AP Calculus AB FRQ Room

Analyzing a Velocity Function with Nested Discontinuities

A particle’s velocity along a line is given by $$v(t)= \frac{(t-1)(t+3)}{(t-1)*\ln(t+2)}$$ for $$t>0

Analyzing Process Data for Continuity

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

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

Applying the Squeeze Theorem with Trigonometric Function

Consider the function $$ f(x)= x^2 \sin(1/x) $$ for $$x\ne0$$, with $$f(0)=0$$. Use the Squeeze Theo

Arithmetic Sequence in Temperature Data and Continuity Correction

A temperature sensor records the temperature every minute and the readings follow an arithmetic sequ

Continuity of a Composite Function

Let $$g(x) = \sqrt{x+3}$$ and $$h(x) = x^2 - 4$$. Define the composite function $$f(x) = g(h(x))$$.

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},

Ensuring Continuity for a Piecewise-Defined Function

Consider the piecewise function $$p(x)= \begin{cases} ax + 3 & \text{if } x < 2, \\ x^2 + b & \text{

Evaluating a Compound Limit Involving Rational and Trigonometric Functions

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

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 Analysis: Identify Limits and Discontinuities

A graph of a function f(x) is provided in the stimulus. The graph shows a removable discontinuity at

Graph Reading: Left and Right Limits

A graph of a function f is provided below which shows a discontinuity at x = 2. Use the graph to det

Graph-Based Analysis of Discontinuity

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

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

Implicit Differentiation and Tangent Slopes

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

Intermediate Value Theorem in Particle Motion

Consider a particle with position function $$s(t)= t^3 - 7*t+3$$. According to the Intermediate Valu

Limit and Integration in Non-Polynomial Particle Motion

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

Limit Evaluation using Conjugate Multiplication

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

Limits and the Squeeze Theorem Application

Consider two scenarios: (1) A function f(x) satisfying $$ -|x| \le f(x) \le |x| $$ for all x near 0,

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 Radical Functions

Evaluate the limit $$\lim_{x \to 4} \frac{\sqrt{x} - 2}{x - 4}$$.

Limits Involving Radicals and Algebra

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

Limits Near Vertical Asymptotes

Consider the function $$f(x) = \frac{1}{x - 4}$$. (a) Determine $$\lim_{x \to 4^-} f(x)$$. (b) Dete

Oscillatory Behavior and Discontinuity

Consider the function $$f(x)=\begin{cases} x\cos(\frac{1}{x}) & x\neq0 \\ 2 & x=0 \end{cases}$$. Ans

Rational Function Limits and Removable Discontinuities

Consider the function $$f(x)=\frac{(x+3)(x-2)}{(x+3)(x+5)}$$. Answer the following:

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 Limit Evaluation

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

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 with Trigonometric Function

Consider the function \(h(x)=x^2\cos(1/x)\) for \(x\neq0\) with \(h(0)=0\). Answer the following:

Table Analysis for Estimating a Limit

The table below shows values of the function $$g(x)$$ for x near 4. Use this data to answer the foll

Trigonometric Limit Evaluation

Examine the function $$ f(x)= \frac{\sin(3*x)}{x} $$ for $$x\ne0$$.

Analyzing Differentiability of an Absolute Value Function

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

Approximating Derivatives Using Secant Lines

For the function $$f(x)=\ln(x)$$, we want to approximate the derivative at $$x=3$$ using secant line

Car's Position and Velocity

A car’s position is modeled by \(s(t)=t^3 - 6*t^2 + 9*t\), where \(s\) is in meters and \(t\) is in

Concavity and the Second Derivative

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

Derivative Applications in Population Growth

A population of cells is modeled by $$P(t)=100*e^{0.2*t}$$, where $$t$$ is in hours. Answer the foll

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.

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 an Absolute Value Function

Consider the function $$f(x)=|x-3|$$, representing the error margin (in centimeters) in a calibratio

Economic Marginal Revenue

A company's revenue function is given by \(R(x)=x*(50-0.5*x)\) dollars, where \(x\) represents the n

Event Ticket Sales Dynamics

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

Finding Derivatives of Composite Functions

Let $$f(x)= (3*x+1)^4$$.

Implicit Differentiation of a Circle

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

Instantaneous Rate of Change of Temperature

The temperature in a room is modeled by $$T(t)=20+5*\sin\left(\frac{\pi*t}{12}\right)$$, where $$t$$

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: Cosine and Linear Combination

Consider the function $$f(x)=\cos(x)+x$$ defined on the interval $$[0,\frac{\pi}{2}]$$.

Inverse Function Analysis: Sum with Reciprocal

Consider the function $$f(x)=x+\frac{1}{x}$$ defined for $$x\geq1$$.

Kinematics and Position Function Analysis

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

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 $$s(t)$$

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

Position Function from a Logarithmic Model

A particle’s position in meters is modeled by $$s(t)= \ln(t+1)$$ for $$t \geq 0$$ seconds.

Proof of Scaling in Derivatives

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

Quotient Rule Application

Let $$f(x)= \frac{e^{x}}{x+1}$$, a function defined for $$x \neq -1$$, which involves both an expone

Real-World Cooling Process

In an experiment, the temperature (in °C) of a substance as it cools is modeled by $$T(t)= 30*e^{-0.

Related Rates: Balloon Surface Area Change

A spherical balloon has volume $$V=\frac{4}{3}\pi r^3$$ and surface area $$S=4\pi r^2$$. If the volu

Riemann Sums and Derivative Estimation

A car’s position $$s(t)$$ in meters is recorded in the table below at various times $$t$$ in seconds

Tangent Lines and Local Linearization

Consider the function $$f(x)=\sqrt{x}$$.

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

Advanced Implicit and Inverse Function Differentiation on Polar Curves

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

Analyzing a Function and Its Inverse

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

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 Temperature Variation

A metal rod's temperature along its length is given by the function $$T(x)= \cos((4*x+2)^2)$$, 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 and Radical Functions

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

Composite Differentiation of an Inverse Trigonometric Function

Let $$H(x)= \arctan(\sqrt{x+3})$$.

Composite Function and Tangent Line

Consider the function $$f(x)=\sqrt{3*x^2+2*x+1}$$. (Note: All derivatives are to be computed without

Composite Function Chain Reaction

A chemist models the concentration of a reacting solution at time $$t$$ (in seconds) with the compos

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: Engineering Stress-Strain Model

In an engineering context, the stress σ as a function of strain ε is given by $$\sigma(\epsilon) = \

Composite, Implicit, and Inverse Combined Challenge

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

Differentiation of Nested Exponential Functions

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

Implicit Curve Analysis: Horizontal Tangents

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

Implicit Differentiation in a Circle

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

Implicit Differentiation of a Logarithmic-Exponential Equation

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

Implicit Differentiation of a Trigonometric Composite Function

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

Implicit Differentiation on a Circle

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

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 Product Rule

Consider the implicit equation $$x*y+y^2=6$$ which defines $$y$$ as a function of $$x$$. Use implici

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

Let $$f(x)=x^3+x+1$$, a one-to-one function, and let $$g$$ be the inverse of $$f$$. Use inverse func

Inverse Function Differentiation in an Exponential Context

Let $$f(x)= e^(3*x) - 2$$ and let $$g(x)$$ be the inverse function of f. Answer the following:

Inverse Trigonometric Function Differentiation

Consider the function $$y=\arctan(2*x)$$. Answer the following:

Manufacturing Optimization via Implicit Differentiation

A manufacturing cost relationship is given implicitly by $$x^2*y + x*y^2 = 1000$$, where $$x$$ repre

Multilayer Composite Function Differentiation

Let $$y=\cos(\sqrt{5*x+3})$$. Answer the following:

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

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 Piecewise Function with Discontinuities

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

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

Complex Piecewise Function Analysis

Consider the function $$f(x)=\begin{cases}\frac{\sin(x)}{x} & x<\pi \\ 2 & x=\pi \\ 1+\cos(x-\pi) &

Cooling Coffee: Exponential Decay Model

A cup of coffee cools according to $$T(t) = 70 + 50e^{-0.1t}$$, where $$T(t)$$ (in °F) is the temper

Economic Cost Function Linearization

A company's production cost is modeled by $$C(x)= 0.02*x^3 - 1.5*x^2 + 40*x + 200$$ dollars, where $

Elasticity of Demand Analysis

A product’s demand function is given by $$Q(p) = 150 - 10p + p^2$$, where $$p$$ is the price, and $$

Error Estimation in Pendulum Period

The period of a simple pendulum is given by $$T=2\pi\sqrt{\frac{L}{g}}$$, where $$L$$ is the length

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

Expanding Balloon: Related Rates with a Sphere

A spherical balloon is being inflated so that its volume increases at a constant rate of $$dV/dt = 1

FRQ 1: Vessel Cross‐Section Analysis

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

FRQ 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 4: Revenue and Cost Implicit Relationship

A company’s revenue (R) and cost (C) are related by the equation $$R^2 + 3*R*C + C^2 = 1000$$. Treat

FRQ 12: Airplane Climbing Dynamics

An airplane’s altitude is modeled by the equation $$y = 0.1*x^2$$, where x (in km) is the horizontal

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

Linearization and Differential Approximations

Let $$f(x)=x^4$$. Use linearization to approximate $$f(3.98)$$ near $$x=4$$.

Linearization and Differentials

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

Marginal Analysis in Economics

A company’s cost function is given by $$C(x)=0.5*x^3 - 3*x^2 + 5*x + 8$$, where $$x$$ represents the

Open-top Box Optimization

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

Optimization in Packaging

An open-top box with a square base is to be constructed so that its volume is fixed at $$1000\;cm^3$

Particle Motion Analysis

A particle moves along a straight line with an acceleration given by $$a(t) = 6 - 4*t$$, where $$t$$

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

Projectile Motion Analysis

The height of a projectile is modeled by the function $$h(t) = -4.9t^2 + 20t + 2$$, where $$t$$ is i

Projectile Motion with Velocity Components

A projectile is launched from the ground with a constant horizontal velocity of 15 m/s and a vertica

Related Rates in a Conical Tank

Water is being poured into a conical tank at a rate of $$\frac{dV}{dt}=10$$ cubic meters per minute.

Shadow Length Problem

A person 1.80 m tall walks away from a 3.0 m tall lamppost at a rate of 1.2 m/s. Let $$x$$ be the di

Temperature Change Analysis

The temperature of a chemical solution is recorded over time. Use the table below, where $$T(t)$$ (i

Vehicle Deceleration Analysis

A car's position function is given by $$s(t)= 3*t^3 - 12*t^2 + 5*t + 7$$, where $$s(t)$$ is measured

Vehicle Position and Acceleration

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

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

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 the Extreme Value Theorem

Consider the function $$f(x) = \sqrt{x} + (8-x)$$ defined on the interval $$[0, 8]$$. Answer the fol

Area Bounded by $$\sin(x)$$ and $$\cos(x)$$

Consider the functions $$f(x)= \sin(x)$$ and $$g(x)= \cos(x)$$ on the interval $$[0, \frac{\pi}{2}]$

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

Composite Function with Piecewise Exponential and Logarithmic Parts

Consider the function $$ f(x) = \begin{cases} e^{x}-1, & x < 2, \\ \ln(x+1), & x \ge 2. \end{cases}

Concavity Analysis of a Trigonometric Function

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

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 Enclosure along a River

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

Discontinuity and Derivative in a Hybrid Piecewise Function

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

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 8: Mean Value Theorem and Non-Differentiability

Examine the function $$f(x)=|x|$$ on the interval [ -1, 1 ].

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 12: Optimization in Manufacturing: Minimizing Cost

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

FRQ 20: Profit Analysis Combining MVT and Optimization

A company’s profit function is given by $$P(x)= -2*x^3 + 18*x^2 - 48*x + 40$$, where $$x$$ (in thous

Increasing/Decreasing Behavior in a Financial Model

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

Instantaneous Velocity Analysis via the Mean Value Theorem

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

Inverse Analysis of a Piecewise Function

Consider the piecewise function $$f(x)=\begin{cases}2*x+1 & x\le 0,\\ 3*x+1 & x>0\end{cases}$$. Ans

Jump Discontinuity in a Piecewise Linear Function

Consider the piecewise function $$ f(x) = \begin{cases} 2x + 1, & x < 3, \\ 2x - 4, & x \ge 3. \end

Mean Value Theorem for a Cubic Function

Consider the function $$f(x)= x^3 - 2*x^2 + x$$ on the closed interval $$[0,2]$$. In this problem, y

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

Monotonicity and Inverse Function Analysis

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

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 in Production with Exponential Cost Function

A manufacturer’s cost function is modeled by $$C(x)= 200 + 50*x + 100*e^{-0.1*x}$$ where $$x$$ repre

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

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 $$t$$

Rational Function Optimization

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

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 Sediment Accumulation

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

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 Chemical Concentration

A scientist observes that the rate of change of chemical concentration in a solution is given by $$r

Accumulated Water Volume in a Tank

A water tank is being filled at a rate given by $$R(t) = 4*t$$ (in cubic meters per minute) for $$0

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:

Analyzing Work Done by a Variable Force

An object is acted on by a force given by $$F(x)= 3*x^2 - x + 2$$ (in newtons), where $$x$$ is the d

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 Under a Parabola

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

Area Under a Piecewise Function

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

Bacterial Growth Modeling with Antibiotic Administration

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

Chemical Accumulation in a Reactor

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

Comparing Riemann Sum and the Fundamental Theorem

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

Convergence of Riemann Sum Estimations

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

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

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

Elevation Profile Analysis on a Hike

A hiker records the elevation (in meters) along a trail at various distances. Use this data to analy

Evaluating a Definite Integral Using U-Substitution

Compute the integral $$\int_{0}^{3} (2*t+1)^5\,dt$$ using u-substitution.

Exploring the Fundamental Theorem of Calculus

Let the function $$F(x) = \int_{1}^{x} \frac{1}{t^2+1}\,dt$$ represent an accumulation function. Ans

Finding the Area of a Parabolic Arch

An architect designs an arch described by the parabola $$y = 10 - \frac{x^{2}}{5}$$. The arch spans

FRQ5: Inverse Analysis of a Non‐Elementary Integral Function

Consider the function $$ P(x)=\int_{0}^{x} e^{t^2}\,dt $$ for x ≥ 0. Answer the following parts.

FRQ6: Inverse Analysis of a Displacement Function

Let $$ S(t)=\int_{0}^{t} (6-2*u)\,du $$ for t in [0, 3], representing displacement in meters. Answer

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 \

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.

FRQ20: Inverse Analysis of a Function with a Piecewise Continuous Integrand

Consider the function $$ I(x)= \begin{cases} \int_{0}^{x}\cos(t)\,dt, & 0 \le x \le \pi/2 \\ \int_{0

Function Transformations and Their Integrals

Let $$f(x)= 2*x + 3$$ and consider the transformed function defined as $$g(x)= f(2*x - 1)$$. Analyze

Medication Concentration and Absorption Rate

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

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

Motion Analysis from Velocity Data

A particle moves along a straight line with the following velocity data (in m/s) recorded at specifi

Net Change in Population Growth

A town's population grows at a rate given by $$P'(t)=0.1*t+50$$ (in individuals per year) where $$t$

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

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). T

Rainfall and Evaporation in a Greenhouse

In a greenhouse, rainfall is modeled by $$R(t)= 8\cos(t)+10$$ mm/hr, while evaporation occurs at a c

Related Rates: Expanding Balloon

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

Riemann Sum Approximation of f(x) = 4 - x^2

Consider the function $$f(x)=4-x^2$$ on the interval $$[0,2]$$. Use Riemann sums to approximate the

Seismic Data Analysis: Ground Acceleration

A seismograph records ground acceleration (in m/s²) during an earthquake. Use the data in the table

Ski Lift Passengers: Boarding and Alighting Rates

On a ski lift, passengers board at a rate $$B(t)= 3e^{-t/2}$$ persons/min and alight at a constant r

Tabular Riemann Sums for Electricity Consumption

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

Trigonometric Integral with U-Substitution

Evaluate the definite integral $$\int_{0}^{\frac{\pi}{4}} \sec^2(t)\tan(t)\,dt$$.

Trigonometric Integration via U-Substitution

Evaluate the integral $$I=\int_{0}^{\frac{\pi}{4}} \tan(x)*\sec^2(x)\,dx.$$ Answer the following par

U-Substitution in a Trigonometric Integral

Evaluate the integral $$\int \sin(2*x) * \cos(2*x)\,dx$$ using u-substitution.

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

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

Bernoulli Differential Equation via Substitution

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

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

Carbon Dating and Radioactive Decay

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

Chemical Reaction Rate

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

Cooling with a Time-Dependent Coefficient

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

Differential Equation in Business Profit

A company's profit $$P(t)$$ changes over time according to $$\frac{dP}{dt} = 100\,e^{-0.5t} - 3P$$.

Falling Object with Air Resistance

A falling object experiences air resistance proportional to the square of its velocity. Its velocity

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

Investment Growth with Continuous Contributions

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

Investment Growth with Continuous Deposits

An investment account accrues interest continuously at an annual rate of 0.05 and receives continuou

Linear Differential Equation using Integrating Factor

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

Logistic Growth Model

A population is modeled by the logistic differential equation $$\frac{dP}{dt}=0.5*P\left(1-\frac{P}{

Mixing Problem in a Salt Solution Tank

A 100-liter tank initially contains a solution with 10 kg of salt. Brine with a salt concentration o

Mixing Problem in a Tank

A tank initially contains 200 L of water with 10 kg of dissolved salt. Brine with a salt concentrati

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

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

Population Dynamics with Harvesting

A wildlife population (in thousands) is monitored over time and is subject to harvesting. The popula

Radioactive Decay

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

Radioactive Decay

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

Radioactive Decay and Half-Life

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

Sand Pile Dynamics

Sand is added to a pile at a constant rate of $$15$$ kg/min while some sand is simultaneously lost d

Seasonal Temperature Variation

The temperature $$T(t)$$ in a region is modeled by the differential equation $$\frac{dT}{dt} = -0.2\

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

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

Slope Field Analysis and Asymptotics

Consider the differential equation $$\frac{dy}{dx}=\frac{x}{1+y^2}$$. Solve the equation and analyze

Slope Field Exploration

Consider the differential equation $$\frac{dy}{dx} = \sin(x)$$. The provided slope field (see stimul

Vehicle Deceleration

A vehicle undergoing braking has its speed $$v$$ (in m/s) recorded over time (in seconds) as shown.

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 Parabola and a Line

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

Area Between a Parabolic Curve and a Line

Consider the functions $$f(x)=5*x-x^2$$ and $$g(x)=x$$ on the interval $$[0,4]$$. The table below sh

Area Between Cost Functions in a Business Analysis

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

Area Between Curves: Revenue and Cost Analysis

A company’s revenue and cost are modeled by the functions $$f(x)=10-x^2$$ and $$g(x)=2*x$$, where $$

Area Between Parabolic Curves

Consider the curves $$f(x)=x^2$$ and $$g(x)=4*x-x^2$$. Determine the area of the region bounded by t

Average of a Logarithmic Function

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

Average Speed from a Velocity Function

A car’s velocity is given by $$v(t)=t^2-4*t+5$$ (in m/s) for $$0 \le t \le 5$$. Assume that $$v(t)$$

Average Speed from Variable Acceleration

A car accelerates along a straight road with acceleration given by $$a(t)=2*t-1$$ (in m/s²) for $$t\

Average Temperature Analysis

A local weather station recorded the temperature throughout a day using the model $$T(t)=-0.5*t+35$$

Average Value of a Polynomial Function

Consider the function $$f(x)= 3*x^2 - 2*x + 1$$ defined on the interval $$[0,4]$$. Use the concept o

Cell Phone Battery Consumption

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

Center of Mass of a Lamina with Variable Density

A thin lamina occupies the interval $$[0,4]$$ along the x-axis and has a variable density $$\delta(x

Charity Donations Over Time

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

Comparing Sales Projections

A company’s projected sales (in thousands of dollars) are modeled by the function $$f(x)=5*x-x^2$$ w

Designing an Open-Top Box

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

Determining Field Area from Intersection of Curves

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

Discontinuities in a Piecewise Function

Consider the function $$f(x)=\begin{cases} \frac{\sin(x)}{x} & \text{if } x \neq 0 \\ 2 & \text{if }

Discounted Cash Flow

A company projects that its annual cash flow will grow according to a geometric sequence. The initia

Economics: Consumer Surplus Calculation

Given the demand function $$d(p)=100-2p$$ and the supply function $$s(p)=20+3p$$, determine the cons

Hollow Rotated Solid

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

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).

Implicit Differentiation in an Electrical Circuit

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

Inverse for a Quadratic Function

Consider the function $$f(x)=x^2+4$$ defined for $$x \ge 0$$. Analyze its inverse function.

Optimization of Average Production Rate

A manufacturing process has a production rate modeled by the function $$P(t)=50e^{-0.1*t}+20$$ (unit

Particle Motion on a Parametric Path

A particle moves along a path given by the parametric equations $$x(t)= t^2 - t$$ and $$y(t)= 3*t -

Pipeline Installation Cost Analysis

The cost to install a pipeline along a route is given by $$C(x)=100+5*\sin(x)$$ (in dollars per mete

Population Growth and Average Rate

A town's population is modeled by the function $$P(t)=1000*e^{0.03*t}$$, where $$t$$ is measured in

Rebounding Ball

A ball is dropped from a height of $$16$$ meters. Each time the ball bounces, its maximum height is

Resource Consumption in an Ecosystem

The rate of consumption of a resource in an ecosystem is given by $$C(t)=50*\ln(1+t)$$ (in units per

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 Calculation via Cross-Sectional Areas

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

Volume of a Solid Using the Washer Method

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

Volume of a Solid with Square Cross Sections

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

Work Calculation from an Exponential Force Function

An object is acted upon by a force modeled by $$F(x)=5*e^{-0.2*x}$$ (in newtons) along a displacemen