AP Calculus BC FRQ Room

Absolute Value Function Limit Analysis

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

Analyzing a Function with a Removable Discontinuity

Consider the function $$r(x)=\frac{x^2-9}{x-3}$$ for $$x\neq3$$ and $$r(3)=2.$$ Answer the follow

Analyzing Limits Using Tabular Data

A function $$f(x)$$ is described by the following table of values: | x | f(x) | |------|------|

Asymptotic Behavior and Horizontal Limits

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

Composite Function in Water Level Modeling

Suppose the water volume in a tank is given by a composite function \(V(t)=f(g(t))\) where $$g(t)=\f

Compound Interest and Loan Repayment

A simplified model for a loan repayment assumes that a borrower owes $$10,000$$ dollars and the rema

Continuity Analysis Involving Logarithmic and Polynomial Expressions

Consider the function $$f(x)= \begin{cases} \frac{\ln(x+2)}{x} & \text{if } x<0 \\ (x+1)^2 & \text{i

Continuity in Piecewise Defined Functions

Consider the piecewise function $$f(x)= \begin{cases} x^2+1, & \text{if } x \leq 3 \\ 2*x+k, & \text

Determining Continuity via Series Expansion

Consider the function $$f(x)= \frac{e^x - \ln(1+x) - x - 1}{x^2}$$ for $$x \neq 0$$ with $$f(0)=L$$.

Economic Equilibrium and Limit Analysis

An economist examines market behavior using a demand function $$D(p)= 100-5*p$$ and a supply functio

Economic Growth and Continuity

The function $$E(t)$$ represents an economy's output index over time (in years). A table provides th

End Behavior and Horizontal Asymptote Analysis

Consider the function $$f(x)=\frac{3*x^3-5*x+2}{2*x^3+4*x^2-1}$$. Answer the following:

End Behavior of an Exponential‐Log Function

Consider the function $$f(x)= e^{-x} \ln(1+x)$$. Analyze its behavior by investigating the limit as

Examining Continuity with an Absolute Value Function

Consider the function defined by $$f(x)=\frac{|x-2|}{x-2}$$ for $$x \neq 2$$. (a) Evaluate $$\lim_{x

Exploring Removable and Nonremovable Discontinuities

Consider the function $$f(x)=\frac{(x-2)(x+3)}{(x-2)}$$ for $$x\neq2$$ and $$f(2)=7$$. Answer the fo

Graphical Analysis of Removable Discontinuity

A graph of a function f is provided (see stimulus). The graph shows that f has a hole at (2, 4) whil

Horizontal Asymptote of a Rational Function

Consider the rational function $$f(x)= \frac{2*x^3+5*x^2-3}{x^3-4*x+1}$$. Answer the questions regar

Limits of Composite Trigonometric Functions

Let $$p(x)= \frac{\sin(3x)}{\sin(5x)}$$.

Manufacturing Process Tolerances

A manufacturing company produces components whose dimensional errors are found to decrease as each c

One-Sided Limits and Jump Discontinuity Analysis

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

Oscillatory Behavior and Squeeze Theorem

Consider the function $$h(x)= x^2 \cos(1/x)$$ for $$x \neq 0$$ with $$h(0)=0$$.

Rational Functions and Limit at Infinity

Consider the rational function $$r(x)= \frac{2x^2+3x-1}{x^2-4}$$.

Related Rates with an Expanding Spherical Balloon

A spherical balloon is being inflated so that its volume increases at a rate of $$\frac{dV}{dt}=100\

Sine over x Function with Altered Value

Consider the function $$ f(x)=\begin{cases} \frac{\sin(x)}{x} & \text{if } x \neq 0, \\ 3 & \text{i

Trigonometric Limits

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

Water Treatment Plant Discontinuity Analysis

A water treatment plant monitors the inflow to a reservoir. Due to sensor calibration, the inflow ra

Analysis of a Piecewise Function's Differentiability

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

Bacteria Culturing in a Bioreactor

In a bioreactor, the bacterial inflow (growth) rate is given by $$B_{in}(t)=\frac{15}{1+e^{-0.3*(t-5

Car Acceleration: Secant and Tangent Slope

A car's position along a straight road is given by $$s(t)= 2t^3 - 9t^2 + 12t$$, where s is in meters

Chain Rule in Biological Growth Models

A biologist models the growth of a bacterial population by the function $$P(t) = (5*t + 2)^4$$, wher

Chemical Reaction Rate Control

During a chemical reaction in a reactor, reactants enter at a rate of $$R_{in}(t)=\frac{10*t}{t+2}$$

Composite Function and Chain Rule Application

Consider the function $$h(x)=\sin(2*x^2+3)$$. Using the chain rule, answer the following parts:

Derivative Estimation from a Graph

A graph of a function $$f(x)$$ is provided in the stimulus. Using the graph, answer the following pa

Derivative Using Limit Definition

Let $$f(x)=\frac{1}{x+2}$$. Using the definition of the derivative, find $$f'(x)$$.

Differentiability of an Absolute Value Function

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

Error Bound Analysis for Cos(x) Approximations in Physical Experiments

In a controlled physics experiment, small angle approximations for $$\cos(x)$$ are critical. Analyze

Evaluating the Derivative Using the Limit Definition

Consider the function $$f(x) = 3*x^2 - 2*x + 1$$. (a) Use the limit definition of the derivative:

Graph Interpretation: Average vs Instantaneous Rates

A function is represented in the table below. Analyze the difference between average and instantaneo

Icy Lake Evaporation and Refreezing

An icy lake gains water from melting ice at a rate of $$M_{in}(t)=5+0.2*t$$ liters per hour and lose

Implicit Differentiation in a Geometric Context

Consider the circle defined by the equation $$x^2+y^2=25.$$ (a) Use implicit differentiation to f

Implicit Differentiation: Conic with Mixed Terms

Consider the curve defined by $$x*y + y^2 = 6$$.

Limit Definition of Derivative for a Rational Function

For the function $$f(x)=\frac{1}{x+1}$$, use the limit definition of the derivative to answer the fo

Maclaurin Polynomial for √(1+x)

A scientist approximates the function $$f(x)=\sqrt{1+x}$$ for small values of x using its Maclaurin

Optimization in a Chemical Reaction

The rate of a chemical reaction is modeled by the function $$R(x)=x*e^{-x}+\ln(x+2)$$, where $$x$$ r

Population Growth Rate

A population is modeled by $$P(t)=\frac{3*t^2 + 2}{t+1}$$, where $$t$$ is measured in years. Analyze

Product and Quotient Rule Application

Consider the function $$f(x)=\frac{x*\ln(x)}{e^{x}+2}$$, defined for $$x>0$$. Analyze its behavior u

Related Rates: Changing Shadow Length

A 1.8 m tall man is walking away from a 5 m tall lamp at a constant speed of 1.2 m/s. The lamp casts

Renewable Energy Storage

A battery storage system experiences charging at a rate of $$C(t)=50+10\sin(0.5*t)$$ kWh and dischar

Secant Line Approximation in an Experimental Context

A temperature sensor records the following data over a short experiment:

Tangent and Normal Lines

Consider the function $$g(x)=\sqrt{x}$$ defined for $$x>0$$. Answer the following:

Temperature Change Rate

The temperature in a chemical reactor is modeled by $$T(t)=\frac{\sin(2*t)}{t}$$ for \(t>0\), where

Temperature Function Analysis

Suppose the temperature over time is modeled by $$T(t)=e^(2*t)*\sin(t)$$, where $$t$$ is measured in

Using the Product Rule in Economics

A company’s revenue function is given by $$R(x)=x*(100-x)$$, where $$x$$ (in hundreds) represents th

Analyzing the Rate of Change in an Economic Model

Suppose the profit function is given by $$P(x)=e^{x}-4*\ln(x+2)$$, where x represents the number of

Chain Rule in a Power Function

Consider the function $$f(x)= (3*x^2 + 2*x + 1)^5$$. Use the chain rule to find its derivative, eval

Composite Chain Rule with Exponential and Trigonometric Functions

Consider the function $$f(x) = e^{\cos(x)}$$. Analyze its derivative and explain the role of the cha

Composite Function Analysis

Consider the function $$f(x)= \sqrt{3*x^2+2*x+1}$$ which arises in an experimental study of motion.

Composite Functions in a Biological Model

In a biological model, the concentration of a substance is given by $$P(x)=e^{-\sqrt{x^2+1}}$$, wher

Composite Functions in Biological Growth

Let a model for bacteria growth be represented by $$f(t)=e^{2*t}$$, and let the effect of nutrient c

Differentiation in a Logistic Population Model

The population of a species is modeled by the logistic function $$P(t)= \frac{1000}{1+e^{-0.3*(t-5)}

Differentiation of an Arctan Composite Function

For the function $$f(x) = \arctan\left(\frac{3*x}{x+1}\right)$$, differentiate with respect to $$x$$

Implicit Differentiation in a Conic Section

Consider the curve defined by $$x^2 + x*y + y^2 = 9$$.

Implicit Differentiation in a Conical Sand Pile Problem

A conical sand pile has a constant ratio between its radius and height given by $$r= \frac{1}{2}*h$$

Implicit Differentiation Involving Logarithms

Consider the equation $$\ln(x+y)=x*y$$ which relates x and y. Answer the following parts:

Implicit Differentiation of an Ellipse

The ellipse is given by $$4*x^2 + 9*y^2 = 36$$.

Implicit Differentiation with Trigonometric Functions

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

Inverse Function Derivative Calculation

Let $$f$$ be a one-to-one differentiable function for which the table below summarizes selected info

Inverse Function Differentiation in a Logarithmic Scenario

Let $$f(x)= \ln(x+2) + x$$, which is a one-to-one differentiable function. It is known that $$f(0)=

Inverse Function Differentiation in a Trigonometric Context

Let $$f(x)= \sin(x) + x$$, defined on the interval $$[0, \frac{\pi}{2}]$$, and let $$g$$ be its inve

Inverse Trigonometric Function in a Navigation Problem

A navigator uses the function $$\theta(x)=\arcsin\left(\frac{x}{10}\right)$$ to determine the angle

Multi-step Differentiation of a Composite Logarithmic Function

Consider the function $$F(x)= \sqrt{\ln\left(\frac{1+e^{2*x}}{1-e^{2*x}}\right)}$$, defined for valu

Multilayer Composite Differentiation in a Climate Model

A climate model gives the temperature $$T(t)$$ (in °C) as a function of time $$t$$ (in years) by $$T

Polar and Composite Differentiation: Arc Slope for a Polar Curve

Consider the polar curve $$r(\theta)=2+\cos(\theta)$$. Answer the following parts:

Projectile Motion and Composite Exponential Functions

A projectile’s height at time $$t$$ (in seconds) is modeled by the function $$h(t)= e^{- (t-2)^2}$$.

Tangent Line to an Ellipse

Consider the ellipse given by $$\frac{x^2}{16} + \frac{y^2}{9} = 1$$. Determine the slope of the tan

Air Conditioning Refrigerant Balance

An air conditioning system is charged with refrigerant at a rate given by $$I(t)=12-0.5t$$ (kg/min)

Analysis of Particle Motion

A particle’s velocity is given by $$v(t)= 4t^3 - 3t^2 + 2$$. Analyze the particle’s motion by invest

Analyzing a Production Cost Function

A company's cost function for producing goods is given by $$C(x)=x^3-12x^2+40x+100$$, where x repres

Analyzing Rate of Approach in a Pursuit Problem

Two cars are traveling on perpendicular roads. Car A is moving east at 60 km/h and is 3 km from the

Approximating Changes with Differentials

Given that the surface area of a sphere is $$A = 4\pi r^2$$, use differentials to approximate the ch

Comparing Rates: Temperature Change and Coffee Cooling

The temperature of a freshly brewed coffee is modeled by $$T(t)=95*e^{-0.05*t}+25$$ (in °F), where $

Deceleration of a Vehicle on a Straight Road

A vehicle travels along a straight road with velocity function $$v(t)=30-4*t$$ (m/s) for $$0 \le t \

Differentials in Engineering: Beam Stress Analysis

The stress S (in Pascals) experienced by an engineering beam under load is modeled by $$S(x)=0.02*x^

Drug Concentration Dynamics

The concentration of a drug in the bloodstream is modeled by $$C(t)= 50*e^{-0.2*t} + 10$$ (in mg/L),

Expanding Circular Ripple

A stone is thrown in a pond, creating circular ripples. The area of the circle defined by the ripple

Graph Interpretation of Experimental Data

A laboratory experiment measured the concentration of a chemical reactant over time. The following g

Implicit Differentiation in Astronomy

The trajectory of a comet is given by the ellipse $$x^2 + 4*y^2 = 16$$, where \(x\) and \(y\) (in as

Implicit Differentiation on a Circle

Consider the circle defined by $$x^2+y^2=25$$, where both $$x$$ and $$y$$ are functions of time $$t$

Inflating Spherical Balloon

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

Inflating Spherical Balloon

A spherical balloon is being inflated such that its volume is given by $$V= \frac{4}{3}*\pi*r^3$$, w

L'Hôpital's Rule in Context

Evaluate the limit $$\lim_{x \to \infty} \frac{5*x^3-4*x^2+1}{7*x^3+2*x-6}$$ using L'Hôpital's Rule.

L'Hospital's Rule for Indeterminate Limits

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

Linear Account Growth in Finance

The amount in a savings account during a promotional period is given by the linear function $$A(t)=1

Linearization of a Radical Function

Consider the function $$f(x)= x^{1/3}$$. Use linearization to approximate function values. Answer th

Linearization of Implicit Equation

Consider the implicit equation $$x^2 + y^2 - 2*x*y = 1$$, which defines $$y$$ as a function of $$x$$

Marginal Analysis in Economics

The cost function for producing $$x$$ items is given by $$C(x)= 0.1*x^3 - 2*x^2 + 20*x + 100$$ dolla

Marginal Cost Analysis

A company’s cost function is given by $$C(x)=100+25*x+4*x^2$$, where $$x$$ represents the number of

Motion along a Curved Path

A particle moves along the curve defined by $$y=\sqrt{x}$$. At the moment when $$x=9$$ and the x-coo

Motion Model Inversion

Suppose that the position of a particle moving along a line is given by $$f(t)=t^3+t$$. Analyze the

Polar Coordinates: Arc Length of a Spiral

Consider the polar curve defined by $$r(\theta)=1+\frac{\theta}{2}$$ for $$0 \le \theta \le \pi$$.

Population Growth and Change: A Nonlinear Model

The population of a bacterial culture is modeled by $$P(t)=\frac{500e^{0.3*t}}{1+e^{0.3*t}}$$, where

Population Growth Rate

The population of a bacteria culture is given by $$P(t)= 500e^{0.03*t}$$, where $$t$$ is in hours. A

Production Cost Analysis

A company’s production cost $$C$$ (in dollars) and production level $$x$$ (in thousands of units) sa

Projectile Motion Analysis

A projectile is launched such that its horizontal and vertical positions are modeled by the parametr

Radical Function Inversion

Let $$f(x)=\sqrt{2*x+5}$$ represent a measurement function. Analyze its inverse.

Reactant Flow in a Chemical Reactor

In a chemical reactor, a reactant is introduced at a rate $$I(t)=15\sin(\frac{t}{2})$$ (grams per mi

Related Rates: Inflating Spherical Balloon

A spherical balloon is being inflated so that its volume, given by $$V= \frac{4}{3}\pi*r^3$$, increa

Related Rates: Pool Water Level

Water is being drained from a circular pool. The surface area of the pool is given by $$A=\pi*r^2$$.

Security Camera Angle Change

A security camera is mounted on a 15 m tall tower. Let $$x$$ denote the horizontal distance from the

Temperature Conversion Model Inversion

The temperature conversion function is given by $$f(x)=\frac{9}{5}*x+32$$, which converts Celsius to

Absolute Extrema via Candidate's Test

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

Analysis of Total Distance Traveled

A particle moves along a line with a velocity function given by $$v(t)=t^2-4*t+3$$ for $$t\in[0,5]$$

Application of Rolle's Theorem

Consider the function $$f(x) = x^2 - 4*x + 4$$ on the interval $$[0,4]$$.

Application of the Mean Value Theorem in Motion

A car's position on a straight road is given by the function $$s(t)=t^3-6*t^2+9*t+5$$, where t is in

Average and Instantaneous Velocity Analysis

A bird’s position is given by $$s(t)=2*t^2-t+1$$ (in meters) for $$t\in[0,3]$$ seconds.

Concavity in an Economic Model

Consider the function $$f(x)= x^3 - 3*x^2 + 2$$, which represents a simplified economic trend over t

Fractal Tree Branch Lengths

A fractal tree is constructed as follows: The trunk has a length of 10 meters. At each generation, e

Fuel Consumption in a Generator

A generator operates with fuel being supplied at a constant rate of $$S(t)=5$$ liters/hour and consu

Function Behavior Analysis

Consider the function \( f(x) = x^4 - 4*x^3 + 6*x^2 - 4*x + 1 \). Answer the following parts:

Inverse Analysis with a Radical Expression

Let $$f(x)= 3*\sqrt{x+4} - 2$$, defined for $$x \ge -4$$, which models a physical process. Answer th

Investigation of a Series with Factorials and Its Operational Calculus

Consider the series $$F(x)=\sum_{n=0}^\infty \frac{x^n}{n!}$$, which represents an exponential funct

Logarithmic-Quadratic Combination Inverse Analysis

Consider the function $$f(x)= \ln(x^2+1)$$ for $$x \ge 0$$. Answer the following parts.

Logistic Growth Model Analysis

Consider the logistic growth model given by $$P(t)=\frac{100}{1+9e^{-0.5*t}}$$. Answer the following

Mean Value Theorem Application

Let $$f(x)=\ln(x)$$ be defined on the interval $$[1, e^2]$$. Answer the following parts using the Me

Mean Value Theorem Application for Mixed Log-Exponential Function

Let $$h(x)= \ln(x) + e^{-x}$$ be defined on the interval [1,3]. Analyze the function using the Mean

Optimization in a Geometric Setting: Garden Design

A farmer is designing a rectangular garden adjacent to a river. No fence is needed along the river s

Optimization in Production Costs

In an economic context, consider the cost function $$C(x)=0.5*x^3-6*x^2+25*x+100$$, where C(x) repre

Projectile Motion Analysis

A projectile is launched vertically with its height given by $$s(t) = -16*t^2 + 64*t + 80$$ (in feet

Rate of Change in a Chemical Reaction

The concentration of a reactant in a chemical reaction is modeled by $$C(t)=10*e^{-0.5*t} + 2$$ (in

Rate of Change in a Logarithmic Temperature Model

A cooling process is modeled by the temperature function $$T(t)= 100 - 20\,\ln(t+1)$$, where t is me

Rate of Reaction: Concentration Change

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

Region Area and Volume: Polynomial and Linear Function

A region in the x-y plane is bounded by the curves $$f(x)=x^2$$ and $$g(x)=2 - x$$. Answer the follo

Series Manipulation and Transformation in an Economic Forecast Model

A forecast model is given by the series $$F(x)=\sum_{n=0}^\infty \frac{(-1)^n}{(n+1)^2} * x^n$$. Ans

Series Representation in a Biological Growth Model

A biological process is approximated by the series $$B(t)=\sum_{n=0}^\infty (-1)^n * \frac{(0.3*t)^n

Square Root Function Inverse Analysis

Consider the function $$f(x)= \sqrt{3*x+4}$$ defined for $$x \ge -\frac{4}{3}$$. Answer the followin

Travel Distance from Speed Data

A traveler’s speed (in km/h) is recorded at various times during a trip. Use the data to approximate

Vector Analysis of Particle Motion

A particle moves in the plane with its position given by the vector function $$\mathbf{r}(t) = \lang

Water Tank Dynamics

A water tank receives water from a pipe at a rate of $$R(t)=3*t+5$$ liters/min and loses water throu

Advanced U-Substitution with a Quadratic Expression

Evaluate the indefinite integral $$\int \frac{2*x}{\sqrt{x^{2}+1}}\,dx$$ using u-substitution.

Area Estimation with Riemann Sums

Consider the function $$f(x)=x^2-4*x+3$$ on the interval $$[1,5]$$. Using a partition of 4 equal sub

Bacteria Growth with Nutrient Supply

A bacterial culture in a laboratory is provided with nutrients at a rate of $$N(t)=6*\ln(t+1)$$ mg/m

Charging a Battery

An electric battery is charged with a variable current given by $$I(t)=4+2*\sin\left(\frac{\pi*t}{6}

Chemical Reaction: Rate of Concentration Change

A chemical reaction features a rate of change of concentration given by $$R(t)= 5*e^{-0.5*t}$$ (in m

Definite Integral Involving an Inverse Function

Evaluate the definite integral $$\int_{1}^{4} \frac{1}{\sqrt{x}}\,dx$$ and explain the significance

Definite Integral via U-Substitution

Evaluate the definite integral $$\int_{1}^{3} (2*x-1)^6\,dx$$ using u-substitution.

Determining Velocity and Displacement from Acceleration

A particle's acceleration is given by $$a(t)=4*t-8$$ (in m/s²) for $$0 \le t \le 3$$ seconds. The in

Distance vs. Displacement from a Velocity Function

A runner's velocity is modeled by $$v(t)=5-0.5*t$$ (in m/s) for $$0\le t\le10$$. The runner may chan

Error Estimation in Riemann Sum Approximations

Consider the function $$f(x)=\sqrt{x}$$ on the interval $$[1,9]$$. When approximating the definite i

Estimating Area Under a Curve Using Riemann Sums

A function $$f(x)$$ is defined on the interval $$[0,6]$$. The following table provides the values of

Finding the Area Between Curves

Find the area of the region bounded by the curves $$y=4-x^2$$ and $$y=x$$.

Integration by Substitution and Inverse Functions

Consider the function $$f(x)= (x-4)^2 + 3$$ for $$x \ge 4$$. Answer the following questions about $$

Integration Involving Inverse Trigonometric Functions

Consider the function $$f(x)= \tan^{-1}(x)$$. Answer the following questions regarding its inverse a

Integration of a Piecewise-Defined Function

Define the function $$f(x)$$ as follows: $$f(x)= \begin{cases} 2*x, & 0\le x < 3 \\ 12, & 3 \le x \

Investigating Partition Sizes

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

Modeling Bacterial Growth Through Accumulated Change

A bacteria population's growth rate is given by $$r(t)=\frac{2*t}{1+t^{2}}$$ (in thousands per hour)

Net Change in Drug Concentration

The rate of change of a drug's concentration in the bloodstream is given by $$R(t)=8*e^{-0.5*t}$$ (i

Non-Uniform Subinterval Riemann Sum

A function $$f(t)$$ is measured at non-uniform time intervals as recorded in the table below: | t (

Optimizing the Inflow Rate Strategy

A municipality is redesigning its water distribution system. The water inflow rate is modeled by $$F

Partial Fractions Integration

Evaluate the integral $$\int_1^3 \frac{4*x-2}{(x-1)(x+2)} dx$$ by decomposing the integrand into p

Particle Motion with Variable Acceleration and Displacement Analysis

A particle moves along a straight line with acceleration given by $$a(t)=4-2*t$$ (in m/s²). At time

Recovering Accumulated Change

A company’s revenue rate changes according to $$R'(t)=8*t-12$$ (in dollars per day). If the revenue

Reservoir Water Level

A reservoir experiences a net water inflow modeled by $$W(t)=40*\sin\left(\frac{\pi*t}{12}\right)-5$

Revenue Accumulation and Constant of Integration

A company's revenue is modeled by $$R(t) = \int_{0}^{t} 3*u^2\, du + C$$ dollars, where t (in years)

Revenue Estimation Using the Trapezoidal Rule

A company recorded its daily revenue (in thousands of dollars) over four days. Use the data in the t

Riemann Sum Approximation with Irregular Intervals

A set of experimental data provides the values of a function $$f(x)$$ at irregularly spaced points a

Riemann Sum from a Table: Plant Growth Data

A function $$f(t)$$ describes the height (in cm) of a plant over time (in days). The measurements ar

Sandpile Accumulation

At an industrial site, sand is continuously added to and removed from a pile. The addition rate is g

Tank Filling Problem

Water flows into a tank at a rate given by $$R(t)=8e^{-0.5*t}+2$$ (in liters per minute) for $$t\geq

Trigonometric Integral via U-Substitution

Evaluate the integral $$\int_{0}^{\pi/2} \sin(2*x)\,dx$$ using an appropriate substitution.

Work Done by a Variable Force

A variable force given by $$F(x)=4*x^2$$ (in Newtons) is applied along a straight line over the disp

Work Done by an Exponential Force

A variable force acting along the x-axis is given by $$F(x)=5 * e^(0.5 * x)$$ (in Newtons) for 0 \(\

Capacitor Charging with Leakage

A capacitor is being charged by a constant current source of $$5$$ A, but it also leaks charge at a

Chemical Reaction Rate

In a chemical reaction, the concentration $$C(t)$$ of a reactant decreases according to the first-or

Cooling Coffee Data Analysis

A cup of coffee cools down in a room according to Newton's Law of Cooling. The temperature $$T(t)$$

Dye Dilution in a Stream

A river has dye added at a constant rate of $$0.5$$ kg/min, and the dye is removed at a rate proport

Epidemic Spread Modeling

In a simplified epidemic model, the number of infected individuals $$I(t)$$ is modeled by the logist

Exact Differential Equations

Consider the differential equation $$ (2*x*y + y^2)\,dx + (x^2+2*x*y)\,dy = 0 $$. Answer the followi

Exponential Growth via Slope Field Analysis

Consider the differential equation $$\frac{dy}{dx} = x * y$$ with the initial condition $$y(0)=2$$.

FRQ 5: Mixing Problem in a Tank

A tank initially contains 100 liters of water with 10 kg of dissolved salt. Brine with a salt concen

FRQ 6: Radioactive Decay

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

Inverse Function Analysis Derived from a Differential Equation Solution

Consider the function $$f(x)=x^3+2$$. Although this function is provided outside of a differential e

Mixing Problem with Differential Equations

A tank initially holds 100 L of a salt solution containing 5 kg of salt. Brine with a salt concentra

Modeling Medication Concentration in the Bloodstream

A patient receives an intravenous drug at a constant rate $$R$$ (mg/min) and the drug is eliminated

Motion along a Line with a Separable Differential Equation

A particle moves along a straight line according to the differential equation $$\frac{dy}{dx} = \fra

Newton's Law of Cooling

An object with an initial temperature of $$80^\circ C$$ is placed in a room at a constant temperatur

Oscillatory Behavior in Differential Equations

Consider the second-order differential equation $$\frac{d^2y}{dx^2}+y=0$$, which describes simple ha

Radioactive Decay

A radioactive substance decays according to the differential equation $$\frac{dA}{dt}=-kA$$, where $

Separable DE with Trigonometric Component

Solve the differential equation $$\frac{dy}{dx}=\sin(x)*\cos(y)$$ with the initial condition $$y(0)=

Simplified Predator-Prey Model

A simplified model for a predator population is given by the differential equation $$\frac{dP}{dt} =

Slope Field Analysis and Solution Curve Sketching for $$\frac{dy}{dx}= x - y$$

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

Slope Field and Sketching a Solution Curve

The differential equation $$\frac{dy}{dx}=x-y$$ has been represented by a slope field. Answer the fo

Temperature Change and Differential Equations

A hot liquid cools in a room at $$20^\circ C$$ according to the differential equation $$\frac{dT}{dt

Water Tank Inflow-Outflow Model

A water tank is subject to an inflow and outflow. The inflow rate is given by $$I(t)=3*t+2$$ liters

Analysis of Particle Motion in the Plane

A particle’s position is given by the vector function $$\mathbf{r}(t)=\langle e^{-t},\,\sin(t)\rangl

Arc Length of a Curve

Consider the curve defined by $$y= \ln(\cos(x))$$ for $$0 \le x \le \frac{\pi}{4}$$. Determine the l

Arc Length of a Logarithmic Curve

Determine the arc length of the curve $$f(x)= \ln(x)$$ on the interval $$[1,e]$$.

Area Between a Function and Its Tangent Line

Let $$f(x)=x^3-x$$. At the point $$x=1$$, find the tangent line to the curve and determine the area

Area Between Curves: Parabolic and Linear Functions

Consider the functions $$f(x)=5*x-x^2$$ and $$g(x)=x$$. Determine the area enclosed between these cu

Area Between Two Curves in a Water Channel

A channel cross‐section is defined by two curves: the upper boundary is given by $$f(x)=12-0.8*x$$ a

Area Under a Curve with a Discontinuity

Consider the function $$f(x)=\frac{1}{x+2}$$ defined on $$[0,3]$$.

Average and Instantaneous Analysis in Periodic Motion

A particle moves along a line with its displacement given by $$s(t)= 4*\cos(2*t)$$ (in meters) for $

Bacterial Decay Modeled by a Geometric Series

A bacterial culture is treated with an antibiotic that reduces the bacterial population by 20% each

Center of Mass of a Rod

A thin rod of length 10 m has a linear density given by $$\rho(x)=3+0.4*x$$ (in kg/m) where $$x$$ is

Determining the Arc Length of a Curve

Consider the curve defined by $$y=\frac{1}{2}*e^{x/2}$$ over the interval $$[0,2]$$.

Electric Charge Distribution Along a Rod

A rod of length 10 m has a linear charge density given by $$\lambda(x) = 3e^{-0.5*x}$$ coulombs per

Kinematics: Motion with Variable Acceleration

A particle moves along a straight line with acceleration $$a(t)=4-2*t$$ (in m/s²). The particle has

Net Cash Flow Analysis

A company’s net cash flow is modeled by $$N(t)=50*\ln(t+1) - 2*t$$ (in thousands of dollars per mont

Particle Motion from Acceleration

A particle has an acceleration given by $$a(t)=3*t-6$$ (m/s²). With initial conditions $$v(0)=2$$ m/

Position and Velocity from Tabulated Data

A particle’s velocity (in m/s) is measured at discrete time intervals as shown in the table. Use the

Projectile Maximum Height

A ball is thrown upward with an acceleration of $$a(t)=-9.8$$ m/s², an initial velocity of $$v(0)=20

Savings Account with Decreasing Deposits

An individual opens a savings account with an initial deposit of $1000 in the first month. Every sub

Volume by Cross-Sectional Area (Square Cross-Sections)

A solid has a base in the xy-plane bounded by the curve $$y=\sqrt{x}$$ and the x-axis for $$x\in[0,4

Volume by Revolution: Washer Method

Consider the region bounded by the curves $$y=x+2$$ and $$y=x^2$$. When this region is rotated about

Volume by Shell Method: Rotating a Region

Consider the region bounded by the curves $$y=x$$ and $$y=x^2$$. This region is rotated about the y-

Volume of a Region via Washer Method

The region in the first quadrant bounded by the curves $$y=x$$ and $$y=x^2$$ is rotated about the x-

Volume of a Solid by the Washer Method

The region bounded by $$y=x^2$$ and $$y=4$$ is rotated about the x-axis, forming a solid with a hole

Volume of a Solid with Variable Cross Sections

A solid has a cross-sectional area perpendicular to the x-axis given by $$A(x)=4-x^2$$ for $$x\in[-2

Volume with Square Cross-Sections

Consider the region under the curve $$y = \sqrt{x}$$ between $$x = 0$$ and $$x = 4$$. Squares are co

Work Done by a Variable Force

A variable force acting along the x-axis is given by $$F(x) = 2 * x + 3$$ (in Newtons). An object mo

Work Done in Lifting a Cable

A cable of length 10 m with a uniform mass density of 5 kg/m hangs vertically from a winch. The winc

Work Done Pumping Water

A water tank is shaped like an inverted circular cone with a height of $$10$$ m and a top radius of

Analysis of Particle Motion Using Parametric Equations

A particle moves in the plane with its position defined by $$x(t)=4*t-2$$ and $$y(t)=t^2-3*t+1$$, wh

Arc Length of a Cycloid

Consider the cycloid defined by the parametric equations $$x(t)= t - \sin(t)$$ and $$y(t)= 1 - \cos(

Arc Length of a Decaying Spiral

Consider the parametric equations $$x(t)= e^{-t}\cos(t)$$ and $$y(t)= e^{-t}\sin(t)$$ for $$t \ge 0$

Arc Length of a Parametric Curve

Consider the parametric equations $$x(t)=\sin(t)$$ and $$y(t)=\cos(t)$$ for $$0\leq t\leq \frac{\pi}

Arc Length of a Vector-Valued Function

Consider the vector-valued function $$\vec{r}(t)= \langle \ln(t+1), \sqrt{t}, e^t \rangle$$ defined

Area Between Polar Curves

Consider the polar curves $$ r_1=2+\cos(\theta) $$ and $$ r_2=1+\cos(\theta) $$. Although the curves

Area Enclosed by a Polar Curve

Consider the polar curve given by $$r = 2*\sin(\theta)$$.

Combined Motion Analysis

A particle’s path is described by the parametric equations $$x(t)= \ln(1+ t^2)$$ and $$y(t)= \sqrt{t

Comprehensive Motion Analysis Using Parametric and Vector-Valued Functions

A particle moves with position given by $$ r(t)=\langle t*e^{-t},\;\ln(1+t) \rangle $$ for $$ t\ge0

Conversion of Parametric to Polar: Motion Analysis

An object moves along a path given by the parametric equations $$x(t)=t^2$$ and $$y(t)=2*t$$ for $$t

Curvature of a Parametric Curve

Consider the curve defined by the parametric equations $$x(t)=\ln(t)$$ and $$y(t)=t^2$$ for \(t>0\).

Drone Altitude Measurement from Experimental Data

A drone’s altitude (in meters) is recorded at various times (in seconds) as shown in the table below

Error Analysis in Taylor Approximations

Consider the function $$f(x)=e^x$$.

Intersection of Parametric Curves

Consider two particles moving along different paths: Particle A: $$x_A(t)= t^2, \quad y_A(t)= 2t +

Intersection Points of Polar Curves

Two polar curves are given by \(r=2\sin(\theta)\) and \(r=1\). Answer the following:

Kinematics in Polar Coordinates

An object moves so that its position in polar coordinates is given by $$r(t)= 4 - t$$ and $$\theta(t

Maclaurin Series for Trigonometric Functions

Let $$f(x)=\sin(x)$$.

Motion Along a Helix

A particle moves along a helix described by the vector-valued function $$\vec{r}(t)=<\cos(t),\, \sin

Motion Along a Helix

A particle moves along a helix defined by $$\mathbf{r}(t)=\langle \cos(t), \sin(t), t \rangle$$.

Parametric Curve Intersection

Two curves are defined parametrically as follows: For curve C1, $$x(t) = t^2$$ and $$y(t) = 2*t + 1$

Parametric Curve: Intersection with a Line

Consider the parametric curve defined by $$ x(t)=t^3-3*t $$ and $$ y(t)=2*t^2 $$. Analyze the proper

Parametric Equations and Intersection Points

Consider the curves defined parametrically by $$x_1(t)=t^2, \; y_1(t)=2t$$ and $$x_2(s)=s+1, \; y_2(

Particle Motion in the Plane

A particle moves in the plane with parametric equations $$x(t)= 3\cos(t)$$ and $$y(t)= 3\sin(t)$$ fo

Polar Coordinates and Area Computation

Examine the polar curve $$r = 2 + \sin(2\theta)$$ and determine the area of the region it encloses.

Projectile Motion: Rocket Launch Tracking

A rocket is launched with its horizontal position given by $$x(t)=100*t$$ (in meters) and its vertic

Tangent Line to a Parametric Curve

Consider the circle parametrized by $$x(t)=3\sin(t)$$ and $$y(t)=3\cos(t)$$ for $$0\le t\le 2\pi$$.

Time of Nearest Approach on a Parametric Path

An object travels along a path defined by $$x(t)=5*t-t^2$$ and $$y(t)=t^3-6*t$$ for $$t\ge0$$. Answe

Vector-Valued Functions in 3D

A space curve is described by the vector function $$\mathbf{r}(t)=\langle e^t,\cos(t),\ln(1+t) \rang

Work Done by a Force along a Path

A force acting on an object is given by the vector function $$\vec{F}(t)= \langle 3t,\; 2,\; t^2 \ra