AP Calculus AB FRQ Room

Advanced Analysis of a Piecewise Function

Consider the function $$f(x)=\begin{cases} x^2*\sin\left(\frac{1}{x}\right) & x\neq 0 \\ 0 & x=0 \en

Analysis of a Rational Function with Exponential and Logarithmic Components

Consider the function $$g(x)=\frac{e^{x}-1-\ln(1+x)}{x}$$ for $$x \neq 0$$. Evaluate the limit as $$

Analysis of Vertical Asymptotes

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

Analyzing a Removable Discontinuity

Consider the function $$f(x) = \frac{x^2 - 4}{x - 2}$$ for $$x \neq 2$$. Notice that f is not define

Analyzing Multiple Discontinuities in a Rational Function

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

Analyzing Process Data for Continuity

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

Continuity Analysis of a Radical Function

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

Continuity and Composition of Functions

Consider two functions: $$ f(x)=\frac{x^2-1}{x-1} $$ for $$x\ne1$$, and the piecewise function $$ g(

Continuity and Limit Comparison for Two Particle Paths

Two particles, A and B, travel along the same line. Their position functions are given by $$s_A(t)=

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 \

Direct Substitution in a Polynomial Function

Consider the polynomial function $$f(x)=2*x^2 - 3*x + 5$$. Answer the following parts concerning lim

Error Analysis in Limit Calculation

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

Estimating Derivatives Using Limit Definitions from Data

The position of an object (in meters) is recorded at various times (in seconds) in the table below.

Evaluating a Limit with Radical Expressions

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

Factoring a Cubic Expression for Limit Evaluation

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

Graphical Estimation of a Limit

The following graph shows the function $$f(x)$$. Use the graph to answer the subsequent questions re

Identifying Discontinuities in a Rational Function

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

Intermediate Value Theorem Application

Consider the polynomial function $$f(x)=x^3-6*x^2+9*x+1$$ on the closed interval [0, 4].

Investigating Discontinuities in a Rational Function

Consider the function $$ h(x)=\frac{x^2-4}{x-2} $$ for $$x\ne2$$.

Jump Discontinuity in a Piecewise Function

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

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}

Limits at Infinity and Horizontal Asymptotes

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

Limits Involving Absolute Value Expressions

Evaluate the limit $$\lim_{x \to 0} \frac{|x|}{x}$$.

Limits Involving Radicals and Algebra

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

Logarithmic Limit Evaluation

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

Modeling Bacterial Growth with a Geometric Sequence

A particular bacterial colony doubles in size every hour. The population at time $$n$$ hours is give

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 Vertical Asymptotes

Consider the function $$ f(x)= \frac{1}{x-4} $$.

Oscillatory Behavior and Non-Existence of Limit

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

Oscillatory Behavior in Damped Trigonometric Functions

Consider the function $$f(x) = x \cos(1/x)$$ for x ≠ 0 and define f(0) = 0. Answer the following:

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

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

Removable Discontinuity in a Rational Function

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

Return on Investment and Asymptotic Behavior

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

Squeeze Theorem 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.

Analyzing a Function's Derivative from its Graph

A graph of a smooth function is provided. Answer the following questions:

Analyzing a Projectile's Motion

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

Derivative Applications in Motion Along a Curve

A particle moves such that its horizontal position is given by $$x(t)= t^2 + 2*t$$ and its vertical

Derivatives in Economics: Cost Functions

A company's production cost is modeled by $$C(q)=500+20*q-0.5*q^2$$, where $$q$$ represents the quan

Differentiating an Absolute Value Function

Consider the function $$f(x)= |3*x - 6|$$.

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

Event Ticket Sales Dynamics

For a popular concert, tickets are sold at a rate of $$f(t)=100-3*t$$ (tickets/hour) while cancellat

Instantaneous Rate and Maximum Acceleration

An object’s position is given by $$s(t)=t^4-4t^3+2t^2$$ (in meters), where t is in seconds. Answer t

Instantaneous Rate of Change from a Graph

A graph of a smooth curve (representing a function $$f(x)$$) is provided with a tangent line drawn a

Inverse Function Analysis: Cubic with Linear Term

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

Inverse Function Analysis: Hyperbolic-Type Function

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

Motion Analysis with Acceleration and Direction Change

A particle moves along a straight line with acceleration given by $$a(t)=12-4*t$$, where $$t$$ is in

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

Polynomial Rate of Change Analysis

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

Population Growth and Instantaneous Rate of Change

A town's population is modeled by $$P(t)= 2000*e^{0.05*t}$$, where $$t$$ is in years. Analyze the ch

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

Rates of Change in Chemical Concentration

In a chemical reaction, the concentration $$C(t)$$ of a substance in a tank is modeled by $$C(t)=\fr

River Pollution Dynamics

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

Slope of a Tangent Line from a Table of Values

Given the table below for a differentiable function $$f(x)$$: | x | f(x) | |-----|------| | 1 |

Tangent Line and Differentiability

Let $$h(x)=\frac{1}{x+2}$$, modeling the concentration of a substance in a chemical solution over ti

Temperature Change Analysis

A temperature sensor records the temperature, $$T(t)=3*t^2 - 2*t + 5$$ (in $$^{\circ}C$$) at time $$

Using Derivative Rules on a Trigonometric Function

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

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 Differentiation with Logarithms

A function is given by $$h(x)=\ln((5*x+1)^2)$$. Use the chain rule to differentiate $$h(x)$$.

Composite Function via Chain Rule in a Financial Context

A company’s profit (in dollars) based on production level (in thousands of units) is modeled by the

Composite Function with Nested Chain Rule

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

Composite Inverse Trigonometric Function Evaluation

Let $$f(x)= \tan(2*x)$$, defined on a restricted domain where it is invertible. Analyze this functio

Concavity Analysis of an Implicit Curve

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

Derivative of an Inverse Function

Let $$f$$ be a differentiable function with an inverse function $$g$$ such that $$f(2)=5$$ and $$f'(

Differentiating an Inverse Trigonometric Function

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

Differentiation of a Complex Implicit Equation

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

Differentiation of Inverse Function with Polynomial Functions

Let \(f(x)= x^3+2*x+1\) be a one-to-one function. Its inverse is denoted by \(f^{-1}\).

Differentiation with Rational Exponents

Let $$y=(3*x+2)^{4/3}$$. Answer the following:

Graph Analysis of a Composite Motion Function

A displacement function representing the motion of an object is given by $$s(t)= \ln(2*t+3)$$. The g

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 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 $$4*x^2+9*y^2=36$$.

Implicit Differentiation in an Elliptical Orbit

Consider the ellipse defined by $$4*x^2 + 9*y^2 = 36$$, which can model the orbit of a satellite.

Implicit Differentiation on a Circle

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

Implicit Differentiation with Exponential-Trigonometric Functions

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

Implicit Differentiation with Mixed Functions

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

Implicit Differentiation with Trigonometric Terms

Consider the implicit equation $$\sin(x*y)+x^2=y^2$$. Find the derivative $$\frac{dy}{dx}$$.

Implicit Trigonometric Equation Analysis

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

Implicitly Defined Inverse Relation

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

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 Derivative of a Sum of Exponentials and Linear Terms

Let $$f(x)= e^(x)+ x$$ and let g be its inverse function satisfying $$g(f(x))= x$$. Answer the follo

Inverse Function Derivative and Recovery

Let $$f(x)=x^3+x$$, which is one-to-one on a suitable interval. Answer the following parts.

Inverse Function Differentiation in Temperature Conversion

Consider the function $$f(x)= \frac{1}{1+e^{-0.5*x}}$$, which converts a temperature reading in Cels

Inverse Trigonometric Differentiation

Let $$L(x)=\arcsin\left(\frac{2*x}{1+x^2}\right)$$.

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,

Analysis of Wheel Rotation

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

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

Cost Efficiency in Production

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

Economic Efficiency in Speed

A vehicle’s fuel consumption per mile (in dollars) is modeled by the function $$C(v)=0.05*v^2 - 3*v

Economics: Marginal Revenue Analysis

A firm’s revenue function is given by $$R(x)=\frac{100x}{x+5}$$ (in dollars), where $$x$$ represents

Error Approximation in Engineering using Differentials

The cross-sectional area of a circular pipe is given by $$A=\pi r^2$$. If the radius is measured as

Expanding Pool Dimensions

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

FRQ 15: Evaluating Limits with L’Hôpital’s Rule

Evaluate the limit $$\lim_{x\to\infty} \frac{5*x^3 - 4*x^2 + 1}{7*x^3 + 2*x - 6}$$ by using L’Hôpita

Growth Rate Estimation in a Biological Experiment

In a biological experiment, the mass $$M(t)$$ (in grams) of a bacteria colony is recorded over time

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.

Linearization for Approximating Square Roots

Let $$f(x)= \sqrt{x}$$. Use linearization to estimate the value of $$\sqrt{16.4}$$, using $$x=16$$ a

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

Minimizing Materials for a Cylindrical Can

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

Optimizing Crop Yield

The yield per acre of a crop is modeled by the function $$Y(p) = 100\,p\,e^{-0.1p}$$, where $$p$$ is

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

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

Radioactive Decay: Rate of Change and Half-life

A radioactive substance decays according to the formula $$N(t)=N_0e^{-kt}$$, where $$N(t)$$ is the a

Rate of Change of Temperature

The temperature of a cooling liquid is modeled by $$T(t)= 100*e^{-0.5*t} + 20$$, where $$t$$ is in m

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

Revenue Sensitivity to Advertising

A firm's revenue (in thousands of dollars) is given by $$R(t)=50\sqrt{t+1}$$, where $$t$$ represents

Seasonal Water Reservoir

A reservoir's water volume (in million m³) changes with the seasons according to $$V(t)=5+2\sin\left

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

Water Flow Rate in a Tank

Water flows into a tank at a rate given by $$r(t)=\frac{2t+1}{t+4}$$ liters per minute, where $$t$$

Absolute Extrema for a Transcendental Function

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

Analysis of a Parametric Curve

Consider the parametric equations $$x(t)= t^2 - 3*t$$ and $$y(t)= 2*t^3 - 9*t^2 + 12*t$$. Analyze th

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 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} $

Application of Rolle's Theorem for a Quadratic Function

Let $$f(x)= x^2 - 4$$ be defined on the interval $$[-2,2]$$. In this problem, you will verify the co

Application of Rolle's Theorem to a Trigonometric Function

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

Concavity Analysis of a Cubic Function

Consider the function $$f(x)= x^3 - 6*x^2 + 9*x + 2$$. Use the second derivative to investigate the

Concavity and Inflection Points of a Cubic Function

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

Cooling of a Cup of Coffee

A cup of coffee cools according to the model $$T(t)= T_{room}+(T_{initial}-T_{room})e^{-kt}$$ with $

Cost Function and the Mean Value Theorem in Economics

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

Cost Minimization in Transportation

A transportation company recorded shipping costs (in thousands of dollars) for different numbers of

Extrema in a Cost Function

A company's cost function is given by $$C(x) = x^3 - 6*x^2 + 12*x + 7$$, where $$x$$ represents the

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 10: First Derivative Test for a Cubic Profit Function

A company’s profit function is given by $$P(x)= x^3 - 9*x^2 + 24*x + 1$$, where $$x$$ represents the

FRQ 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

Garden Fence Optimization Problem

A rectangular garden is to be built adjacent to a building. Fencing is required on only three sides

Increase and Decrease Analysis of a Polynomial Function

Consider the function $$f(x)= x^4 - 4*x^3 + 6*x^2 - 4*x + 1$$ on the interval $$[-1, 3]$$.

Inflection Points and Concavity in a Real-World Cost Function

A company's cost function is given by $$C(x) = 0.5*x^3 - 6*x^2 + 20*x + 100$$, where \( x \) represe

Inverse Analysis of a Linear Function

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

Minimizing Average Cost in Production

A company’s cost function is given by $$C(x)= 0.5*x^3 - 6*x^2 + 20*x + 100$$, where $$x$$ represents

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

Optimization of a Rectangular Enclosure

A rectangular pen is to be constructed along the side of a barn so that only three sides require fen

Optimizing an Open-Top Box from a Metal Sheet

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

Pharmacokinetics: Drug Concentration Decay

A drug in the bloodstream decays according to $$D(t)= D_0*e^{-k*t}$$, where $$t$$ is in hours. Answe

Piecewise Function and the Mean Value Theorem

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

Rational Function Optimization

Consider the rational function $$f(x)= \frac{x^2 + 1}{x - 1}$$ defined on the interval $$[2,6]$$. An

Reservoir Sediment Accumulation

A reservoir experiences sediment deposition from rivers and sediment removal via dredging. The sedim

Slope Analysis for Parametric Equations

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

Solving an Exponential Equation

Solve for $$x$$ in the equation $$e^{2x}= 5*e^{x}$$. Answer the following:

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 $$x^2 + x*y + y^2 = 7$$.

Temperature Regulation in a Greenhouse

A greenhouse is regulated by an inflow of warm air and an outflow of cooler air. The inflow temperat

Traffic Intersection Flow Analysis

At a busy urban intersection, traffic flow is modeled by an inflow rate $$I(t)=30+5*t$$ and an outfl

Water Cooling Tower Efficiency

In a water cooling tower, water is pumped in at a rate $$R_{in}(t)=10+0.5*t^2$$ L/min and discharged

Accumulated Change Function Evaluation

Let $$F(x)=\int_{1}^{x} (2*t+3)\,dt$$ for $$x \ge 1$$. This function represents the accumulated chan

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:

Application of the Fundamental Theorem in a Discounted Cash Flow Model

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

Area Under a Curve Using Riemann Sums

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

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

Composite Functions and Accumulation

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

Cooling of a Liquid Mixture

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

Definite Integral Approximation Using Riemann Sums

Consider the function $$f(x)= x^2 + 3$$ defined on the interval $$[2,6]$$. A table of sample values

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

Environmental Modeling: Pollution Accumulation

The pollutant enters a lake at a rate given by $$P(t)=5*e^{-0.3*t}$$ (in kg per day) for $$t$$ in da

Estimating Distance Traveled Using the Trapezoidal Rule from Speed Data

During a car journey, the speed (in km/hr) is recorded at regular intervals. The table below shows s

Evaluating a Trigonometric Integral Using U-Substitution

Evaluate the integral $$\int_{0}^{\frac{\pi}{2}} \sin(2*x)\,dx$$ using u-substitution.

FRQ7: Inverse Analysis of an Exponential Accumulation Function

Define the function $$ Q(x)=\int_{1}^{x} \left(\ln(t)+\frac{1}{t}\right)\,dt $$ for x > 1. Answer th

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

FRQ13: Inverse Analysis of an Investment Growth Function

An investment's accumulated value is given by $$ G(t)=\int_{0}^{t} \frac{1}{1+u}\,du $$ for t ≥ 0. A

FRQ17: Inverse Analysis of a Biologically Modeled Accumulation Function

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

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

Integration Using U-Substitution

Consider the function $$g(x)= (x-3)^4$$ defined on the interval $$[3,7]$$.

Medication Concentration and Absorption Rate

A patient's blood concentration of a drug (in mg/L) is monitored over time before reaching its peak.

Midpoint Riemann Sum for Temperature Data

A weather station records temperature (in degrees Celsius) at hourly intervals. The data for a 4-hou

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 $$0\le t\le8$$,

Net Change vs Total Accumulation in a Velocity Function

A particle moves with velocity $$v(t)=5-t^2$$ (in m/s) for t in [0,4]. Answer the following:

Net Surplus Calculation

A consumer's satisfaction is given by $$S(x)=100-4*x^2$$ and the marginal cost is given by $$C(x)=30

Oxygen Levels in a Bioreactor

In a bioreactor, oxygen is introduced at a rate $$O_{in}(t)= 7 - 0.5t$$ mg/min and is consumed at a

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 $$a(t)=6-4*t$$ (in m/s²) for $$

Rainfall Accumulation via Integration

A region experiences rain where the rate of rainfall (in inches per hour) is given by $$r(t)=0.5+0.2

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

Temperature Cooling: An Initial Value Problem

An object cools according to the differential equation $$T'(t)=-0.2\,(T(t)-20)$$, where $$T(t)$$ is

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

Trapezoidal Rule Application with Population Growth

A biologist recorded the instantaneous growth rate (in thousands per year) of a species over several

Vehicle Distance Estimation from Velocity Data

A car's velocity (in m/s) is recorded at several time points during a trip. Use the table below for

Volume of a Solid: Exponential Rotation

Consider the region bounded by the curve $$y=e^{-x}$$, the x-axis, and the vertical lines $$x=0$$ an

Water Accumulation in a Tank

Water flows into a tank at a rate given by $$R(t)=2*\sqrt{t}$$ (in m³/min) for t in minutes. Answer

A Separable Differential Equation: Growth Model

Consider the differential equation $$\frac{dy}{dx}=3*x*y^2$$ that models a growth process. Use separ

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

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

Chemical Reaction Rate with Second-Order Decay

A chemical reaction follows the rate law $$\frac{d[A]}{dt}=-k[A]^2$$, where $$[A](t)$$ (in M) is the

Chemical Reactor Mixing

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

Cooling with a Time-Dependent Coefficient

A substance cools according to $$\frac{dT}{dt} = -k(t)(T-25)$$ where the cooling coefficient is give

Cooling with Variable Ambient Temperature

An object cools in an environment where the ambient temperature varies with time. Its temperature $$

Drug Concentration Model

The concentration $$C(t)$$ (in mg/L) of a drug in a patient's bloodstream is modeled by the differen

Economic Growth with Investment Outflow

A company’s investment fund grows continuously at an annual rate of $$5\%$$, but expenses lead to a

Ecosystem Nutrient Cycle

In a forest ecosystem, nitrogen is deposited from the atmosphere at a rate of $$2$$ kg/ha/year while

Environmental Pollution Model

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

Epidemic Spread (Simplified Logistic Model)

In a simplified model of an epidemic, the number of infected individuals $$I(t)$$ (in thousands) is

Epidemic Spread with Limited Capacity

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

Heating and Cooling in an Electrical Component

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

Investment Growth with Withdrawals

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

Linear Differential Equation using Integrating Factor

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

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 in a Tank

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

Mixing Problem with Constant Flow

A tank initially contains 200 liters of water with 10 kg of dissolved salt. Brine with a salt concen

Mixing with Variable Inflow Rate

A 50-liter tank initially contains water with 1 kg of dissolved salt. Water containing 0.2 kg of sal

Modeling Orbital Decay

A satellite’s altitude $$h(t)$$ decreases over time due to atmospheric drag, following $$\frac{dh}{d

Newton's Law of Cooling

An object is heated to $$100^\circ C$$ and left in a room at $$20^\circ C$$. According to Newton's l

Newton's Law of Cooling

A hot object is placed in a room with constant temperature $$20^\circ C$$. Its temperature $$T$$ sat

Nonlinear Differential Equation

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

RC Circuit Charging

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

Reaction Kinetics in a Tank

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

Reversible Chemical Reaction

In a reversible chemical reaction, the concentration $$C(t)$$ of a product is governed by the differ

Separable Differential Equation involving $$y^{1/3}$$

Consider the differential equation $$\frac{dy}{dx} = y^{1/3}$$ with the initial condition $$y(8)=27$

Separable Differential Equation with Initial Condition

Consider the differential equation $$\frac{dy}{dx}= \frac{x^2}{2*y}$$ with the initial condition $$y

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

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

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

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

Slope Field Sketching for $$\sin(x)$$ Model

Given the slope field for the differential equation $$\frac{dy}{dx} = \sin(x)$$, sketch a solution c

Substitution to Linearize

The differential equation $$\frac{dy}{dx} = \frac{x + y}{1 - x*y}$$ appears non-linear. With the sub

Tank Draining Differential Equation

Water drains from a tank at a rate that depends on the square root of the volume, according to $$\fr

Water Level in a Reservoir

A reservoir's water volume $$V$$ (in million m³) is measured at various times $$t$$ (in days) as sho

Area Between an Exponential Function and a Linear Function

Consider the functions $$f(x)=e^{x}$$ and $$g(x)=x+1$$ on the interval $$[0,1]$$.

Area Between Two Curves from Tabulated Data

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

Average Concentration in Medical Dosage

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

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

Average of a Logarithmic Function

Let $$f(x)=\ln(x+2)$$ represent a measured quantity over the interval $$[0,6]$$.

Calculation of Consumer Surplus

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

Cost Analysis with Discontinuous Pricing

A utility company’s billing is modeled by the function $$C(q)=\begin{cases} 3*q & \text{if } 0\le q\

Cost Optimization for a Cylindrical Container

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

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

Finding the Area Between Two Curves

Let the functions $$f(x)=5*x-x^2$$ and $$g(x)=x$$ be given. Find the area of the region bounded by 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

Hollow Rotated Solid

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

Implicit Differentiation in Thermodynamics

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

Interpreting Integrated Quantities in a Changing System

A system is modeled by a rate function given by $$R(t)=t^2-4*t+6$$, where $$t$$ is in minutes. The c

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 $$

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 $$

Position, Velocity, and Acceleration Analysis of a Moving Car

A car moving along a straight line experiences an acceleration given by $$a(t)= 2*t - 4$$ (in m/s²).

Shaded Area between $$y=\sqrt{x}$$ and $$y=\frac{x}{2}$$

Consider the curves $$f(x)=\sqrt{x}$$ and $$g(x)=\frac{x}{2}$$. Use integration to determine the are

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

Volume of a Solid of Revolution using Shells

Consider the region under the curve $$f(x)=e^{-x}$$ for $$x \in [0,1]$$. This region is revolved abo

Volume of a Solid Using the Disc Method

Consider the region in the xy-plane bounded by $$y = \sqrt{x}$$ and $$y=0$$ for $$0 \le x \le 9$$. T

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 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 with Square Cross-Sections

Consider the region bounded by the curve $$y=x^2$$ and the line $$y=4$$ for $$0 \le x \le 2$$. Squar

Work Done by a Variable Force

A variable force acting along a straight line is described by $$F(x)=3*x^2$$ Newtons, where $$x$$ is

Work to Pump Water from a Cylindrical Tank

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