AP Calculus BC FRQ Room

Algebraic Manipulation in Limit Computations

Let $$s(x)=\frac{x^3-8}{x-2}.$$ Answer the following:

Analysis of a Piecewise Function with Multiple Definitions

Consider the function $$h(x)=\begin{cases} \frac{x^2-9}{x-3} & \text{if } x<3, \\ 2*x-1 & \text{if

Application of the Squeeze Theorem with Trigonometric Oscillations

Consider the function $$f(x)= x^2*\sin(1/x)$$ for $$x \neq 0$$ and $$f(0)=0$$. Answer the following

Approximating Limits Using Tabulated Values

The function g(x) is sampled near x = 2 and the following values are recorded: | x | g(x) | |--

Bacterial Growth Experiment

A laboratory experiment involves a bacterial culture whose population at hour $$n$$ is modeled by a

Composite Function and Continuity

Consider the piecewise function $$ g(x)=\begin{cases} x^2 & \text{if } x<2, \\ 3x-2 & \text{if } x\

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

Composite Functions: Limits and Continuity

Let $$f(x)=x^2-1$$, which is continuous for all $$x$$, and let $$g(x)=f(\sqrt{x+1})$$.

Continuity Analysis Using a Piecewise Defined Function

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

Continuity Conditions for a Piecewise-Defined Function

Consider the function defined by $$ f(x)= \begin{cases} 2*x+1, & x < 3 \\ ax^2+ b, & x \ge 3 \end{c

Evaluating Limits Involving Absolute Value Functions

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

Evaluating Limits Involving Exponential and Rational Functions

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

Factorization and Limits

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

Graphical Analysis of a Removable Discontinuity

Consider the function $$f(x)=\frac{x^2-1}{x-1}$$ for x \neq 1, with a defined value of f(1) = 3. Ans

Graphical Analysis of Volume with a Jump Discontinuity

A graph of the water volume \(V(t)\) in a tank shows a jump discontinuity at \(t=6\) minutes. Answer

Graphical Analysis of Water Tank Volume

The water volume in a tank over time is recorded and displayed in the graph provided. Due to a senso

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

Infinite Limits and Vertical Asymptotes

Let $$g(x)=\frac{1}{(x-2)^2}$$. Answer the following:

Limit Evaluation Involving Radicals and Rationalization

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

Limits and Absolute Value Functions

Examine the function $$f(x)= \frac{|x-3|}{x-3}$$ defined for $$x \neq 3$$.

Limits and Removable Discontinuity in Rational Functions

Consider the rational function $$g(x) = \frac{(x-2)(x+3)}{x-2}.$$ Use this expression to answer the

Logarithmic Function Limits

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

Modeling with a Removable Discontinuity

A water reservoir’s inflow rate is modeled by the function $$R(t)=\frac{t^2-16}{t-4}$$ liters per mi

Parameter Determination for Continuity

Let $$h(x)= \begin{cases} \frac{e^{2x} - 1 - a\,\ln(1+bx)}{x} & x \neq 0 \\ c & x = 0 \end{cases}.$$

Piecewise Function Continuity

Consider the function defined by $$ f(x)=\begin{cases} \frac{x^2-4}{x-2}, & x\neq2\\ k, & x=2 \en

Squeeze Theorem with an Oscillating Function

Let $$f(x)=x * \cos(\frac{1}{x})$$ for $$x \neq 0$$, and define $$f(0)=0$$. Answer the following:

Squeeze Theorem with Oscillatory Behavior

Examine the function $$s(x)=x^2*\sin(1/x)$$ for x ≠ 0.

Analysis of a Quadratic Function

Consider the function $$f(x)=3*x^2 - 2*x + 5$$. Using the limit definition of the derivative, answer

Analysis of Higher-Order Derivatives

Let $$f(x)=x*e^{-x}$$ model the concentration of a substance over time. Analyze both the first and s

Analyzing a Removable Discontinuity in a Rational Function

Consider the function defined by $$f(x)= \begin{cases} \frac{x^2-1}{x-1} & x \neq 1 \\ 3 & x = 1 \e

Circular Motion Analysis

An object moves along a circular path with angular position given by $$\theta(t)=2*t-0.1*t^2$$ (in r

Cooling Tank System

A laboratory cooling tank has heat entering at a rate of $$H_{in}(t)=200-10*t$$ Joules per minute an

Epidemiological Rate Change Analysis

In epidemiology, the spread of a disease is sometimes modeled by a combination of logarithmic and ex

Exploration of Derivative Notation and Higher Order Derivatives

Given the function $$f(x)=x^2*e^x$$, analyze its derivatives.

Graphical Derivative Analysis

A series of experiments produced the following data for a function $$f(x)$$:

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 in Circular Motion

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

Implicit Differentiation in Logarithmic Equations

Consider the relation given by $$x*\ln(y)+y*\ln(x)=5$$, where $$x>0$$ and $$y>0$$.

Implicit Differentiation of a Circle

Given the equation of a circle $$x^2 + y^2 = 25$$,

Implicit Differentiation with Inverse Functions

Suppose a differentiable function $$f$$ satisfies the equation $$f(x) + f^(-1)(x) = 2*x$$ for all x

Instantaneous Rate of Change of a Trigonometric Function

Consider the function $$h(t)=\sin(2*t) + \cos(t)$$ which models the displacement (in centimeters) of

Irrigation Reservoir Analysis

An irrigation reservoir has an inflow rate modeled by $$I(t)=\frac{100}{1+e^{-0.5*(t-10)}}$$ liters

Plant Growth Rate Analysis

A plant’s height (in centimeters) after $$t$$ days is modeled by $$h(t)=0.5*t^3 - 2*t^2 + 3*t + 10$$

Population of a Colony: Sum and Derivative Analysis

A colony of cells grows such that the number of cells on the nth day is given by $$a_n= 100(1.2)^{n-

Radioactive Decay and Derivative

A radioactive substance decays according to $$M(t)= 50e^{-0.03t}$$, where t is in years and M(t) is

Related Rates: Draining Conical Tank

Water is draining from an inverted conical tank with a height of 6 m and a top radius of 3 m. The vo

Revenue Change Analysis via the Product Rule

A company’s revenue (in thousands of dollars) is modeled by $$R(x) = (2*x + 3)*(x^2 - x + 4)$$, wher

Savings Account Growth: From Discrete Deposits to Continuous Derivatives

An individual deposits $$P$$ dollars at the beginning of each month into an account that earns a con

Secant vs. Tangent: Approximation and Limit Approach

Consider the function $$f(x)= \sqrt{x}$$. Use both a secant line approximation and the limit definit

Second Derivative and Concavity Analysis

Consider the function $$f(x)=x^3-6*x^2+12*x-5$$. Answer the following:

Second Derivative of a Composite Function

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

Sediment Accumulation in a Dam

Sediment enters a dam reservoir at a rate of $$S_{in}(t)=5\ln(t+1)$$ kg/hour, while sediment is remo

Tangent and Normal Lines

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

Taylor Series Expansion of ln(x) About x = 2

For a financial model, the function $$f(x)=\ln(x)$$ is expanded about $$x=2$$. Use this expansion to

Analyzing a Composite Function with Nested Radicals

Consider the function $$h(x)=\sqrt{1+\sqrt{2+3x}}$$. Answer the following parts:

Chain Rule with Exponential Function

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

Chemical Mixing: Implicit Relationships and Composite Rates

In a chemical mix tank, the solute amount (in grams) and the concentration (in mg/L) are related by

Composite Differentiation in Biological Growth

A biologist models the temperature $$T$$ (in °C) of a culture over time $$t$$ (in hours) by the func

Composite Function with Implicitly Defined Inner Function

Let the function $$h(x)$$ be defined implicitly by the equation $$h(x) - \ln(h(x)) = x$$, and consid

Composite Temperature Function and Its Second Derivative

A temperature profile is modeled by a composite function: $$T(t) = h(m(t))$$, where $$m(t)= 3*t^2 +

Differentiation of an Inverse Exponential Function

Let $$f(x)=e^{2*x}-7$$, and let g denote its inverse function. Answer the following parts.

Differentiation of an Inverse Trigonometric Composite Function

Let $$y = \arcsin(\sqrt{x})$$. Answer the following:

Financial Flow Analysis: Investment Rates

An investment fund experiences deposits at a rate modeled by the composite function $$D(t)=g(h(t))$$

Implicit Differentiation for an Elliptical Path

An ellipse is given by the equation $$4*x^2 + y^2 = 16$$. Answer the following parts.

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 of an Ellipse

Consider the ellipse defined by $$4*x^2+9*y^2=36$$. Use implicit differentiation to determine the sl

Implicit Differentiation on an Ellipse

Consider the ellipse defined by $$ 4*x^2+9*y^2=36 $$.

Implicit Differentiation with an Exponential Function

Given the equation $$ e^{x*y}= x+y $$, use implicit differentiation.

Implicit Differentiation: Circle and Tangent Line

The equation $$x^2 + y^2 = 25$$ represents a circle. Use implicit differentiation to find the deriva

Indoor Air Quality Control

In a controlled laboratory environment, the rate of fresh air introduction is modeled by the composi

Inverse Differentiation of a Trigonometric Function

Consider the function $$f(x)=\arctan(2*x)$$ defined for all real numbers. Analyze its inverse functi

Inverse Function Analysis for Exponential Functions

Let $$f(x)=e^{2*x}+1$$ and let g be the inverse function of f. Answer the following parts.

Inverse Function Derivative with Logarithms

Let $$f(x)= \ln(x+2) + x$$ with inverse function $$g(x)$$. Find the derivative $$g'(y)$$ in terms of

Inverse of a Composite Function

Let $$f(x)=\sqrt{3*x+1}$$ and $$g(x)=x^2-1$$, and define $$h(x)=f(g(x))$$. Analyze the invertibility

Logarithmic and Exponential Composite Function with Transformation

Let $$g(x)=\ln((3*x+1)^2)-e^{x}$$. Answer the following questions.

Optimization in Manufacturing Material

A manufacturer is designing a closed box with a square base of side length $$x$$ and height $$h$$ th

Related Rates in an Inflating Balloon

The volume of a spherical balloon is given by $$V= \frac{4}{3}*\pi*r^3$$, where r is the radius. Sup

Shadow Length and Related Rates

A 1.8 m tall person walks away from a 4 m tall lamppost at a speed of 1.2 m/s. Let $$x$$ be the dist

Temperature Modeling and Composite Functions

A weather balloon ascends and the temperature at altitude x (in kilometers) is modeled by $$T(x) = \

Approximating Changes with Differentials

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

Arc Length Calculation

Consider the curve $$y = \sqrt{x}$$ for $$x \in [1, 4]$$. Determine the arc length of the curve.

Boat Crossing a River: Relative Motion

A boat must cross a 100 m wide river. The boat's speed relative to the water is 5 m/s (directly acro

Circular Motion and Angular Rate

A point moves along a circle of radius 5 meters. Its angular position is given by $$\theta(t)=2*t^2-

Conical Tank Filling

A conical water tank has a height of $$10$$ m and a top radius of $$4$$ m. The water in the tank for

Differentials and Function Approximation

Consider the function $$f(x)=x^{1/3}$$. At $$x=8$$, answer the following parts.

Economic Optimization: Profit Maximization

A company's profit (in thousands of dollars) is modeled by $$P(x) = -2x^2 + 40x - 150$$, where $$x$$

Economics: Cost Function and Marginal Analysis

A company's cost function is given by $$C(x)= 0.5*x^3 - 4*x^2 + 10*x + 100$$, where $$x$$ represents

Estimation Error with Differentials

Let $$f(x)=x^3$$. Use differentials to estimate the value of $$f(2.05)$$ and determine the approxima

Evaluating Limits Using L'Hospital's Rule

Analyze the function $$f(x)=\frac{5x^3-4x^2+1}{7x^3+2x-6}$$ as $$x \to \infty$$. Using a calculator

Exponential Cooling Rate Analysis

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

Firework Trajectory Analysis

A firework is launched and its height (in meters) is modeled by the function $$h(t)=-4.9t^2+30t+5$$,

Implicit Differentiation: Tangent to a Circle

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

Ladder Sliding Down a Wall

A 10-meter ladder leans against a vertical wall and begins to slide. The bottom slides away from the

Linearization in Inverse Function Approximation

Let $$f(x)=x^5+2*x+1$$ be a one-to-one function. Although its inverse cannot be found explicitly, li

Minimum Time to Cross a River

A person must cross a 100-meter-wide river. They can swim at 2 m/s and run along the bank at 5 m/s.

Motion on a Straight Line with a Logarithmic Position Function

A particle moves along a straight line with its position given by $$s(t)=\ln(t+2)+t^2$$ (in meters),

Optimization in Related Rates: Expanding Circular Oil Spill

An oil spill spreads out on a water surface forming a circle. At a certain moment, the area of the s

Optimizing Area of a Rectangular Field

A farmer has 100 meters of fencing to enclose three sides of a rectangular field (the fourth side be

Polar Curve: Slope of the Tangent Line

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

Population Growth Rate Analysis

A population grows exponentially according to $$P(t)=1200e^{0.15t}$$, where t is measured in months.

Related Rates in a Circular Pool

A circular pool is being filled such that the surface area increases at a constant rate of $$10$$ ft

Revenue and Marginal Analysis

A company’s revenue function is given by $$R(p)= p*(1000 - 5*p)$$, where $$p$$ is the price per unit

Series Convergence and Approximation for f(x) Centered at x = 2

Consider the function $$f(x)= \sum_{n=0}^{\infty} \frac{(-1)^n (x-2)^{2*n}}{n+1}$$. Answer the follo

Series Identification and Approximation

Consider the series $$F(x)= \sum_{n=0}^{\infty} \frac{(-3)^n (x-1)^n}{n!}$$. Answer the following:

Surface Area of a Solid of Revolution

Consider the curve $$y = \ln(x)$$ for $$x \in [1, e]$$. Find the surface area of the solid formed by

Temperature Change in Coffee Cooling

The temperature of a cup of coffee is modeled by $$T(t)=70+50e^{-0.1*t}$$, where $$T(t)$$ is in °F a

Temperature Change of Coffee: Exponential Cooling

The temperature of a cup of coffee is modeled by the function $$x(t)= 70 + 50e^{-0.1*t}$$, where $$t

Trigonometric Implicit Relation

Consider the implicit equation $$\sin(x*y) + x - y = 0$$.

Volume Measurement Inversion

The volume of a sphere is given by $$f(x)=\frac{4}{3}*\pi*x^3$$, where $$x$$ is the radius. Analyze

Air Pollution Control in an Enclosed Space

In an enclosed environment, contaminated air enters at a rate of $$I(t)=15-\frac{t}{2}$$ m³/min and

Analysis of a Rational Function and Its Inverse

Consider the function $$f(x)= \frac{2*x+3}{x-1}$$ defined for $$x \neq 1$$. Answer the following par

Analysis of an Exponential Function

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

Analyzing The Behavior of a Log-Exponential Function Over a Specified Interval

Consider the function $$h(x)= \ln(x) + e^{-x}$$. A portion of its values is given in the following t

Application of the Mean Value Theorem

Consider the function $$f(t)=t^3-3*t^2+2*t+5$$ representing the position (in meters) of a car along

Area Between Curves and Rates of Change

An irrigation canal has a cross-sectional shape described by \( y=4-x^2 \) for \( |x| \le 2 \). The

Chemical Reaction Rate

During a chemical reaction, the concentration of a reagent (in M) is measured over time (in minutes)

Convergence and Series Approximation of a Simple Function

Consider the function defined by the power series $$f(x)=\sum_{n=0}^\infty \frac{(-1)^n}{n+1} * x^n$

Determining Convergence and Error Analysis in a Logarithmic Series

Investigate the series $$L(x)=\sum_{n=1}^\infty (-1)^{n+1} * \frac{(x-1)^n}{n}$$, which represents a

Extreme Value Analysis of a Logarithmic-Exponential-Polynomial Function

Examine the function $$f(x)= x^2\,e^{-x} + \ln(x)$$ defined for $$x > 0$$. Apply methods to find its

Instantaneous vs. Average Rates in a Real-World Model

A company’s monthly revenue is modeled by $$ R(t)=0.5t^3-4t^2+12t+100, \quad 0 \le t \le 6,$$ where

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.

Logistic Growth in Biology

The logistic growth of a species is modeled by $$P(t) = \frac{1}{1 + e^{-0.5*(t-4)}}$$, where t is i

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 with a Trigonometric Function

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

Optimal Timing via the Mean Value Theorem

A particle’s position is given by $$s(t)=t^2e^{-t}+3$$ for $$t\in[0,3]$$.

Optimization in Particle Routing

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

Optimization Problem: Designing a Box

A company needs to design an open-top box with a square base that has a volume of $$32\,m^3$$. The c

Population Growth Model Analysis

A population of organisms is modeled by the function $$P(t)= -2*t^2+20*t+50$$, where $$t$$ is measur

Population Growth Modeling

A region's population (in thousands) is recorded over a span of years. Use the data provided to anal

Profit Maximization in Business

A company’s profit function is given by $$P(x) = -2*x^3 + 30*x^2 - 100*x + 150$$, where x represents

Radiocarbon Dating in Artifacts

An archaeological artifact contains a radioactive isotope with an initial concentration of 100 units

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

Reservoir Sediment Accumulation

A reservoir accumulates sediment at a rate of $$S_{in}(t)=3*t$$ tonnes/day but also loses sediment v

Trigonometric Function and its Inverse

Consider the function $$f(x)= \sin(x) + x$$ defined on the interval $$[-\pi/2, \pi/2]$$. Answer the

Accumulated Displacement from Acceleration

A particle moving along a straight line has an acceleration of $$a(t)=6-4*t$$ (in m/s²), with an ini

Accumulated Population Change from a Growth Rate Function

A population changes at a rate given by $$P'(t)= 0.2*t^2 - 1$$ (in thousands per year) for t between

Antiderivatives and the Fundamental Theorem

Suppose a continuous function $$h(x)$$ is defined on [2, 8] and its graph (provided) shows that it i

Area Under a Parametric Curve

A curve is defined parametrically by $$x(t)=t^2$$ and $$y(t)=t^3-3*t$$ for $$t \in [-2,2]$$.

Average Value of a Function on an Interval

Let $$f(x)=\ln(x)$$ be defined on the interval $$[1,e]$$. Determine the average value of $$f(x)$$ on

Cost Accumulation from Marginal Cost Function

A company’s marginal cost function $$MC(q)$$ (in dollars per unit) for producing $$q$$ units is give

Error Bound Analysis for the Trapezoidal Rule

For the function $$f(x)=\ln(x)$$ on the interval $$[1,2]$$, the error bound for the trapezoidal rule

Evaluating a Complex Integral

Evaluate the integral $$\int_{0}^{2} 2*x*(x+1)^3\,dx$$ using an appropriate substitution method.

Heat Energy Accumulation

The rate of heat transfer into a container is given by $$H(t)= 5\sin(t)$$ kJ/min for $$t \in [0,\pi]

Improper Integral Evaluation

Evaluate the integral $$\int_{1}^{\infty} \frac{1}{t^2}\, dt$$ by answering the following parts.

Midpoint Riemann Sum Estimation

The function $$f(x)$$ is sampled at the following (possibly non-uniform) x-values provided in the ta

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

Numerical Approximation: Trapezoidal vs. Simpson’s Rule

The function $$f(x)=\frac{1}{1+x^2}$$ is to be integrated over the interval [-1, 1]. A table of valu

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 Increase from a Discontinuous Growth Rate

A sudden migration event alters the population growth rate. The growth rate (in individuals per year

Rainfall Accumulation Over Time

A storm produces rainfall at a rate modeled by the function $$r(t)=6 * t^(1/2)$$, where $$0 \le t \l

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:

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 Sums and Inverse Analysis from Tabular Data

Let $$f(x)= 2*x + 1$$ be defined on the interval $$[0,5]$$. Answer the following questions about $$f

Volume Accumulation in a Reservoir

A reservoir is being filled at a rate given by $$R(t)= e^{2*t}$$ liters per minute. Determine the t

Work Done by a Variable Force

A variable force given by $$F(x)= 3*x^2$$ (in Newtons) acts on an object as it moves along a straigh

Analysis of a Piecewise Function with Potential Discontinuities

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

Analyzing a Rational Differential Equation

Consider the differential equation $$\frac{dy}{dx} = \frac{x^2-1}{y}$$.

Bacteria Culture with Regular Removal

A bacterial culture has a population $$B(t)$$ that grows at a continuous rate of $$12\%$$ per hour,

Chemical Reaction Rate and Series Approximation

A chemical reaction is modeled by the differential equation $$\frac{dC}{dt} = -0.2 * C^2$$ with the

Cooling of a Metal Rod

A metal rod cools according to the differential equation $$\frac{dT}{dt}=-k\,(T-25)$$ with an initia

Cooling of an Object Using Newton's Law of Cooling

An object cools in a room with constant ambient temperature. The cooling process is modeled by Newto

Cooling with Time-Varying Ambient Temperature

An object cools according to the modified Newton's Law of Cooling given by $$\frac{dT}{dt}= -k*(T-T_

Differential Equation with Exponential Growth and Logistic Correction

Consider the modified differential equation $$\frac{dy}{dt} = a*y - b*y^2$$, where $$a$$ and $$b$$ a

Economic Growth Model

An economy's output \(Y(t)\) is modeled by the differential equation $$\frac{dY}{dt}= a\,Y - b\,Y^2$

Economic Model: Differential Equation for Cost Function

A company’s marginal cost is represented by $$MC(x)=\frac{dC}{dx}=3+2*x$$, where $$x$$ is the number

Electrical Circuit Analysis Using an RL Circuit

An RL circuit is described by the differential equation $$L\frac{di}{dt}+R*i=E$$, where $$L$$ is the

Euler's Method and Differential Equations

Consider the initial value problem $$\frac{dy}{dx} = x*y$$ with the initial condition $$y(0)=1$$. Eu

Euler's Method Approximation

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

Exact Differential Equation

Examine the differential equation $$ (2*x*y + y^2)\,dx + (x^2 + 2*x*y)\,dy = 0 $$. Determine if the

FRQ 2: Separable Differential Equation with Initial Condition

Consider the differential equation $$\frac{dy}{dx} = \frac{4*x}{y}$$ with the initial condition $$y(

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 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

Integrating Factor for a Non-Exact Differential Equation

Consider the differential equation $$ (y - x)\,dx + (y + 2*x)\,dy = 0 $$. This equation is not exact

Investment Growth Model

An investment account grows continuously at a rate proportional to its current balance. The balance

Logistic Model in Population Dynamics

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

Mixing Problem in a Tank

A tank initially contains 100 L of water with 5 kg of dissolved salt. Brine containing 0.1 kg of sal

Optimization in Construction: Minimizing Material for a Container

A manufacturer is designing an open-top cylindrical container with fixed volume $$V$$. The material

Parameter Identification in a Cooling Process

The temperature of an object cooling in an environment at $$20^\circ C$$ is modeled by Newton's Law

Particle Motion with Variable Acceleration

A particle moving along a straight line has an acceleration given by $$a(t) = 6 - 4*t$$ (in m/s²). A

Pollutant Concentration in a Lake

A lake receives a pollutant at a constant rate of $$5$$ kg/day and the pollutant is removed at a rat

Slope Field Analysis and DE Solutions

Consider the differential equation $$\frac{dy}{dx} = x$$. The equation has a slope field as represen

Slope Field and Solution Curve Sketching

Consider the differential equation $$\frac{dy}{dx} = x - 1$$. A slope field for this differential eq

Solution and Analysis of a Linear Differential Equation with Equilibrium

Consider the differential equation $$\frac{dy}{dx} = 3*y - 2$$, with the initial condition $$y(0)=1$

Solving a Separable Differential Equation

Solve the differential equation $$\frac{dy}{dx} = \frac{4*x}{y}$$ with the initial condition $$y(0)=

Advanced Parameter-Dependent Integration Problem

Consider the function $$g(x)=e^{-a*x}$$, where $$a>0$$ and $$x$$ lies within $$[0,b]$$. The average

Analyzing Acceleration Data from Discrete Measurements

A vehicle’s acceleration (in m/s²) is recorded at discrete time intervals as given in the table. Use

Analyzing the Inverse of an Exponential Function

Let $$f(x)=\ln(2*x+1)$$, defined for $$x\ge0$$.

Average Chemical Concentration Analysis

In a chemical reaction, the concentration of a reactant (in M) is recorded over time as given in the

Average Speed from a Variable Acceleration Scenario

A particle moves along the x-axis under an acceleration given by $$a(t)= 3*t - 2$$ (in m/s²) and has

Average Value of a Piecewise Function

A function $$f(x)$$ is defined piecewise over the interval $$[0,6]$$ as follows: $$f(x)=\begin{case

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 Plate

A lamina occupies the rectangular region defined by $$0 \le x \le 2$$ and $$0 \le y \le 3$$, with a

Center of Mass of a Rod with Variable Density

A rod extending along the x-axis from $$x=0$$ to $$x=10$$ meters has a density given by $$\rho(x)=2+

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

Comparing Average and Instantaneous Rates of Change

For the quadratic function $$f(x)= 3*x^2 - 4*x + 1$$ on the interval $$[1,3]$$, investigate both its

Consumer Surplus Analysis

The demand function for a product is given by $$D(p)=120-2*p$$, where \(p\) is the price in dollars.

Determining Average Value of a Velocity Function

A runner’s velocity is given by $$v(t)=2*t+3$$ (in m/s) for $$t\in[0,5]$$ seconds.

Determining the Arc Length of a Curve

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

Economic Analysis: Consumer and Producer Surplus

In a market analysis, the demand and supply for a product are modeled by the equations: Demand: $$D(

Force on a Submerged Plate

A vertical rectangular plate is submerged in water. The plate is 3 m wide and extends from a depth o

Integration in Cost Analysis

In a manufacturing process, the cost per minute is modeled by $$C(t)=t^2 - 4*t + 7$$ (in dollars per

Optimization and Integration: Maximum Volume

A company designs open-top cylindrical containers to hold $$500$$ liters of liquid. (Recall that $$1

Particle Motion: Position, Velocity, and Acceleration

A particle moves along a straight line with acceleration given by $$a(t)=3*t-4$$ (in m/s²). At time

Population Growth: Cumulative Increase

A bacterial culture grows at a rate given by $$r(t)=3*e^{0.2*t}$$ (in thousands per hour), where $$t

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

Temperature Modeling: Applying the Average Value Theorem

The temperature of a chemical solution in a tank is modeled by $$T(t)=20+5\cos(0.5*t)$$ (°C) for $$t

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 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 of a Solid with Elliptical Cross Sections

Consider a solid whose base is the region bounded by $$y=x^2$$ and $$y=4$$. Cross sections perpendic

Volume of an Arch Bridge Support

The arch of a bridge is modeled by $$y=12-\frac{x^2}{4}$$ for $$x\in[-6,6]$$. Cross-sections perpend

Volume with Equilateral Triangle Cross Sections

The region bounded by $$y=4-x^2$$ and $$y=0$$ for $$x\in[-2,2]$$ serves as the base of a solid. Cros

Work Done by a Variable Force

A variable force is applied along a frictionless surface and is given by $$F(x)=6-0.5*x$$ (in Newton

Work to Pump Water from a Tank

A cylindrical tank of radius 2 ft and height 10 ft is partially filled with water to a depth of 8 ft

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

Analyzing Concavity for a Polar Function

Consider the polar function given by \(r=5-2\sin(\theta)\). Answer the following:

Arc Length and Curvature Comparison

Consider two curves given by: $$C_1: x(t)=\ln(t),\, y(t)=\sqrt{t}$$ for $$1\leq t\leq e$$, and $$C_2

Arc Length in Polar Coordinates

A polar curve is defined by $$r(\theta)=1+\cos(\theta)$$ for $$0 \leq \theta \leq \pi$$.

Arc Length of a Parametrically Defined Curve

A curve is defined by the parametric equations $$x(t)=t^2$$ and $$y(t)=\frac{t^3}{3}$$ for $$0 \leq

Arc Length of a Polar Curve

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

Area between Two Polar Curves

Given the polar curves $$R(\theta)=3$$ and $$r(\theta)=2$$ for $$0 \le \theta \le 2\pi$$, find the a

Area Between Two Polar Curves

Consider the two polar curves $$r_1(θ)= 3+\cos(θ)$$ and $$r_2(θ)= 1+\cos(θ)$$. Answer the following:

Comparing Representations: Parametric and Polar

A curve is represented by the parametric equations $$x(t)=3\cos(t)-\sin(t)$$ and $$y(t)=3\sin(t)+\co

Conversion between Polar and Cartesian Coordinates

The polar equation $$r = 2 + 2\cos(\theta)$$ describes a limaçon. Analyze this curve by converting i

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

Displacement from a Vector-Valued Velocity Function

A particle's velocity is given by $$\vec{v}(t)=\langle \cos(t), \sin(t), t \rangle$$ for $$t \in [0,

Enclosed Area of a Parametric Curve

A closed curve is given by the parametric equations $$x(t)=3*\cos(t)-\cos(3*t)$$ and $$y(t)=3*\sin(t

Implicit Differentiation and Curves in the Plane

The curve defined by $$x^2y + xy^2 = 12$$ describes a relation between $$x$$ and $$y$$.

Inner Loop of a Limaçon in Polar Coordinates

The polar curve given by \(r=1+2\cos(\theta)\) forms a limaçon with an inner loop. Answer the follow

Logarithmic Exponential Transformations in Polar Graphs

Consider the polar equation $$r=2\ln(3+\cos(\theta))$$. Answer the following:

Modeling Projectile Motion with Parametric Equations

A projectile is launched with an initial speed of \(20\) m/s at an angle of \(45^\circ\) above the h

Motion Along a Parametric Curve

Consider the parametric equations $$x(t)= 2*t - 3$$ and $$y(t)= t^2$$, where $$t$$ represents time i

Motion in the Plane: Logarithmic and Radical Components

A particle’s position in the plane is given by the vector-valued function $$\mathbf{r}(t)=\langle \l

Parametric Curve Intersection

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

Parametric Plotting and Cusps

Let the parametric equations be $$ x(t)=t-\sin(t) $$ and $$ y(t)=1-\cos(t) $$ for $$ 0 \le t \le 2\p

Particle Motion in Circular Motion

A particle moves along a circular path given by the parametric equations $$x(t)= 5\cos(t)$$ and $$y(

Polar Boundary Conversion and Area

A region in the polar coordinate plane is defined by $$1 \le r \le 3$$ and $$0 \le \theta \le \frac{

Polar Plots and Intersection Points in Design

A designer creates a pattern using the polar equations $$r=5\cos(θ)$$ and $$r=5\sin(θ)$$. Analyze th

Polar Spiral: Area and Arc Length

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

Projectile Motion using Parametric Equations

A projectile is launched with an initial speed of $$v_0 = 20\,\text{m/s}$$ at an angle of $$30^\circ

Projectile Motion: Rocket Launch Tracking

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

Related Rates: Expanding Balloon

A spherical balloon is being inflated so that its volume increases at a constant rate of $$30\,cm^3/

Spiral Motion in Polar Coordinates

A particle moves in polar coordinates with \(r(\theta)=4-\theta\) and the angle is related to time b

Symmetry and Self-Intersection of a Parametric Curve

Consider the parametric curve defined by $$x(t)= \sin(t)$$ and $$y(t)= \sin(2*t)$$ for $$t \in [0, \

Synthesis of Parametric, Polar, and Vector Concepts

A drone's flight path is given in polar coordinates by $$r(\theta)= 5+ 2\sin(\theta)$$. It is parame

Tangent Line to a 3D Vector-Valued Curve

Let $$\textbf{r}(t)= \langle t^2, \sin(t), \ln(t+1) \rangle$$ for $$t \in [0,\pi]$$. Answer the foll

Tangent Line to a Parametric Curve

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

Taylor/Maclaurin Series: Approximation and Error Analysis

Let $$f(x)=\ln(1+x)$$. Without using a calculator, generate the third-degree Maclaurin polynomial fo

Vector-Valued Function of Particle Trajectory

A particle in space follows the vector function $$\mathbf{r}(t)=\langle t, t^2, \sqrt{t} \rangle$$ f

Work Done Along a Path in a Force Field

A force field is given by \(\mathbf{F}(x,y)=\langle x,\,y \rangle\), and a particle moves along a pa