AP Calculus AB FRQ Room

Absolute Value Function and Discontinuity

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

Algebraic Manipulation in Limit Evaluation

Evaluate the limit $$\lim_{x \to 1} \frac{x^3 - 1}{x - 1}$$.

Analyzing a Piecewise Function’s Limits and Continuity

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

Analyzing Continuity in a Piecewise Function

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

Analyzing Multiple Discontinuities in a Rational Function

Let $$f(x)= \frac{(x^2-9)(x+4)}{(x-3)(x^2-16)}$$.

Application of the Intermediate Value Theorem

Let the function $$f(x)= x^3 - 4*x - 1$$ be continuous on the interval $$[0, 3]$$. Answer the follow

Capstone Problem: Continuity and Discontinuity in a Compound Piecewise Function

Consider the function $$f(x)=\begin{cases} \frac{x^2-1}{x-1} & x<2 \\ \frac{x^2-4}{x-2} & x\ge2 \end

Continuity of a Sine-over-x Function

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

Continuity of Composite Functions

Let $$f(x)=x+2$$ for all x, and let $$g(x)=\begin{cases} \sqrt{x}, & x \geq 0 \\ \text{undefined},

Determining Horizontal Asymptotes of a Log-Exponential Function

Examine the function $$s(x)=\frac{e^{x}+\ln(x+1)}{x}$$, which is defined for $$x > 0$$. Determine th

Determining Parameters for a Continuous Log-Exponential Function

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

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 Sequential Limits in Particle Motion

A particle’s velocity is given by the function $$v(t)= \frac{(t-2)(t+4)}{t-2}$$ for $$t \neq 2$$, an

Exponential Limit Parameter Determination

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

Graph-Based Analysis of Discontinuity

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

Intermediate Value Theorem in Temperature Modeling

A continuous function $$ f(x) $$ describes the temperature (in °C) throughout a day, with $$f(0)=15$

Jump Discontinuity Analysis

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

Limit Evaluation in a Parametric Particle Motion Context

A particle’s position in the plane is given by the parametric equations $$x(t)= \frac{t^2-4}{t-2}, \

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 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 Absolute Value

Consider the function $$f(x) = \frac{|x - 3|}{x - 3}$$. (a) Evaluate $$\lim_{x \to 3^-} f(x)$$ and

Modeling Temperature Change: A Real-World Limit Problem

A scientist records the temperature (in °C) of a chemical reaction during a 24-hour period using the

One-Sided Limits and Continuity of a Piecewise Function

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

Real-world Application: Economic Model of Inventory Growth

A company monitors its inventory \(I(t)\) (in units) over time (in months) using the rate function $

Removable Discontinuity and Limit

Consider the function $$ f(x)=\frac{x^2-9}{x-3} $$ for $$ x\ne3 $$, which is not defined at $$ x=3 $

Removal of Discontinuity by Redefinition

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

Robotic Arm and Limit Behavior

A robotic arm moving along a linear axis has a velocity function given by $$v(t)= \frac{t^3-8}{t-2}$

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

Squeeze Theorem Application with Trigonometric Functions

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

Squeeze Theorem with Bounded Function

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

Vertical Asymptote Analysis

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

Advanced Implicit Differentiation

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

Analysis of Motion in the Plane

A particle moves in the plane with its position given by $$\mathbf{s}(t)=\langle t^2 - 4*t,\, 3*t +

Analyzing a Function with a Removable Discontinuity

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

Analyzing a Projectile's Motion

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

Application of Derivative in Calculating Slope of a Curve

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

Approximating Small Changes with Differentials

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

Chain Rule Application

Consider the composite function $$f(x)=\sqrt{1+4*x^2}$$, which may describe a physical dimension.

Concavity and the Second Derivative

Consider the function $$f(x)=x^4-4*x^3+6*x^2$$. Answer the following:

Derivative from Definition for a Piecewise Function

Consider the piecewise function $$ f(x)= \begin{cases} e^{x} & x < 0 \\ \ln(x+1) + 1 & x \ge

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

Deriving the Derivative from First Principles for a Reciprocal Square Root Function

Let $$f(x)=\frac{1}{\sqrt{x}}$$ for $$x > 0$$. Using the definition of the derivative, show that $$f

Differentiability of a Piecewise Function

Consider the function defined by $$ f(x)= \begin{cases} x^2 & x \le 2 \\ 4*x-4 & x > 2 \end{cases

Differentiating a Product of Linear Functions

Let $$f(x) = (2*x^2 + 3*x)\,(x - 4)$$. Use the product rule to find $$f'(x)$$.

Differentiation of a Composite Motion Function

A particle’s position is given by $$s(t) = t^2 * \ln(t)$$ for $$t > 0$$. Use differentiation to anal

Differentiation of a Log-Linear Function

Consider the function $$f(x)= 3 + 2*\ln(x)$$ which might model a process with a logarithmic trend.

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

Implicit Differentiation of a Circle

Consider the equation $$x^2 + y^2 = 25$$ representing a circle with radius 5. Answer the following q

Inverse Function Analysis: Exponential Transformation

Consider the function $$f(x)=3*e^x-2$$ 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: Rational Decay Function

Consider the function $$f(x)=\frac{1}{1+x^2}$$ defined for $$x\geq0$$.

Inverse Function Analysis: Restricted Rational Function

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

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 $$

Polynomial Rate of Change Analysis

Consider the function $$f(x)= x^3 - 2*x^2 + x$$, which models a physical process. Analyze the rates

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

River Pollution Dynamics

A factory discharges pollutants into a river at a rate of $$f(t)=20+3*t$$ (kg/hour), while the river

Sand Pile Accumulation

A sand pile is being formed by a conveyor belt that drops sand at a rate of $$f(t)=5+0.5*t$$ (kg/min

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 to a Curve

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

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

Chain and Product Rules in a Rate of Reaction Process

In a chemical reaction, the rate is modeled by the function $$R(t)= \sqrt{t+3}*\cos(2*t)$$, where $$

Chain Rule in an Economic Model

In an economic model, the cost function for producing a good is given by $$C(x)=(3*x+1)^5$$, where $

Chain Rule in an Economic Model

In a manufacturing process, the cost function is given by $$C(q)= (\ln(1+3*q))^2$$, where $$q$$ is t

Chain Rule in Angular Motion

An object rotates such that its angular position is given by $$\theta(t)= \arctan(3*t^2)$$, where $$

Chain Rule in Particle Motion

A particle's position is given by $$s(t)=\cos(5*t^2)$$ in meters, where time $$t$$ is measured in se

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 Logarithms

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

Chain Rule with Multiple Nested Functions in a Physics Model

In a physics experiment, the displacement of a particle is modeled by the function $$s(t)=\sqrt{\cos

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 Differentiation with Nested Functions

Differentiate the function $$F(x)=\sqrt{\cos(4*x^2+1)}$$ using the chain rule. Your answer should re

Composite Function Analysis in Temperature Change

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

Composite Function from an Implicit Equation

Consider the implicit equation $$x^2 + y^2 + x*y = 7$$, which defines $$y$$ as an implicit function

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 Nested Chain Rule

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

Composite Log-Exponential Function Analysis

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

Concavity Analysis of an Implicit Curve

Consider the relation $$x^2+xy+y^2=7$$.

Differentiation of an Inverse Trigonometric Composite Function

Consider the function $$y = \arctan(\sqrt{3x})$$.

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 Physics Trajectory

A projectile follows a trajectory described by the implicit equation $$y^2 + 2*x*y + 3*x^2 = 50$$.

Implicit Differentiation Involving Trigonometric Functions

For the relation $$\sin(x) + \cos(y) = 1$$, consider the curve defined implicitly.

Implicit Differentiation of a Logarithmic Equation

Given the equation $$\ln(x) + \ln(y) = \ln(10)$$, answer the following parts.

Implicit Differentiation of an Ellipse in Navigation

A flight path is modeled by the ellipse $$\frac{x^2}{16}+\frac{y^2}{9}=1$$.

Implicit Differentiation with Chain and Product Rules

Consider the curve defined implicitly by $$e^{xy} + x^2y = 10$$. Assume that the point $$(1,2)$$ lie

Implicit Differentiation with Exponentials and Logarithms

Consider the curve defined implicitly by $$x*e^(y) + \ln(y)= e$$. It is given that the point $$(1, 1

Implicit Differentiation with Product Rule

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

Implicitly Defined Inverse Relation

Consider the relation $$y + \ln(y)= x.$$ Answer the following:

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

Inverse Function Differentiation in a Piecewise Scenario

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

Inverse Function Differentiation in an Exponential Model

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

Inverse Function Differentiation in Logarithmic Functions

Let $$f(x)=\ln(x+2)$$, which is one-to-one and has an inverse function $$g(y)$$. Answer the followin

Inverse Function Differentiation in Mixing Solutions

Let the function $$f(x)=2*x^3+x-5$$ model the concentration of a solution as a function of a paramet

Inverse Function Differentiation with a Logarithmic Function

Consider the function $$f(x)= \ln(5*x+3)$$. This logarithmic function is used to model various pheno

Inverse Function Differentiation with Logarithmic Function

Let $$f(x) = x + \ln(x)$$ and let g denote its inverse function. Answer the following parts.

Inverse Trigonometric Differentiation in a Geometry Problem

Consider a scenario in which an angle $$\theta$$ in a geometric configuration is given by $$\theta=\

Inverse Trigonometric Function Differentiation

Let $$f(x)=\arcsin\left(\frac{2*x}{5}\right)$$, with the understanding that $$\left|\frac{2*x}{5}\ri

Nested Composite Function Differentiation

Consider the function $$f(x)= \sqrt{\ln(3*x^2+2)}$$, where $$\sqrt{\ }$$ denotes the square root. So

Population Dynamics via Composite Functions

A biological population is modeled by $$P(t)= \ln\left(20*e^(0.1*t^2)+ 5\right)$$, where t is measur

Projectile Motion and Composite Function Analysis

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

Second Derivative via Implicit Differentiation

Consider the ellipse given by $$\frac{x^2}{4}+\frac{y^2}{9}=1$$. Answer the following:

Accelerating Car Motion Analysis

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

Airplane Altitude Change

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

Analysis of a Function Combining Polynomial and Exponential Terms

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

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

Balloon Altitude and Temperature

A hot air balloon rises such that its altitude is given by $$h(t)=3*t^{2/3}$$ meters, where t is in

Balloon Inflation Analysis

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

Blood Drug Concentration

In a pharmacokinetic study, the concentration of a drug in the bloodstream is modeled by $$D(t)=\fra

Chemical Reaction Rate Analysis

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

Coffee Cooling Analysis Revisited

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

Demand Function Inversion and Analysis

The product demand is modeled by $$p(q)=\frac{100}{q+1}+20$$, where p is the price (in dollars) and

Designing a Flower Bed: Optimal Shape

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

Estimating Function Change Using Differentials

Let $$f(x)=x^{1/3}$$. Use differentials to approximate the change in $$f(x)$$ when $$x$$ increases f

Expanding Pool Dimensions

The surface area of a circular pool is increasing at a rate of $$20$$ ft²/s. The area of a circle is

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

FRQ 3: Ladder Sliding Problem

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

FRQ 6: Particle Motion Analysis on a Straight Line

A particle moving along a straight line has its velocity described by $$v(t) = 3*t^2 - 4*t + 2$$, wh

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

Implicit Differentiation in Related Rates

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

Inflating Spherical Balloon

A spherical balloon is being inflated such that its volume increases at a constant rate of $$\frac{d

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 Application

Evaluate the limit $$\lim_{x\to0}\frac{e^{2*x}-1}{3*x}$$.

Linear Approximation of ln(1.05)

Let $$f(x)=\ln(x)$$. Use linearization at x = 1 to approximate the value of $$\ln(1.05)$$.

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 in Medicine Dosage

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

Medicine Dosage: Instantaneous Rate of Change

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

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

Optimization & Linearization in Engineering Design

A material's strength is modeled by the function $$S(x)= 50*x^2 - 3*x^3$$, where $$x$$ (in centimete

Optimizing Road Construction Costs

An engineer is designing a road that connects a point on a highway to a town located off the highway

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 Analysis

A particle moves along a straight line with displacement given by $$s(t)=t^3-6t^2+9t+2$$ for $$0\le

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

Reaction Rates in Chemistry

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

Related Rates in Shadows: A Lamp and a Tree

A lamp post 5 m tall casts a shadow of a tree. At a certain moment, the tree is 3 m from the lamp an

Savings Account Growth Modeled by a Geometric Sequence

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

Vehicle Position and Acceleration

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

Volume Change Analysis in a Swimming Pool

The volume of a pool is given by $$V(t)=8t^2-32t+4$$, where V is in gallons and t in hours. Analyze

Application of Rolle's Theorem

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

Area and Volume: Polynomial Boundaries

Let $$f(x)= x^2$$ and $$g(x)= 4 - x^2$$. Consider the region bounded by these two curves.

Car Speed Analysis via MVT

A car's position is given by $$f(t) = t^3 - 3*t^2 + 2*t$$ (in meters) for $$t$$ in seconds on the cl

Chemical Mixing in a Tank

A 200-liter tank initially contains pure water. A salt solution with a concentration of 0.5 kg/L flo

Chemical Reactor Temperature Optimization

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

Comprehensive Analysis of a Rational Function

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

Designing an Optimal Can

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

FRQ 11: Particle Motion with Non-Constant Acceleration

A particle moves along a straight line with acceleration given by $$a(t)= 12*t - 6$$ (in m/s²). If t

FRQ 16: Finding Relative Extrema for a Logarithmic Function

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

Inverse Analysis of a Function with Square Root and Linear Term

Consider the function $$f(x)=\sqrt{3*x+1}+x$$. Answer the following questions regarding its inverse.

Inverse Analysis of an Exponential Function

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

Investment with Continuous Compounding and Variable Rates

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

Mean Value Theorem Applied to Car Position Data

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

Mean Value Theorem for a Piecewise Function

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

Monotonicity and Inverse Function Analysis

Consider the function $$f(x)= x + e^{-x}$$ defined for all real numbers. Investigate its monotonicit

Motion Analysis: A Runner's Performance

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

Piecewise Function with Trigonometric and Constant Segments

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

Predicting Fuel Efficiency in Transportation

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

Profit Function Concavity Analysis

A company’s profit is modeled by $$P(x) = -2*x^3 + 18*x^2 - 48*x + 10$$, where $$x$$ is measured in

Projectile Motion and Derivatives

A projectile is launched so that its height is given by $$h(t) = -4.9*t^2 + 20*t + 1$$, where $$t$$

Related Rates in an Evaporating Reservoir

A reservoir’s volume decreases due to evaporation according to $$V(t)= V_0*e^{-a*t}$$, where $$t$$ i

Reservoir Evaporation and Rainfall

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

Sand Pile Dynamics

A sand pile is being formed on a surface where sand is both added and selectively removed. The inflo

Tangent Line and MVT for ln(x)

Consider the function $$f(x)= \ln(x)$$ on the interval $$[1,e]$$.

Tangent Line to an Implicitly Defined Curve

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

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 Bacteria Growth

A laboratory observes a bacterial colony whose rate of growth (in bacteria per hour) is modeled by t

Accumulation and Inflection Points

Suppose a function's rate of change is given by $$f'(x)=3*x^2-12*x+9.$$ Answer the following parts:

Antiderivative via U-Substitution

Evaluate the antiderivative of the function $$f(x)=(3*x+5)^4$$ and use it to compute a definite inte

Antiderivatives and the Constant of Integration in Modelling

A moving car has its velocity modeled by $$v(t)= 5 - 2*t$$ (in m/s). Answer the following parts to o

Application of the Fundamental Theorem of Calculus

A particle moves along a straight line with an instantaneous velocity given by $$v(t)=3*t^2+2*t$$ (i

Area Between Curves

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

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 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

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].

Chemical Accumulation in a Reactor

A chemical reactor has a net accumulation rate given by $$R(t)=5*\cos(t) + 2$$ (in kg/hour), where $

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 Methods for a Complex Function

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

Composite Functions and Accumulation

Let the accumulation function be defined by $$F(x)=\int_{2}^{x} \sqrt{t+1}\,dt.$$ Answer the followi

Definite Integral as an Accumulation Function

A generator consumes fuel at a rate given by $$f(t)=t*e^{-t}$$ (in liters per second). Answer the fo

Economic Accumulation of Revenue

The marginal revenue (MR) for a company is given by $$MR(x)=50*e^{-0.1*x}$$ (in dollars per item), w

Estimating Area Under a Curve Using Riemann Sums

Consider the function whose values are given in the table below. Use the table to estimate the area

Evaluating an Integral with a Trigonometric Function

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

FRQ1: Analysis of an Accumulation Function and its Inverse

Consider the function $$ F(x)=\int_{1}^{x} (2*t+3)\,dt $$ for $$ x \ge 1 $$. Answer the following pa

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 \

FRQ10: Inverse Analysis of a Production Accumulation Function

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

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

Fundamental Theorem and Net Change

A function $$f$$ has a derivative given by $$f'(x)=3\sqrt{x}$$ for $$0\le x\le 9$$, and it is known

Graphical Analysis of an Accumulation Function

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

Integration of Exponential Functions with Shifts

Evaluate the integral $$\int_{0}^{2} e^{2*(x-1)}\,dx$$ using an appropriate substitution.

Integration to Determine Work Done by a Variable Force

A variable force along a straight line is given by $$F(x)=4*x^2 - 3*x + 2$$ (in Newtons). Determine

Modeling Accumulated Revenue over Time

A company’s revenue rate is given by $$R(t)=100*e^{0.1*t}$$ dollars per month, where t is measured i

Net Change in Salaries: An Accumulation Function

A company models its annual bonus savings with the rate function $$B'(t)= 500*(1+\sin(t))$$ dollars

Pollutant Concentration in a River

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

Population Growth and Accumulation

A rabbit population grows in an enclosed field at a rate given by the differential equation $$P'(t)=

Rainfall Accumulation Analysis

The rainfall intensity at a location is modeled by the function $$i(t) = 0.5*t$$ (inches per hour) f

River Flow Volume Calculation

A river has a flow rate given by $$Q(t)=4+\sin(t)$$ (in m³/s), where t is time in hours. Compute the

Tabular Riemann Sums for Electricity Consumption

A household's daily electricity consumption (in kWh) over 5 consecutive days is recorded in the tabl

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

Temperature Change in a Room

The rate of change of the temperature in a room is given by $$\frac{dT}{dt}=0.5*t+1$$, where $$T$$ i

Total Distance from Velocity Data

A car’s velocity, in meters per second, is recorded over time as given in the table below: | Time (

Trapezoidal Rule in Estimating Accumulated Change

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

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

Volume of a Solid by Washer Method

A region is bounded by the curves $$y=\sqrt{x}$$ and $$y=\frac{x}{2}$$. This region, between the cur

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

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 Mixing

In a continuously stirred chemical reactor, a reactant is added at a rate that results in an inflow

Chemical Reactor Temperature Profile

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

Exponential Growth: Separable Equation

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

Falling Object with Air Resistance

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

Implicit IVP with Substitution

Solve the initial value problem $$\frac{dy}{dx}=\frac{y}{x}+\frac{x}{y}$$ with $$y(1)=2$$. (Hint: Us

Implicit Solution for $$\frac{dy}{dx}=\frac{x+2}{y+1}$$

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

Investigating a Piecewise Function's Discontinuity

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

Modeling Continuous Compound Interest

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

Nonlinear Cooling of a Metal Rod

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

Nonlinear Differential Equation

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

Population Model with Harvesting

A fish population in a lake is modeled by the differential equation $$\frac{dP}{dt}=0.3*P\left(1-\fr

Radioactive Decay Differential Equation

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

Radioactive Decay with Production

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

Reaction Kinetics in a Tank

In a chemical reactor, the concentration $$C(t)$$ of a reactant decreases according to the different

Separable Differential Equation with Trigonometric Factor

Consider the differential equation $$\frac{dy}{dx}=(2y+3)\cos(x)$$. Answer each part using separatio

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

Accumulated Electrical Charge from a Current Function

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

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 a Cubic and a Linear Function

Consider the functions $$f(x)=x^3-3*x$$ and $$g(x)=x$$. Use integration to determine the area of the

Average Concentration Calculation

In a continuous stirred-tank reactor (CSTR), the concentration of a chemical is given by $$c(t)=5+3*

Average Drug Concentration in the Bloodstream

The concentration of a drug in the bloodstream is modeled by $$C(t)=\frac{20*t}{1+t^2}$$ (in mg/L) f

Determining Velocity and Position from Acceleration

A particle moves along a line with acceleration given by $$a(t)=4-2*t$$ (in $$m/s^2$$). At time $$t=

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$

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

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

Implicit Differentiation in an Electrical Circuit

In an electrical circuit, the voltage $$V$$ and current $$I$$ are related by the equation $$V^2 + (3

Implicit Differentiation in Thermodynamics

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

Piecewise Function Analysis

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

Population Dynamics in a Wildlife Reserve

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

Revenue Optimization via Integration

A company’s revenue is modeled by $$R(t)=1000-50*t+2*t^2$$ (in dollars per hour), where $$t$$ (in ho

River Current Analysis

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

Temperature Average Calculation

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

Voltage and Energy Dissipation Analysis

The voltage across an electrical component is modeled by $$V(t)=12*e^{-0.1*t}*\ln(t+2)$$ (in volts)

Volume by Discs: Revolved Region

Consider the region bounded by the curves $$y=x$$ and $$y=x^2$$ between their intersection points. T

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 with Rectangular Cross Sections

A solid has a base on the x-axis from $$x=0$$ to $$x=3$$. The cross-sectional areas (in m²) perpendi

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

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$

Work Done Stretching a Spring with Variable Constant

A spring does not follow Hooke's law exactly: its effective spring constant varies with displacement