AP Calculus BC FRQ Room

Analyzing Limits Using Tabular Data

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

Application of the Squeeze Theorem with Trigonometric Functions

Let $$f(x)= x^2 \sin\left(\frac{1}{x}\right)$$ for $$x\neq0$$, and $$f(0)=0$$. Analyze the behavior

Applying Algebraic Techniques to Evaluate Limits

Examine the limit $$\lim_{x\to4} \frac{\sqrt{x+5}-3}{x-4}$$. Answer the following: (a) Evaluate the

Applying the Squeeze Theorem

Let $$f(x)=x^2\sin(1/x)$$ for $$x \neq 0$$ and define $$f(0)=0$$. Use the Squeeze Theorem to evaluat

Comparing Methods for Limit Evaluation

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

Compound Function Limits and Continuity Involving a Logarithm

Consider the function $$f(x)= \ln(|x-5|)$$, defined for $$x \neq 5$$. Analyze its behavior near x =

Computing a Limit Using Algebraic Manipulation

Evaluate the limit $$\lim_{x\to2} \frac{x^2-4}{x-2}$$ using algebraic manipulation.

Continuity Analysis of a Rational-Piecewise Function

Consider the function $$r(x)=\begin{cases} \frac{x^2-1}{x-1} & x<0, \\ 2*x+c & x\ge0. \end{cases}$$

Continuity Analysis Using a Piecewise Defined Function

Let $$f(x)=\begin{cases} x^2, & x \leq 1 \\ 2x-1, & x> 1 \end{cases}$$.

Direct Substitution in Polynomial Functions

Consider the function $$f(x)=2*x^2-3*x+1$$. Answer the following:

Evaluating a Logarithmic Limit

Given the limit $$\lim_{x \to 2} \frac{\ln(x-1)}{x^2-4} = k$$, find the value of $$k$$ using algebra

Evaluating Limits Involving Exponential and Rational Functions

Consider the limits involving exponential and polynomial functions. (a) Evaluate $$\lim_{x\to\infty}

Evaluating Limits via Rationalizing Techniques

Let $$f(x)=\frac{\sqrt{2*x+9}-3}{x}.$$ Answer the following parts.

Experimental Data Limit Estimation from a Table

Using the table below, estimate the behavior of a function f(x) as x approaches 1.

Exploring Infinite and Vertical Asymptotes in Rational Functions

Consider the function $$q(x)= \frac{2x^3-x}{x^2-1}$$.

Exploring the Squeeze Theorem

Define the function $$ f(x)= \begin{cases} x^2*\cos\left(\frac{1}{x}\right), & x\neq0 \\ 0, & x=0

Exponential Inflow with a Shift in Outflow Rate

A water tank receives water at a rate given by $$R_{in}(t)=20\,e^{-t}$$ liters per minute. The water

Higher‐Order Continuity in a Log‐Exponential Function

Define $$ f(x)= \begin{cases} \frac{e^x - 1 - \ln(1+x)}{x^3}, & x \neq 0 \\ D, & x = 0, \end{cases}

Horizontal Asymptote of a Rational Function

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

Investigating a Function with a Removable Discontinuity

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

Limit Behavior in a Container Optimization Problem

A manufacturer designs a closed cylindrical container with a fixed volume $$V$$ (in cubic units). Th

Limits Involving Radicals

Consider the function $$f(x)=\frac{\sqrt{x+4}-2}{x}$$ defined for $$x \neq 0$$. Answer the following

Logarithmic Function Limits

Consider the function $$f(x)=\frac{\ln(1+3*x)}{x}$$ for $$x \neq 0$$. Answer the following:

One-Sided Limits and Discontinuities

Consider the function $$p(x)=\begin{cases} x^2+1, & x<2, \\ 4*x-3, & x\ge2. \end{cases}$$ Answer t

Parameter Determination for Continuity

Let $$f(x)= \frac{x^2-1}{x-1}$$ for $$x \neq 1$$, and suppose that $$f(1)=m$$. Answer the following:

Pendulum Oscillations and Trigonometric Limits

A pendulum’s angular displacement from the vertical is given by $$\theta(t)= \frac{\sin(2*t)}{t}$$ f

Pond Ecosystem Nutrient Levels

In a pond ecosystem, nutrient input occurs from periodic runoff events. Each runoff adds 20 kg of nu

Related Rates: Changing Shadow Length

A streetlight is mounted at the top of a 12 m tall pole. A person 1.8 m tall walks away from the pol

Removable Discontinuity in a Cubic Function

Consider the function $$f(x)=\frac{x^3-8}{x-2}$$ defined for $$x\neq2$$. Answer the following:

Removable Discontinuity in a Trigonometric Function

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

Saturation of Drug Concentration in Blood

A patient is given a drug with each dose containing 50 mg. However, due to metabolism, only 20% of t

Temperature Change Analysis

The function $$T(t)$$ represents the temperature (in $$^\circ C$$) in a chemical reactor as a functi

Water Tank Flow Analysis

A water tank receives water from an inlet and drains water through an outlet. The inflow rate is giv

Analyzing Motion Through Derivatives

A car’s position as a function of time is given by $$s(t) = 4*t^3 - 12*t^2 + 9*t + 5,$$ where $$s

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

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 from the Limit Definition: Function $$f(x)=\sqrt{x+2}$$

Consider the function $$f(x)=\sqrt{x+2}$$ for $$x \ge -2$$. Using the limit definition of the deriva

Differentiation in Biological Growth Models

In a biological experiment, the rate of resource consumption is modeled by $$R(t)=\frac{\ln(t^2+1)}{

Exponential Population Growth in Ecology

A certain species in a reserve is observed to grow according to the function $$P(t)=1000*e^{0.05*t}$

Finding and Interpreting Critical Points and Derivatives

Examine the function $$f(x)=x^3-9*x+6$$. Determine its derivative and analyze its critical points.

Finding the Derivative of a Logarithmic Function

Consider the function $$g(x)=\ln(3*x+1)$$. Answer the following:

Heat Transfer in a Rod: Modeling and Differentiation

The temperature distribution along a rod is given by $$T(x)= 100 - 2x^2 + 0.5x^3$$, where x is in me

Implicit Differentiation and Tangent Line Slope

Consider the curve defined by $$x^2 + x*y + y^2 = 7$$. Answer the following:

Implicit Differentiation in Circular Motion

A particle moves along the circle defined by $$x^2 + y^2 = 25$$. Answer the following parts.

Investment Return Rates: Continuous vs. Discrete Comparison

An investment's value grows continuously according to $$V(t)= 5000e^{0.07t}$$, where t is in years.

Maclaurin Series for e^x Approximation

Consider the function $$f(x)=e^x$$, which models many growth processes in nature. Use its Maclaurin

Population Growth Approximation using Taylor Series

A biologist models population growth with the exponential function $$P(t)=e^{0.05*t}$$. To estimate

Rate Function Involving Logarithms

Consider the function $$h(x)=\ln(x+3)$$.

Rate of Change Analysis in a Temperature Model

A temperature model is given by $$T(t)=25+4*t-0.5*t^2$$, where $$t$$ is time in hours. Analyze the t

Rational Function Derivative Using Quotient Rule

Consider the function $$g(x)=\frac{5*x-7}{x+2}$$. Find its derivative and analyze its critical featu

Related Rates in a Conical Tank

Water is draining from a conical tank. The tank has a total height of 10 m and its radius is always

Secant and Tangent Lines: Analysis of Rate of Change

Consider the function $$f(x)=x^3-6*x^2+9*x+1$$. This function represents a model of a certain proces

Secants and Tangents in Profit Function

A firm’s profit is modeled by the quadratic function $$f(x)=-x^2+6*x-8$$, where $$x$$ (in thousands)

Second Derivative and Concavity

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

Tangent Line Approximation

Consider the function $$f(x)=\cos(x)$$. Answer the following:

Tangent Line Approximation for a Parabolic Arch

Engineers design a parabolic arch described by $$y(x)= -0.5*x^2 + 4*x$$.

Tracking a Car's Velocity

A car moves along a straight road according to the position function $$s(t)=t^3 - 6*t^2 + 9*t + 2$$,

Using Taylor Series to Approximate the Derivative of sin(x²)

A physicist is analyzing the function $$f(x)=\sin(x^2)$$ and requires an approximation for its deriv

Vibration Analysis: Rate of Change in Oscillatory Motion

The displacement of a vibrating string is given by $$f(t)= 3\sin(4t) - 2\cos(4t)$$, where t is in se

Widget Production Rate

A widget manufacturing plant produces widgets according to the function $$P(t)=4*t^2 - 3*t + 10$$ wh

Analyzing a Composite Function from a Changing Systems Model

The displacement of an object is given by $$s(t) = \sqrt{t^2 + 4*t + 5}$$ (in meters), where $$t$$ i

Biological Growth Model Differentiation

In a biological model, the concentration of a chemical is modeled by $$C(t)=e^{-0.5*t}+\ln(2*t+3)$$.

Calculating an Inverse Trigonometric Derivative in a Physics Context

A pendulum's angle is modeled by $$\theta = \arcsin(0.5*t)$$, where $$t$$ is time in seconds and $$\

Chain Rule in Economic Utility Functions

A consumer's utility function is given by $$U(x,y)=\sqrt{x+y^2}$$, where x and y represent quantitie

Chain Rule with Exponential Function

Consider the function $$h(x)= e^{\sin(4*x)}$$ which models a process with exponential growth modulat

Composite Function with Exponential and Radical

Consider the function $$ f(x)= \sqrt{e^{5*x}+x^2} $$.

Composite Population Growth Function

A population model is given by $$P(t)= e^{3\sqrt{t+1}}$$, where $$t$$ is measured in years. Analyze

Differentiation in an Economic Cost Function

The cost of producing $$q$$ units is modeled by $$C(q)= (5*q)^{3/2} + 200*\ln(1+q)$$. Differentiate

Fuel Tank Dynamics

A fuel storage tank is being filled by a pump at a rate given by the composite function $$P(t)=(4*t+

Higher Order Implicit Differentiation in a Nonlinear Model

Assume that \(x\) and \(y\) are related by the nonlinear equation $$e^{x*y} + x - \ln(y) = 5$$ with

Ice Cream Storage Dynamics

An ice cream storage facility receives ice cream at a rate given by the composite function $$I(t)=d(

Implicit Differentiation in a Non-Standard Function

Consider the equation $$x^2*y + \sin(y) = x$$, which implicitly defines $$y$$ as a function of $$x$$

Implicit Differentiation in a Pressure-Temperature Experiment

In a chemistry experiment, the pressure $$P$$ (in atm) and temperature $$T$$ (in °C) of a system sat

Implicit Differentiation in Economic Equilibrium

In a market, the relationship between the price $$x$$ (in dollars) and the demand $$y$$ (in thousand

Implicit Differentiation with Logarithmic Functions

Let $$x$$ and $$y$$ be related by the equation $$\ln(x*y) + x - y = 0$$.

Implicit Differentiation with Product and Chain Rule in a Thermal Expansion Model

A material's length $$L$$ (in meters) under thermal expansion satisfies the equation $$L - \sin(L *

Inverse Analysis of an Exponential-Linear Function

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

Inverse Analysis of Cubic Plus Linear Function

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

Inverse Function Derivative in an Exponential Model

Let $$f(x)= e^{2*x} + x$$. Given that $$f$$ is one-to-one and differentiable, answer the following p

Inverse Function Differentiation for a Trigonometric-Polynomial Function

Let $$f(x)= \sin(x) + x^2$$ be defined on the interval $$[0, \pi/2]$$ so that it is invertible, with

Inverse Function Differentiation in a Sensor

A sensor produces a reading described by the function $$f(t)= \ln(t+1) + t^2$$, where $$t$$ is in se

Inverse Function Differentiation with Composite Trigonometric Functions

Let $$f(x)= \sin(2*x) + x$$, which is differentiable and one-to-one. It is given that $$f(\pi/6)= 1$

Inverse Trigonometric Differentiation

Differentiate the function $$ y= \arctan\left(\frac{2*x}{1-x}\right) $$.

Investigating the Inverse of a Rational Function

Consider the function $$f(x)=\frac{2*x-1}{x+3}$$ with $$x \neq -3$$. Analyze its inverse.

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

Navigation on a Curved Path: Boat's Eastward Velocity

A boat's location in polar coordinates is described by $$r(t)= \sqrt{4*t+1}$$ and its direction by $

Revenue Model and Inverse Analysis

A company's revenue (in millions) is modeled by $$R(x)=\sqrt{x+4}+3$$, where x represents production

Water Tank Composite Rate Analysis

A water tank receives water from an inflow pipe where the inflow rate is given by the composite func

Approximating Function Values Using Linearization

Consider the function $$f(x)=x^4$$. Use linearization at x = 4 to approximate the value of $$f(3.98)

Chemical Reaction Temperature Change

In a laboratory experiment, the temperature T (in °C) of a reacting mixture is modeled by $$T(t)=20+

Differentiation and Concavity for a Non-Motion Problem: Water Filling a Tank

The volume of water in a tank is given by $$V(t)=4*t^3-12*t^2+9*t+15$$, where $$V$$ is in gallons an

Draining Conical Tank

Water is draining from a conical tank at a rate of $$5$$ m³/min. The tank has a height of $$10$$ m a

Horizontal Tangents on Cubic Curve

Consider the curve defined by $$x^3 + y^3 - 6*x*y = 0$$.

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

Interpreting Position Graphs: From Position to Velocity

A graph of position (in meters) versus time (in seconds) is provided in the stimulus. The graph show

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 for Approximating Function Values

Let $$f(x)= \sqrt{x}$$. Use linearization at $$x=10$$ to approximate $$\sqrt{10.1}$$. Answer the fol

Linearization of Trigonometric Implicit Function

Consider the implicit equation $$\tan(x + y) = x - y$$, which implicitly defines $$y$$ as a function

Logistic Population Model Inversion

Consider the logistic population model given by $$f(t)=\frac{50}{1+e^{-0.3*(t-5)}}$$. This function

Maximizing the Area of an Inscribed Rectangle

A rectangle is inscribed in a semicircle of radius $$R$$, with its base along the diameter. The rect

Optimizing a Cylindrical Can Design

A manufacturer wants to design a cylindrical can with a fixed surface area of $$600\pi$$ cm² in orde

Ozone Layer Recovery Simulation

In a simulation of ozone layer dynamics, ozone is produced at a rate of $$I(t)=\frac{25}{t+1}$$ (Dob

Particle Motion with Changing Acceleration

A particle moves along a straight line with an acceleration given by $$a(t)=8-3*t$$ (in m/s²). Given

Particle Motion with Measured Positions

A particle moves along a straight line with its velocity given by $$v(t)=4*t-1$$ for $$t \ge 0$$ (in

Polar Curve: Slope of the Tangent Line

Consider the polar curve defined by $$r(\theta)=10e^{-0.1*\theta}$$.

Quadratic Function Inversion with Domain Restriction

Let $$f(x)=x^2+4$$. Since quadratic functions are not one-to-one over all real numbers, consider an

Rational Function Particle Motion Analysis

A particle moves along a straight line with its position given by $$s(t)=\frac{t^2+1}{t-1}$$, where

Revenue Function and Marginal Revenue

A company’s revenue (in thousands of dollars) is modeled as a function of units sold (in thousands)

Seasonal Reservoir Dynamics

A reservoir receives water inflow influenced by seasonal variations, modeled by $$I(t)=50+30\sin\Big

Spherical Balloon Inflation

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

Analysis of a Decay Model with Constant Input

Consider the concentration function $$C(t)= 30\,e^{-0.5t} + \ln(t+1)$$, where t is measured in hours

Analysis of a Quartic Function as a Perfect Power

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

Car Motion: Velocity and Total Distance

A car’s position along a straight road is modeled by $$s(t) = t^3 - 6*t^2 + 9*t + 15$$ (in meters),

Combining Series and Integration to Analyze a Population Model

A population's growth rate is approximated by the series $$P'(t)=\sum_{n=0}^\infty \frac{t^n}{(n+1)!

Concavity and Inflection Points in Particle Motion

Consider the position function of a particle $$s(x)=x^3-6*x^2+9*x+2$$.

Derivative Analysis of a Rational Function

Consider the function $$s(x)=\frac{x}{x^2+1}$$. Answer the following parts:

Determining Absolute Extrema in a Motion Context

A particle’s position is modeled by $$s(t)=-t^3+6*t^2-9*t+2$$, where $$t\in[0,5]$$ seconds.

Drug Dosage Infusion

A patient receives an intravenous drug infusion at a rate given by $$D(t)=4*\exp(-0.2*t)$$ mg/min. A

Echoes in an Auditorium

In an auditorium, an audio signal produces echoes. The first echo has an intensity that is 70% of th

Economic Optimization: Maximizing Profit

The profit function for a product is given by $$P(x) = -2*x^3 + 27*x^2 - 108*x + 150$$, where \(x\)

Expanding Oil Spill - Related Rates

A circular oil spill is expanding such that its area is given by $$A(t) = \pi*[r(t)]^2$$. The radius

Extreme Value Theorem in Temperature Variation

A metal rod’s temperature (in °C) along its length is modeled by the function $$T(x) = -2*x^3 + 12*x

Fractal Tree Branch Lengths

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

Inverse Analysis for a Function with Multiple Transformations

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

Maximizing the Area of an Inscribed Rectangle

A rectangle is inscribed in a circle of radius $$5$$. Determine the dimensions of the rectangle that

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

Mean Value Theorem in River Flow

A river cross‐section’s depth (in meters) is modeled by the function $$f(x) = x^3 - 4*x^2 + 3*x + 5$

Mean Value Theorem on a Quadratic Function

Consider the function $$f(x)=x^2-4*x+3$$ defined on the closed interval $$[1, 5]$$. Verify that the

Motion with a Piecewise-Defined Velocity Function

A particle travels along a line with a piecewise velocity function defined by $$ v(t)=\begin{cases}

Optimization in Particle Routing

A delivery robot’s distance along its predetermined path is described by $$s(t)=\sin(t)+0.5*t$$, whe

Optimizing Fencing for a Rectangular Garden

A homeowner plans to build a rectangular garden adjacent to a river (so the side along the river nee

Particle Motion on a Curve

A particle moves along a straight-line path with its position given by \( s(t)=t^3 - 6*t^2 + 9*t + 1

Pharmaceutical Dosage and Metabolism

A patient is administered a medication with an initial dose of 50 mg. Due to metabolism, the amount

Population Growth vs. Harvest Model

A fish population in a lake grows naturally at a rate given by $$G(t)=\frac{50}{1+t}$$ (in fish/mont

Radius of Convergence and Series Manipulation in Substitution

Let $$f(x)=\sum_{n=0}^\infty c_n * (x-2)^n$$ be a power series with radius of convergence $$R = 4$$.

Relative Extrema Using the First Derivative Test

Consider the function $$ f(x)=e^{-x^2}.$$ Answer the following parts:

Retirement Savings with Diminishing Deposits

Alex starts a retirement savings plan where the deposit each month forms a geometric sequence. In th

Revenue Optimization in Business

A company’s price-demand function is given by $$P(x)= 50 - 0.5*x$$, where $$x$$ is the number of uni

River Sediment Transport

Sediment enters a river from a landslide at a rate of $$S_{in}(t)=4*\exp(0.2*t)$$ tonnes/day and is

Rolle's Theorem: Modeling a Car's Journey

An object moves along a straight line and its position is given by $$s(t)= t^3-6*t^2+9*t$$ for $$t$$

Salt Tank Mixing Problem

In a mixing tank, salt is added at a constant rate of $$A(t)=10$$ grams/min while the salt solution

Second Derivative Test for Critical Points

Consider the function $$f(x)=x^3-9*x^2+24*x-16$$.

Series Convergence and Differentiation in Thermodynamics

In a thermodynamic process, the temperature $$T(x)=\sum_{n=0}^\infty \frac{(-2)^n}{n+1} * (x-5)^n$$

Temperature Variations

The daily temperature of a city (in °C) is recorded at various times during the day. Use the tempera

Water Tank Rate of Change

The volume of water in a tank is modeled by $$V(t)= t^3 - 6*t^2 + 9*t$$ (in cubic meters), where $$t

Accumulation Function in an Investment Model

An investment has an instantaneous rate of return given by $$r(t)=0.05*t+0.02$$ (per year). The accu

Approximating Energy Consumption Using Riemann Sums

A household’s power consumption (in kW) is recorded over an 8‐hour period. The following table shows

Area Between the Curves: Linear and Quadratic Functions

Consider the curves $$y = 2*t$$ and $$y = t^2$$. Answer the following parts to find the area of th

Average Value and Accumulated Change

For the function $$f(x)= x^2+1$$ defined on the interval [0, 4], find the average value of the funct

Definite Integral Evaluation via the Fundamental Theorem of Calculus

Let the function be $$f(x)=3*x^2+2*x$$. Evaluate the definite integral from $$x=1$$ to $$x=4$$.

Evaluating a Piecewise Function with a Removable Discontinuity

Consider the function $$ f(x)= \begin{cases} \frac{x^2-4}{x-2} & \text{if } x \neq 2,\\ 3 & \text{if

Fuel Consumption Estimation with Midpoint Riemann Sums

A vehicle’s fuel consumption rate (in liters per hour) over a trip is recorded at various times. The

Graphical Transformations and Inverse Functions

Consider the linear function $$f(x)= \frac{1}{2}*x + 5$$ defined for all real $$x$$. Answer the foll

Improper Integral Convergence

Examine the convergence of the improper integral $$\int_1^\infty \frac{1}{x^p}\,dx$$.

Integration of a Trigonometric Function by Two Methods

Evaluate the definite integral $$\int_0^{\frac{\pi}{2}} \sin(x)*\cos(x)\,dx$$ using two different me

Integration Using U-Substitution

Evaluate the indefinite integral $$\int (4*x+2)^5\,dx$$ using u-substitution.

Logarithmic Functions in Ecosystem Models

Let \(f(t)= \ln(t+2)\) for \(t \ge 0\) model an ecosystem measurement. Answer the following question

Marginal Cost and Production

A factory's marginal cost function is given by $$MC(x)= 4 - 0.1*x$$ dollars per unit, where $$x$$ is

Modeling a Car's Journey with a Time-Dependent Velocity

A car's velocity is modeled by $$ v(t)= \begin{cases} 4t, & 0 \le t < 3, \\ 12, & 3 \le t \le 5, \en

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)

Particle Displacement and Total Distance

A particle moves along a straight line with a velocity function $$v(t)=3*t^2-2*t+1$$ for $$0\le t\le

Population Model Inversion and Accumulation

Consider the logistic-type function $$f(t)= \frac{8}{1+e^{-t}}$$, representing a population model, d

Region Bounded by a Parabola and a Line: Area and Volume

Consider the region bounded by the curves $$y=x^{2}$$ and $$y=2*x+3$$. Answer the following:

Related Rates: Expanding Circular Ripple

A stone is dropped into a still pond, producing a circular ripple. The radius $$r$$ of the ripple (i

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)

Scooter Motion with Variable Acceleration

A scooter's acceleration is given by $$a(t)= 2*t - 1$$ (m/s²) for $$t \in [0,5]$$, with an initial v

Bacteria Growth with Antibiotic Treatment

A bacterial culture has a population $$N(t)$$ that grows at a rate proportional to its size, given b

Coffee Cooling: Differential Equation Application

A cup of coffee is cooling according to Newton's Law of Cooling. The following table provides measur

Compound Interest and Investment Growth

An investment account grows according to the differential equation $$\frac{dA}{dt}=r*A$$, where the

Constructing and Interpreting a Slope Field

Consider the differential equation $$\frac{dy}{dx} = \sin(x) - y$$. Answer the following:

Epidemic Spread Modeling

An epidemic in a closed population of $$N=10000$$ individuals is modeled by the logistic equation $$

Estimating Total Change from a Rate Table

A car's velocity (in m/s) is recorded at various times according to the table below:

Euler's Method Approximation

Let the differential equation be $$\frac{dy}{dx} = x+y$$ with the initial condition $$y(0)=1$$. Usin

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 and Decay

A bacterial population grows according to the differential equation $$\frac{dy}{dt}=k\,y$$ with an i

Forced Oscillation in a Damped System

Consider the differential equation $$\frac{dx}{dt}=-0.2*x+\sin(t)$$ with initial condition $$x(0)=1$

FRQ 1: Slope Field Analysis for $$\frac{dy}{dx}=x$$

Consider the differential equation $$\frac{dy}{dx}=x$$. Answer the following parts.

FRQ 10: Cooling of a Metal Rod

A metal rod cools in a room according to Newton's Law of Cooling: $$\frac{dT}{dt} = -k (T - A)$$. Th

FRQ 13: Cooling of a Planetary Atmosphere

A planetary atmosphere cools according to Newton's Law of Cooling: $$\frac{dT}{dt}=-k(T-T_{eq})$$, w

FRQ 15: Cooling of a Beverage in a Fridge

A beverage cools according to Newton's Law of Cooling, described by $$\frac{dT}{dt}=-k(T-A)$$, where

Implicit Solution of a Separable Differential Equation

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

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

Logistic Growth: Time to Half-Capacity

Consider a logistic population model governed by the differential equation $$\frac{dP}{dt}=kP\left(1

Mixing Problem in a Tank

A tank initially contains 50 liters of pure water. A brine solution with a salt concentration of $$3

Mixing Problem in a Water Tank

A tank initially contains $$100$$ liters of saltwater with $$5$$ kg of salt dissolved in it. Pure wa

Mixing Problem with Time-Dependent Inflow

A tank initially contains $$100$$ L of salt water with a salt concentration of $$0.5$$ kg/L. Pure wa

Mixing Problem: Salt Water Tank

A tank initially contains $$1000$$ liters of pure water with $$50$$ kg of salt dissolved in it. Brin

Newton's Law of Cooling

A hot liquid is cooling in a room. The temperature $$T(t)$$ (in degrees Celsius) of the liquid at ti

Radioactive Decay with External Source

A radioactive substance decays while receiving a constant external activation. Its behavior is model

Rainfall in a Basin: Differential Equation Model

During a rainstorm, the depth of water $$h(t)$$ (in centimeters) in a basin is modeled by the differ

Second-Order Differential Equation in a Mass-Spring System

A mass-spring system without damping is modeled by the differential equation $$m\frac{d^2x}{dt^2}+kx

Separable DE with Exponential Function

Solve the differential equation $$\frac{dy}{dx}=y\cdot\ln(y)$$ for y > 0 given the initial condition

Separable DE with Trigonometric Component

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

Separable Differential Equation and Maclaurin Series Approximation

Consider the differential equation $$\frac{dy}{dx} = e^{x} * \sin(y)$$ with the initial condition $$

Separable Differential Equation with Absolute Values

Consider the differential equation $$\frac{dy}{dx} = \frac{|x|}{y}$$ with the condition that $$y>0$$

Slope Field and Equilibrium Analysis for $$\frac{dy}{dx}= y*(1-y)$$

Consider the autonomous differential equation $$\frac{dy}{dx}= y*(1-y)$$. Answer the following:

Solving a Linear Differential Equation using an Integrating Factor

Consider the linear differential equation $$\frac{dy}{dx} + \frac{2}{x} * y = \frac{\sin(x)}{x}$$ wi

Solving a Nonlinear Differential Equation by Separation

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

Water Pollution with Seasonal Variation

A river receives a pollutant with a time-varying influx modeled by $$I(t)=20+5\cos(0.5*t)$$ kg/day,

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 and Average Speed for a Parametric Curve

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

Arc Length of a Suspension Cable

A suspension bridge uses a cable that hangs along a curve modeled by $$y=100+\frac{1}{50}x^2$$ for $

Area Between Curves in a Business Context

A company’s revenue and cost (in dollars) for producing items (in hundreds) are modeled by the funct

Area Between Curves in a Physical Context

The heights of two particles moving along parallel tracks are given by $$h_1(t)=t^2$$ and $$h_2(t)=4

Area Under a Curve with a Discontinuity

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

Average Car Speed Analysis from Discrete Data

A car's speed (in km/h) is recorded at equal time intervals over a 1-hour journey. Analyze the car's

Average Population Density on a Road

A town's population density along a road is modeled by the function $$P(x)=50*e^{-0.1*x}$$ (persons

Average Power Consumption

A household's power consumption is modeled by the function $$P(t)=3+2*\sin\left(\frac{\pi}{12}*t\rig

Average Reaction Concentration in a Chemical Reaction

In a chemical reaction, the concentration of a reactant is modeled by $$C(t)=20*\exp(-0.5*t)$$ (in m

Average Temperature Analysis

A meteorological station recorded the temperature in a region as a function of time given by $$T(t)

Average Temperature of a Day

In a certain city, the temperature during the day is modeled by a continuous function $$T(t)$$ for $

Average Value of a Velocity Function

The velocity of a car is modeled by $$v(t)=3*t^2-12*t+9$$ (m/s) for $$t\in[0,5]$$ seconds. Answer th

Bonus Payout: Geometric Series vs. Integral Approximation

A company issues monthly bonuses that decrease by 20% each month. The bonus in the first month is $5

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

Center of Mass of a Thin Rod

A thin rod extends from $$x=0$$ to $$x=4$$ m and has a density function $$\lambda(x)=1+\frac{\ln(x+2

Environmental Contaminant Spread Analysis

A contaminant enters a lake at a rate given by $$r(t)=4e^{-0.5*t}$$ kilograms per day, where $$t$$ i

Error Analysis in Taylor Polynomial Approximations

Let $$h(x)= \cos(3*x)$$. Analyze the error involved when approximating $$h(x)$$ by its third-degree

Mass of a Wire with Variable Density

A thin wire lies along the curve $$y= \sqrt{x}$$ for $$0 \le x \le 4$$. The wire has a linear densit

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

Net Change and Direction of Motion

A particle’s velocity is given by $$v(t)=\sin(t)-\frac{1}{2}*t$$ for $$0 \le t \le 6$$.

Series and Integration Combined: Error Bound in Integration

Consider the integral $$\int_{0}^{0.5} \frac{1}{1+x^2} dx$$. Use the Taylor series expansion of the

Total Distance Traveled with Changing Velocity

A runner’s velocity is given by $$v(t)=3*(t-1)*(t-4)$$ m/s for $$0 \le t \le 5$$ seconds. Note that

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 Wavy Dome

An auditorium roof has a varying cross-sectional area given by $$A(x)=\pi*(1 + 0.1*\sin(x))^2$$ (in

Volume Using Washer Method

Consider the region bounded by the curves $$y=\sqrt{x}$$ and $$y=\frac{x}{2}$$. This region is rotat

Work Done by a Variable Force

A force acting on an object is given by $$F(x)=5+3*x$$ (in newtons), where $$x$$ is the displacement

Work Done by a Variable Force

A force acting on an object along a displacement is given by $$F(x)=3*x^2 -2*x+1$$ (in Newtons), whe

Work Done with a Discontinuous Force Function

A force acting on an object is defined piecewise by $$F(x)=\begin{cases} x^2, & 0 \le x < 4,\\ 16, &

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 Parametric Curve with Logarithms

Consider the curve defined parametrically by $$x(t)=\ln(t+1)$$ and $$y(t)=\sqrt{t}$$ for $$0 \le t \

Comparing Arc Lengths in Parametric and Polar Systems

Consider the curve given in parametric form by $$x(t)=\cos(2*t)$$ and $$y(t)=\sin(2*t)$$ for $$0\le

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

Conversion of Polar to Parametric Form

A particle’s motion is given in polar form by the equations $$r = 4$$ and $$\theta = \sqrt{t}$$ wher

Curvature of a Space Curve

Let the space curve be defined by $$r(t)= \langle t, t^2, \ln(t+1) \rangle$$ for $$t > -1$$.

Differentiability of a Piecewise-Defined Vector Function

Consider the vector-valued function $$\textbf{r}(t)= \begin{cases} \langle t, t^2 \rangle & \text{i

Helical Motion with Damping

A particle moves along a helical path with damping, described by the vector function $$\vec{r}(t)= \

Intersections in Polar Coordinates

Two polar curves are given by $$r = 3 - 2*\sin(\theta)$$ and $$r = 1 + \cos(\theta)$$.

Motion Along a Helix

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

Optimization of Walkway Slope with Fixed Arc Length

A walkway is designed with its shape given by the parametric equations $$x(t)= t$$ and $$y(t)= c*t*(

Optimization on a Parametric Curve

A curve is described by the parametric equations $$x(t)= e^{t}$$ and $$y(t)= t - e^{t}$$.

Parametric Motion and Change of Direction

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

Parametric Oscillations and Envelopes

Consider the family of curves defined by the parametric equations $$x(t)=t$$ and $$y(t)=e^{-t}\sin(k

Polar and Parametric Form Conversion

A curve is given in polar form by $$r(\theta)=\frac{2}{1+\cos(\theta)}$$. This curve represents a co

Polar to Cartesian Conversion and Tangent Slope

Consider the polar curve $$r=2*(1+\cos(\theta))$$. Answer the following parts.

Projectile Motion with Parametric Equations

A ball is launched from ground level with an initial speed of $$20 \text{ m/s}$$ at an angle of $$\f

Vector-Valued Kinematics

A particle follows a path in space described by the vector-valued function $$r(t) = \langle \cos(t),

Work Done by a Force along a Vector Path

A force field is given by $$\mathbf{F}(t)=\langle2*t,\;3\sin(t)\rangle$$ and an object moves along a