(x)= \frac{e^{x}-e}{\ln(x)-\ln(1)}$$ for $$x \neq 1$$. The function is unde

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

Continuity of a Sine-over-x Function

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

Evaluating Limits Near Vertical Asymptotes

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

Exponential Limit Parameter Determination

Consider the function $$f(x)=\frac{e^{3*x} - e^{k*x}}{x}$$ for $$x \neq 0$$, and define $$f(0)=L$$,

Factorization and Limit Evaluation

Consider the function $$f(x) = \frac{x^3 - 8}{x - 2}$$. (a) Factor the numerator and simplify the e

Factorization and Removable Discontinuity

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

Graph-Based Analysis of Discontinuity

Examine the graph of a function that appears to be defined by $$f(x)= 3x - 2$$ for all $$x \neq 2$$,

Horizontal Asymptote of a Rational Function

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

Implicit Differentiation with Rational Exponents

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

Intermediate Value Theorem and Root Existence

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

Intermediate Value Theorem with an Exponential-Logarithmic Function

Consider the function $$u(x)=e^{x}-\ln(x+2)$$, defined for $$x > -2$$. Since $$u(x)$$ is continuous

Jump Discontinuity in a Piecewise Function

Consider the function $$g(x)=\begin{cases} \frac{x^2-4}{x-2} & x<2\\ 5 & x=2\\ x+3 & x>2 \end{cases}

Limit Involving a Square Root and Removable Discontinuity

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

Limit with Square Root and Removable Discontinuity

Consider the function $$f(x)=\begin{cases} \frac{\sqrt{4*x+8}-4}{x-2} & x\neq2 \\ 1 & x=2 \end{cases

Limits at Infinity and Horizontal Asymptotes

Examine the function $$f(x)=\frac{3x^2+2x-1}{6x^2-4x+5}$$ and answer the following:

Limits Involving a Removable Discontinuity

Consider the function $$g(x)= \frac{(x+3)(x-2)}{x-2}$$ defined for $$x \neq 2$$. Answer the followin

Limits Involving Trigonometric Functions in Particle Motion

A particle moves along a line with velocity given by $$v(t)= \frac{\sin(2*t)}{t}$$ for $$t > 0$$. An

Limits of Absolute Value Functions

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

Logarithmic Function Continuity

Consider the function $$g(x)=\frac{\ln(2*x+3)-\ln(5)}{x-1}$$ for $$x \neq 1$$. To make $$g(x)$$ cont

Optimization and Continuity in a Manufacturing Process

A company designs a cylindrical can (without a top) for which the cost function in dollars is given

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{

Particle Motion with Removable Discontinuity

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

Piecewise Function Continuity Analysis

The function f is defined by $$ f(x) = \begin{cases} \frac{x^2 - 1}{x - 1}, & x \neq 1 \\ k, & x

Piecewise Function Continuity and IVT

Let $$f(x)=\begin{cases} x^2, & x \leq 1 \\ a*x+b, & x > 1 \end{cases}$$. Determine constants a and

Real-World Application: Temperature Sensor Calibration

A temperature sensor in a lab records temperatures (in °C) according to the function $$f(t)= \frac{t

Removable Discontinuity and Redefinition

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

Squeeze Theorem for an 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

Air Quality and Pollution Removal

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

Approximating Small Changes with Differentials

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

Approximating the Tangent Slope

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

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

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

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 of the Square Root Function via Limit Definition

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

Derivative using the Limit Definition for a Linear Function

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

Differentiation Using the Quotient Rule

Consider the function \(q(x)=\frac{3*x^2+5}{2*x-1}\). Answer the following parts.

Exponential Growth Rate

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

Finding Derivatives with Product and Quotient Rule

Let $$f(x)=\sin(x)*\frac{x^2+1}{x}$$ for $$x \neq 0$$. Answer the following questions:

Finding the Second Derivative

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

Graph Interpretation of the Derivative

Consider the function $$f(x)= x^3 - 6*x^2 + 9*x + 1$$. A graph of this function is provided below.

Higher Order Derivatives and Concavity

Let \(f(x)=x^3 - 3*x^2 + 5*x - 2\). Answer the following parts.

Identifying Horizontal Tangents

A continuous function $$f(x)$$ has a derivative $$f'(x)$$ such that $$f'(4)=0$$ and $$f'(x)$$ change

Instantaneous Rate of Change in Motion

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

Instantaneous Rate of Temperature Change in a Coffee Cup

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

Inverse Function Analysis: Cubic Function

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

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

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 $$s(t)$$

Marginal Cost Function in Economics

A company’s cost function is given by $$C(x)=200+8*x+0.05*x^2$$, where $$C(x)$$ is in dollars and $$

Mountain Stream Flow Adjustment

A mountain stream receives additional water from snowmelt at a rate of $$f(t)=4*t$$ (cubic feet/seco

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

Rate of Chemical Reaction

The concentration of a reactant in a chemical reaction is modeled by \(C(t)=10*e^{-0.3*t}\), where \

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 Application: Temperature Change in a Chemical Reaction

The temperature (in $$\degree C$$) during a chemical reaction is modeled by $$T(t)= 25 - 2*t + \frac

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

Secant Line Slope Approximations in a Laboratory Experiment

In a chemistry lab, the concentration of a solution is modeled by $$C(t)=10*\ln(t+1)$$, where $$t$$

Secant Slope from Tabulated Data

A table below gives values of a function $$f(x)$$ representing the concentration of a solution at di

Slope of a Tangent Line from Experimental Data

Experimental data recording the distance traveled by an object over time is provided in the table be

Using the Limit Definition to Derive the Derivative

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

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

Analyzing Composite Functions Involving Inverse Trigonometry

Let $$y=\sqrt{\arccos\left(\frac{1}{1+x^2}\right)}$$. Answer the following:

Analyzing Motion in the Plane using Implicit Differentiation

A particle moves in the xy-plane along a path defined implicitly by $$x^2+x*y+y^2=7$$. Determine the

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 with Exponential and Trigonometric Functions

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

Chain Rule with 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

Chain Rule with Trigonometric Function

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

Composite and Rational Function Differentiation

Let $$P(x)=\frac{x^2}{\sqrt{1+x^2}}$$.

Composite Function Differentiation Involving Product and Chain Rules

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

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 with Inverse Trigonometric Components

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

Composite Function with Inverse Trigonometric Outer Function

Consider the function $$H(x)=\arctan(\sqrt{x^2+1})$$. Answer the following parts.

Composite Temperature Model

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

Composite Trigonometric Function Analysis in Pendulum Motion

A pendulum's angular displacement is modeled by the function $$\theta(t)= \sin(\sqrt{2*t+1})$$.

Composite, Implicit, and Inverse Combined Challenge

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

Designing a Tapered Tower

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

Differentiation Involving Exponentials and Inverse Trigonometry

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

Differentiation of a Complex Implicit Equation

Consider the equation $$\sin(xy) + \ln(x+y) = x^2y$$.

Implicit Differentiation in a Biochemical Reaction

Consider a biochemical reaction modeled by the equation $$x*e^{y} + y*e^{x} = 10$$, where $$x$$ and

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 a Population Growth Model

Consider the model $$e^{x*y} + x - y = 5$$ that relates time \(x\) to a population scale value \(y\)

Implicit Differentiation in an Ellipse

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

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 with Logarithmic and Radical Components

Consider the equation $$\ln(x+y)=\sqrt{x*y}$$.

Implicit Differentiation: Combined Product and Chain Rules

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

Inverse Function Derivative for a Log-Linear Function

Let $$f(x)= x+ \ln(x)$$ for $$x > 0$$ and let g be the inverse of f. Solve the following parts:

Inverse Function Derivative for a Logarithmic Function

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

Inverse Trigonometric Function Differentiation

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

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,

Nested Trigonometric Function Analysis

A physics experiment produces data modeled by the function $$h(x)=\cos(\sin(3*x))$$, where $$x$$ is

Related Rates in a Circular Colony

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

Temperature Change Model Using Composite Functions

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

Airplane Altitude Change

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

Analysis of Wheel Rotation

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

Analyzing Cost Functions Using Derivatives

A cost function for producing $$x$$ units is given by $$C(x)=0.1x^3 - 2x^2 + 20x + 100$$. This funct

Bacterial Growth Analysis

The number of bacteria in a culture is given by $$P(t)=500e^{0.2*t}$$, where $$t$$ is measured in ho

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

Differentiability of a Piecewise Function

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

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

Estimating Small Changes using Differentials

In an electrical circuit, the resistance is given by $$R(x) = \frac{1}{1+x^2}$$, where x is a parame

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

Falling Object's Velocity Analysis

A rock is thrown upward from the top of a building with a velocity function $$v(t)= 20 - 9.8*t$$ (in

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 2: Balloon Inflation Analysis

A spherical balloon is being inflated. Its volume is given by $$V = \frac{4}{3}\pi r^3$$, and the ra

FRQ 11: Shadow Length Change

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

FRQ 17: Water Heater Temperature Change

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

Graphing a Function via its Derivative

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

Inverse Function Analysis in a Real-World Model

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

L'Hôpital's Rule in Chemical Kinetics

In a chemical kinetics experiment, the reaction rate is modeled by the function $$f(x)=\frac{\ln(1+3

Limit Evaluation Using L'Hôpital's Rule

Evaluate the limit $$\lim_{x \to \infty} \frac{5x^3 - 4x^2 + 1}{7x^3 + 2x - 6}$$.

Linear Approximation in Production Cost Estimation

A company's cost function is given by $$C(x)=0.02x^2+10x+500$$, where $$x$$ (in thousands) is the nu

Linear Approximations: Estimating Function Values

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

Linearization and Differentials

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

Linearization for Function Estimation

Use linear approximation to estimate the value of $$\ln(4.1)$$. Let the function be $$f(x)=\ln(x)$$

Linearization in Medicine Dosage

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

Linearization of a Nonlinear Function

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

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 Analysis from Velocity Function

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

Particle Acceleration and Direction of Motion

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

Particle Motion with Changing Acceleration

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

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

Rates of Change in Economics: Marginal Cost

A company's cost function is given by $$C(q)= 0.5*q^2 + 40$$, where q (in units) is the quantity pro

Related Rates: Expanding Circular Ripple

A ripple in a still pond expands in the shape of a circle. The area of the ripple is given by $$A=\p

Related Rates: Expanding Oil Spill

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

Related Rates: Inflating Balloon

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

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(

Tangent Line and Linearization Approximation

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

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

Train Motion Analysis

A train’s acceleration is given by $$a(t)=\sin(t)+0.5$$ (m/s²) for $$0 \le t \le \pi$$ seconds. The

Vehicle Deceleration Analysis

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

Vehicle Position and Acceleration

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

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

Absolute Extrema via the Candidate's Test

Consider the function $$f(x)= \sqrt{x} - x$$ on the closed interval $$[0,4]$$. Use the candidate's t

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 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, &

Analyzing Endpoints and Discontinuities for an Absolute Value Function

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

Analyzing the Function $$f(x)= x*\ln(x) - x$$

Consider the function $$f(x)= x*\ln(x) - x$$ defined for $$x > 0$$.

Bacterial Culture Growth: Identifying Critical Points from Data

A microbiologist records the population of a bacterial culture (in millions) at different times (in

Behavior Analysis of a Logarithmic Function

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

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

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

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

Designing an Optimal Can

A closed cylindrical can is to have a volume of $$600$$ cubic centimeters. The surface area of the c

FRQ 3: Relative Extrema for a Cubic Function

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

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 18: Marginal Cost Analysis and Concavity

The cost per unit of producing $$x$$ units is given by $$C(x)= 100 + 20*x - 0.5*x^2$$ for $$0 \le x

Increasing/Decreasing Behavior in a Financial Model

A financial analyst models the performance of an investment with the function $$f(x)= \ln(x) - \frac

Inverse Analysis of an Exponential Function

Consider the function $$f(x)=2*e^(x)+3$$. Analyze its inverse function as instructed in the followin

Investigating a Piecewise Function with a Vertical Asymptote

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

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,

Liquid Cooling System Flow Analysis

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

Mean Value Theorem for a Logarithmic Function

Consider the function $$f(x)= \ln(x)$$ defined on the interval $$[1, e^2]$$. Use the Mean Value Theo

Motion Analysis via Derivatives

A particle moves along a straight line with its position described by $$s(t)= t^3 - 6*t^2 + 9*t + 5$

Predicting Fuel Efficiency in Transportation

A vehicle’s performance was studied by recording the miles traveled and the corresponding fuel consu

Quartic Polynomial Concavity Analysis

Consider the quartic function $$f(x)= x^4 - 6*x^3 + 11*x^2 - 6*x$$, defined on the interval $$[0,4]$

Reservoir Evaporation and Rainfall

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

Revenue Optimization in Economics

A company's revenue is modeled by the function $$R(x)= x*e^{-0.1*x}$$, where $$x$$ (in thousands) re

Sign Analysis of f'(x)

The first derivative $$f'(x)$$ of a function is known to have the following behavior on $$[-2,2]$$:

Volume of Solid with Square Cross-Sections

Consider the region between $$f(x)= \sin(x)$$ and the x-axis on the interval $$[0, \pi]$$. A solid i

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

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

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

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:

Evaluating an Integral with a Trigonometric Function

Evaluate the definite integral $$\int_{0}^{\frac{\pi}{2}} \cos(x)*\sin(x)\,dx$$ using an appropriate

Exact Area Under a Transformed Function Using U-Substitution

Evaluate the area under the curve described by the integral $$\int_{1}^{5} 2*(x-1)^{3}\,dx$$ using u

FRQ16: Inverse Analysis of an Integral Function via U-Substitution

Let $$ U(x)=\int_{0}^{x} 2*(t-3)^2\,dt $$ for x ≥ 3. Answer the following parts.

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

Growth of Investment with Regular Contributions and Withdrawals

An investment account receives contributions at a rate of $$C(t)= 100e^{0.05t}$$ dollars per year an

Logistically Modeled Accumulation in Biology

A biologist is studying the growth of a bacterial culture. The rate at which new bacteria accumulate

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 $$0\le t\le8$$,

Population Growth: Accumulation through Integration

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

Rainwater Collection in a Reservoir

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

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

Total Distance Traveled from Velocity Data

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

Total Water Volume from a Flow Rate Function

A river’s flow rate (in cubic meters per second) is modeled by the function $$Q(t)=4+2*t$$, where $$

Volume Accumulation in a Leaking Tank

Water leaks from a tank at a rate given by $$R(t)=3-0.5*t$$ (in liters per minute) for t in [0,6]. I

Water Flow in a Tank

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

Bacterial Growth under Logistic Model

A bacterial culture grows according to the logistic differential equation $$\frac{dB}{dt}=rB\left(1-

Bacterial Growth with Constant Removal

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

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 Challenge

Consider the nonlinear differential equation $$\frac{dy}{dt} - y = -y^3$$ with the initial condition

Bernoulli Differential Equation via Substitution

Consider the differential equation $$\frac{dy}{dx}=y+x*y^2$$. Recognize that this is a Bernoulli equ

Carbon Dating and Radioactive Decay

Carbon dating is based on the radioactive decay model given by $$\frac{dC}{dt}=-kC$$. Let the initia

Charging a Capacitor in an RC Circuit

In an RC circuit, the charge $$Q$$ on a capacitor satisfies the differential equation $$\frac{dQ}{dt

Chemical Reaction Rate and Concentration Change

The rate of a chemical reaction is described by the differential equation $$\frac{dC}{dt}=-0.3*C^2$$

Chemical Reactor Temperature Profile

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

Combined Cooling and Slope Field Problem

A cooling process is modeled by the equation $$\frac{d\theta}{dt}=-0.07\,\theta$$ where $$\theta(t)=

Cooling of a Hot Beverage

According to Newton's Law of Cooling, the temperature $$T(t)$$ of a hot beverage satisfies $$\frac{d

Drug Concentration with Continuous Infusion

A drug is administered intravenously such that its blood concentration $$C(t)$$ (in mg/L) follows th

Drug Infusion and Elimination

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

Exponential Growth: Separable Equation

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

Implicit Differentiation and Slope Analysis

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

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 Problem with Evaporation and Drainage

A tank initially contains 200 L of water with 20 kg of pollutant. Water enters the tank at 2 L/min w

Newton's Law of Cooling with Variable Ambient Temperature

An object is cooling according to Newton's Law of Cooling, but the ambient temperature is not consta

Nonlinear Cooling of a Metal Rod

A thin metal rod cools in an environment at $$15^\circ C$$ according to the differential equation $$

Oil Spill Cleanup Dynamics

To mitigate an oil spill, a cleanup system is employed that reduces the volume of oil in contaminate

Particle Motion with Variable Acceleration

A particle moves along a straight line with acceleration $$a(t)=3-2*t$$ (in m/s²). Its initial veloc

Population Dynamics with Harvesting

A fish population is governed by the differential equation $$\frac{dP}{dt} = 0.4*P\left(1-\frac{P}{1

Radioactive Decay Differential Equation

A radioactive substance decays according to the differential equation $$\frac{dM}{dt}=-k*M$$. If the

RC Circuit Charging

In a resistor-capacitor (RC) circuit, the charge $$Q(t)$$ on the capacitor is modeled by the differe

Sketching Solution Curves on a Slope Field

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

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

Traffic Flow Dynamics

On a highway, the density of cars, \(D(t)\) (in cars), changes over time due to a constant inflow of

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

Analysis of a Rational Function's Average Value

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

Area Between Curves: 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

Arithmetic Savings Account

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

Average Concentration in Medical Dosage

A patient’s bloodstream concentration of a medication is modeled by the function $$C(t)=\frac{100}{1

Average Growth Rate in a Biological Process

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

Average Temperature Analysis

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

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 $$

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

Average Value of a Trigonometric Function

Consider the function $$f(x)=\sin(x)+1$$ defined on the interval $$[0,\pi]$$. This function models a

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

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

Charity Donations Over Time

A charity receives monthly donations that form an arithmetic sequence. The first donation is $$50$$

Download Speeds Improvement

An internet service provider increases its download speeds as part of a new promotional plan such th

Exponential Decay Function Analysis

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

Hiking Trail: Position from Velocity

A hiker's velocity is given by $$v(t)=3\cos(t/2)+1$$ (in km/h) for 0 ≤ t ≤ 2π. Assuming the hiker st

Ice Rink Design: Volume and Area

An ice rink is designed with a cross-sectional profile given by $$y=4-x^2$$ (with y=0 as the base).

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

Motion Experiment with Sinusoidal Acceleration

A particle has an acceleration given by $$a(t)=2\sin(t)$$ (in m/s²) for 0 ≤ t ≤ 2π. The initial cond

Net Change and Total Distance in Particle Motion

A particle has acceleration $$a(t)=12-8*t$$ (in $$m/s^2$$) for $$t \ge 0$$, with initial velocity $$

Pharmacokinetic Analysis

A drug concentration in the bloodstream is modeled by $$C(t)=15*e^{-0.2*t}+2$$, where $$t$$ is in ho

Pollutant Accumulation in a River

Along a 20 km stretch of a river, a pollutant enters the water at a rate described by $$p(x)=0.5*x+2

Population Growth with Variable Growth Rate

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

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 $$

Radioactive Decay Accumulation

A radioactive substance decays at a rate given by $$r(t)= C*e^{-k*t}$$ grams per day, where $$C$$ an

Related Rates: Shadow Length Change

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

River Discharge Analysis

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

Stress Analysis in a Structural Beam

A beam in a building experiences a stress distribution along its length given by $$\sigma(x)=100-15*

Tank Filling Process Analysis

Water flows into a tank at a rate modeled by $$R(t)=5+0.5*t$$ (in liters per minute) for $$0 \le t \

Total Distance from a Runner's Variable Velocity

A runner’s velocity (in m/s) is modeled by the function $$v(t)=t^2-10*t+16$$ for $$0 \le t \le 10$$

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 of a Solid of Revolution Rotated about a Line

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

Volume of a Solid of Revolution 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 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 a Hole Using the Washer Method

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

Volume of a Solid with Square Cross Sections

A solid has a base in the xy-plane given by the region bounded by the curves $$y=x$$ and $$y=\sqrt{x

Volume with Semicircular Cross‐Sections

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

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 Filling with Graduated Inflow

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