AP Calculus BC FRQ Room

Analysis of a Jump Discontinuity

Consider the function $$f(x)=\begin{cases} 3*x+1, & x<4 \\ 2*x-3, & x\geq4 \end{cases}$$.

Application of the Squeeze Theorem

Let $$f(x)=x^2 * \sin(\frac{1}{x})$$ for $$x \neq 0$$. Answer the following:

Approaching Vertical Asymptotes

Consider the function $$g(x)=\frac{3}{(x-2)^2}-\frac{1}{x-2}.$$ Answer the following:

Asymptotic Behavior in Rational Functions

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

Composite Function Involving Logarithm and Rational Expression

Consider the piecewise function $$ f(x)=\begin{cases} \frac{1}{x-1} & \text{if } x<2, \\ \ln(x-1) &

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

Drainage Rate with a Removable Discontinuity

A drainage system is modeled by the function $$R_{out}(t)=\frac{t^2-2\,t-15}{t-5}$$ liters per minut

Economic Equilibrium and Limit Analysis

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

Epsilon-Delta Style (Conceptual) Analysis

Consider the function $$f(x)=\begin{cases} 3*x+2, & x\neq1, \\ 6, & x=1. \end{cases}$$ Answer the

Establishing Continuity in a Piecewise Function

Consider the piecewise-defined function $$p(x)= \begin{cases} \frac{x^2-4}{x-2} & x \neq 2, \\ k & x

Evaluating Limits Involving Radical Expressions

Consider the function $$h(x)= \frac{\sqrt{4x+1}-3}{x-2}$$.

Exploring Infinite and Vertical Asymptotes in Rational Functions

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

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

Graph Analysis of a Discontinuous Function

Examine a function $$f(x)=\frac{x^2-4}{x-2}$$. A graph of the function is provided in the attached s

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

Internet Data Packet Transmission and Error Rates

In a data transmission system, an error correction protocol improves the reliability of transmitted

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:

Left-Hand and Right-Hand Limits for a Sign Function

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

Limit Evaluation Involving Radicals and Rationalization

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

Limits Involving Absolute Value Functions

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

Limits Involving Trigonometric Functions

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

Limits with Composite Logarithmic Functions

Consider the function $$t(x)=x*\ln(x)$$ defined for x > 0.

Mixed Function Inflow Limit Analysis

Consider the water inflow function defined by $$R(t)=10+\frac{\sqrt{t+4}-2}{t}$$ for \(t\neq0\). Det

One-Sided Limits and Jump Discontinuities

Consider the piecewise function defined by: $$ f(x)=\begin{cases} 2-x, & x<1\\ 3*x-1, & x\ge1 \en

Piecewise Function Continuity

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

Radical Function Limit via Conjugate Multiplication

Consider the function $$f(x)=\frac{\sqrt{2*x+9}-3}{x}$$ defined for $$x \neq 0$$. Answer the followi

Real-World Temperature Sensor Analysis

A temperature sensor is modeled by the function $$T(t)=\frac{t^2-9}{t-3}$$ for t ≠ 3 (with t in minu

Squeeze Theorem with a Log Function

Let $$f(x)= x\,\ln\Bigl(1+\frac{1}{x}\Bigr)$$ for $$x > 0$$. Use the Squeeze Theorem to determine $$

Analysis of Increasing and Decreasing Intervals

Let $$f(x)=x^4 - 8*x^2$$. Answer the following parts.

Chemical Mixing Tank

In an industrial process, a mixing tank receives a chemical solution at a rate of $$C_{in}(t)=25+5*t

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

Determining Rates of Change with Secant and Tangent Lines

A function is graphed by the equation $$f(t)=t^3-3*t$$ and its graph is provided. Use the graph to a

Drug Concentration in Bloodstream: Differentiation Analysis

A drug's concentration in the bloodstream is modeled by $$C(t)= 50e^{-0.25t} + 5$$, where t is in ho

Exploration of Derivative Notation and Higher Order Derivatives

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

Exponential Growth and Its Derivative

A culture of bacteria grows according to the model $$P(t)= 100*e^{0.03*t},$$ where \(P(t)\) is th

Implicit Differentiation with Trigonometric Functions

Consider the curve defined by $$\sin(x*y) = x + y$$.

Logarithmic Differentiation

Let $$T(x)= (x^2+1)^{3*x}$$ model a quantity with variable growth characteristics. Use logarithmic d

Logarithmic Differentiation: Equating Powers

Consider the equation $$y^x = x^y$$ that relates $$x$$ and $$y$$ implicitly.

Maclaurin Series for arctan(x) and Error Estimate

An engineer in signal processing needs the Maclaurin series for $$g(x)=\arctan(x)$$ and an understan

Population Growth Approximation using Taylor Series

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

Population Model Rate Analysis

A city's population is modeled by $$P(x)=2000+500\ln(x)$$, where $$x$$ represents years since a base

Production Output Rate Analysis Using a Product Function

A factory's production output (in items per hour) is modeled by $$P(t) = t^2*(20 - t)$$, where t (in

Profit Rate Analysis in Economics

A firm’s profit function is given by $$\Pi(x)=-x^2+10*x-20$$, where $$x$$ (in hundreds) represents t

Reservoir Management Problem

A reservoir used for irrigation receives water at a rate of $$I(t)=20+2\sin(t)$$ liters per hour and

Satellite Orbit Altitude Modeling

A satellite’s altitude (in kilometers) above Earth is modeled by $$a(t)= 500*\cos\left(\frac{\pi}{6}

Secant vs. Tangent: Approximation and Limit Approach

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

Tangent Line Approximation

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

Tangent Lines and Related Approximations

For the function $$f(x)= \sin(x) + x^2,$$ (a) Compute \(f'(0)\). (b) Write the equation of the t

Taylor Series for Cos(x) in Temperature Modeling

An engineer uses the cosine function to model periodic temperature variations. Approximate $$\cos(x)

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 Trigonometric Light Intensity Model

A light sensor records the intensity of light according to the function $$I(x) = \cos(\sqrt{3*x + 2}

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

Composite Differentiation in Polynomial Functions

Consider the function $$f(x)= (2*x^3 - x + 1)^4$$. Use the chain rule to differentiate f(x).

Composite Functions in a Biological Growth Model

A biologist models the substrate concentration by the function $$ g(t)= \frac{1}{1+e^{-0.5*t}} $$ an

Composite Temperature Change in a Chemical Reaction

A chemical reaction in a laboratory is modeled by the composite temperature function $$R(t)= f(g(t))

Dam Water Release and River Flow

A dam releases water into a river at a rate given by the composite function $$R(t)=c(b(t))$$, where

Differentiation Involving Absolute Values and Composite Functions

Consider the function $$f(x)= \sqrt{|2*x - 3|}$$. Answer the following:

Differentiation of an Inverse Trigonometric Form

Consider the function $$f(x)=\arcsin\left(\frac{2*x}{1+x^2}\right)$$. Answer the following parts.

Exponential Composite Function Differentiation

Consider the function $$f(x)= e^{3*x^2+2*x}$$.

Higher-Order Derivatives via Implicit Differentiation

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

Implicit Differentiation and Inverse Functions Combined

Consider the function defined implicitly by the equation $$\sin(y) + y\cos(x) = x.$$ Answer the fo

Implicit Differentiation for a Spiral Equation

Consider the curve given by the equation $$x^2 + y^2 = 4*x*y$$. Analyze its derivative using implici

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 in a Hyperbola-like Equation

Consider the equation $$ x*y = 3*x - 4*y + 12 $$.

Implicit Differentiation in a Nonlinear Equation

Consider the equation $$x*y + y^3 = 10$$, which defines y implicitly as a function of x.

Implicit Differentiation in Circular Motion

Consider the circle described by $$x^2+y^2=49$$, representing a particle's path. Answer the followin

Implicit Differentiation in Exponential Equation

Consider the equation $$e^{x*y}+x^2-y^3=0$$ that relates x and y. Answer the following parts:

Implicit Differentiation of a Circle

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

Implicit Differentiation with Trigonometric Functions

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

Inverse Analysis of a Log-Polynomial Function

Consider the function $$f(x)=\ln(x^2+1)$$. Analyze its one-to-one property on the interval $$[0,\inf

Inverse Function Derivative in a Cubic Function

Let $$f(x)= x^3+ 2*x - 1$$, a one-to-one differentiable function. Its inverse function is denoted as

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 Function Differentiation in Exponential-Linear Model

Let $$f(x)= x + e^{-x}$$, which is invertible with inverse function $$g(x)$$. Use the inverse functi

Inverse Function Differentiation in Navigation

A vehicle’s distance traveled is modeled by $$f(t)= t^3 + t$$, where $$t$$ represents time in hours.

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

Inverse Trigonometric Functions: Analysis and Application

Consider the function $$f(x) = \arctan(3*x)$$. Analyze its rate of change and the equation of the ta

Logarithmic Differentiation of a Variable Exponent Function

Consider the function $$y= (x^2+1)^{\sqrt{x}}$$.

Optimization in Manufacturing Material

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

Parameter Dependent Composite Function

The temperature in a metal rod is modeled by $$T(x)= e^{a*x}$$, where the parameter $$a$$ changes wi

Particle Motion with Composite Position Function

A particle moves along a line with its position given by $$s(t)= \sin(t^2)$$, where $$s$$ is in mete

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

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

Water Tank Composite Rate Analysis

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

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

Cooling Coffee Temperature

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

Data Table Inversion

A function $$f(x)$$ is represented by the following data table. Use the data to analyze the inverse

Economic Rates: Marginal Profit Analysis

A manufacturer’s profit (in dollars) from producing $$x$$ items is modeled by $$P(x)=500*x-2*x^2$$.

Exponential and Trigonometric Bounded Regions

Let the region in the xy-plane be bounded by $$y = e^{-x}$$, $$y = 0$$, and the vertical line $$x =

Graphical Analysis of Derivatives

A function $$f(x)$$ is plotted on the graph provided below. Using this graph, answer the following:

Graphical Interpretation of Slope and Instantaneous Rate

A graph (provided below) displays a linear function representing a physical quantity over time. Use

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$

L'Hôpital’s Rule in Chemical Reaction Rates

In a chemical reaction, the ratio of certain concentrations is modeled by $$R(x)=\frac{3*x^2-2*x+1}{

Ladder Sliding Problem

A 10-meter ladder is leaning against a vertical wall. The bottom of the ladder is pulled away from t

Linearization Approximation Problem

Given the function $$f(x)=\sqrt{x+4}$$, use linearization to approximate the value of $$f(5.1)$$.

Linearization in Engineering Load Estimation

In an engineering project, the load on a beam is modeled by $$L(x)=100*x^2+50*x+200$$, where $$L(x)$

Linearization of Trigonometric Implicit Function

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

Marginal Cost and Revenue Analysis

A company’s cost function is given by $$C(x)=100+25*x+4*x^2$$, where $$C(x)$$ is measured in dollars

Modeling Cooling: Coffee Temperature with Logarithmic Decline

A cup of coffee cools according to the model $$T(t)= 90 - 20\ln(1+t)$$, where $$T$$ is in degrees Ce

Optimizing Factory Production with Log-Exponential Model

A factory's production is modeled by $$P(x)=200x^{0.3}e^{-0.02x}$$, where x represents the number of

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

Series Approximation for Investment Growth

An investment accumulation function is modeled by $$A(t)= 1 + \sum_{n=1}^{\infty} \frac{(0.07t)^n}{n

Series Approximation in an Exponential Population Model

A population is modeled by $$P(t)= 1000 \times \sum_{n=0}^{\infty} \frac{(0.05t)^n}{n!}$$, which is

Urban Traffic Flow Analysis

An urban highway ramp experiences an inflow of cars at a rate of $$I(t)=40+2t$$ (cars per minute) an

Vehicle Motion on a Curved Path

A vehicle moving along a straight road has its position given by $$s(t)= 4*t^3 - 24*t^2 + 36*t + 5$$

Analyzing Extrema for a Rational Function

Let $$f(x)= \frac{x^2+2}{x+1}$$ be defined on the interval $$[0,4]$$. Use calculus methods to analyz

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

Curve Sketching Using Derivatives

For the function $$f(x)=\frac{x}{x^2+1}$$, use its derivatives to sketch a rough graph of the functi

Investigating a Composite Function Involving Logarithms and Exponentials

Let $$f(x)= \ln(e^x + x^2)$$. Analyze the function by addressing the following parts:

Linear Approximation and Differentials

Let \( f(x) = \sqrt{x} \). Use linear approximation to estimate \( \sqrt{10} \). Answer the followin

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

MVT Application: Rate of Temperature Change

The temperature in a room is modeled by $$T(t)= -2*t^2+12*t+5$$, where $$t$$ is in hours. Analyze th

Optimization in Particle Routing

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

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

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

Projectile Motion Analysis

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

Relative Motion in Two Dimensions

A point moves in the plane along the path defined by $$x(t)=t^2-4*t+3$$ and $$y(t)=\ln(t+1)$$ for $$

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

Accumulated Change via U-Substitution

Evaluate the definite integral representing the accumulated amount of a substance given by $$\int_{1

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

Advanced Inflow/Outflow Dynamics

A reservoir receives water from a river at a rate given by $$f(t)=50*(1+0.1*t)$$ cubic meters per ho

Analyzing and Integrating a Function with a Removable Discontinuity

Consider the function $$ f(x)= \begin{cases} \frac{x^2-9}{x-3} & \text{if } x \neq 3,\\ 4 & \text{if

Arc Length of a Power Function

Find the arc length of the curve $$y=\frac{1}{3}*x^{3/2}$$ on the interval $$[0,9]$$.

Area Between Inverse Functions

Consider the functions $$f(x)=\sqrt{x}$$ and $$g(x)=x-2$$.

Area Under a Piecewise Function

A function is defined piecewise as follows: $$f(x)=\begin{cases} x & 0 \le x \le 2,\\ 6-x & 2 < x \

Bacteria Population Accumulation

A bacteria culture grows at a rate given by $$r(t)=0.5*t^2-t+3$$ (in thousand bacteria per hour) for

Consumer Spending Accumulation

Consumers' marginal spending over a 10-hour day is modeled by $$S(t)= 100*e^{-0.2*t}$$ dollars per h

Convergence of an Improper Integral Representing Accumulation

Consider the improper integral $$\int_{1}^{\infty} \frac{1}{t^2}dt$$, which can be interpreted as th

Cyclist's Displacement from Variable Acceleration

A cyclist's acceleration is given by $$a(t)= 7 - 3*t$$ (in m/s²). At time $$t=0$$, the cyclist has a

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 $$x=4$$.

Definite Integral via the Fundamental Theorem of Calculus

Consider the linear function $$f(x)=2*x+3$$ defined on the interval $$[1,4]$$. A graph of the functi

Displacement and Distance from a Velocity Function

A particle moves along a straight line with its velocity given by $$v(t)=3\sin(t)$$ (in m/s) for $$t

Error Estimation in Riemann Sum Approximations

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

Filling a Tank: Antiderivative with Initial Value

Water is entering a tank at a rate given by $$r(t)= \frac{2}{t+1}$$ liters per minute. The initial

Integration by Parts: Logarithmic Function

Evaluate the definite integral $$\int_{1}^{3} x*\ln(x) dx$$ using integration by parts. Answer the f

Integration of a Piecewise Function for Total Area

Consider the piecewise function $$f(x)$$ defined by: $$f(x)=3-x$$ for $$0 \le x \le 3$$, and $$f(x)=

Integration Using U-Substitution

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

Inverse Functions in Economic Models

Consider the function $$f(x) = 3*x^2 + 2$$ defined for $$x \ge 0$$, representing a demand model. Ans

Investigating Partition Sizes

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

Marginal Cost and Total Cost in Production

A company's marginal cost function is given by $$MC(q)=12+2*q$$ (in dollars per unit) for $$q$$ in t

Mechanical Systems: Total Change and Inverse Analysis

Consider the function \(f(x)= x^3 + 3*x\) defined for all real \(x\), modeling a mechanical system.

Minimizing Material for a Container

A company wants to design an open-top rectangular container with a square base that must have a volu

Motion and Accumulation: Particle Displacement

A particle moving along a straight line has a velocity function given by $$v(t)=3*t^{2} - 12*t + 5$$

Particle Motion: Accumulated Position Function

A particle moves along a line with velocity given by $$v(t)= t^2 - 4*t + 3$$. Its position functio

Population Growth from Birth Rate

In a small town, the birth rate is modeled by $$B(t)= \frac{100}{1+t^2}$$ people per year, where $$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

Temperature Change Analysis

A series of temperature readings (in °C) are recorded over the day as shown in the table. Analyze th

U-Substitution in Accumulation Functions

In a chemical reactor, the accumulation rate of a substance is given by $$R(x)= 3*(x-2)^4$$ units pe

U-Substitution Integration

Evaluate the definite integral $$\int_1^5 (2*x-3)^4 dx$$ using the method of u-substitution.

U-Substitution Integration Challenge

Evaluate the integral $$\int_0^2 (2*x+1)\,(x^2+x+3)^5\,dx$$ using an appropriate u-substitution.

Volume of a Solid with Known Cross-sectional Area

A solid extends from $$x=0$$ to $$x=5$$, and its cross-sectional area perpendicular to the x-axis is

Water Accumulation in a Reservoir

A reservoir receives water at an inflow rate modeled by $$r(t)=\frac{20}{t+1}$$ (in cubic meters per

Analysis of a Nonlinear Differential Equation

Consider the nonlinear differential equation $$\frac{dy}{dx} = y^3-3*y$$.

Chemical Reaction Rate and Series Approximation

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

Coffee Cooling: Differential Equation Application

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

Combined Differential Equations and Function Analysis

Consider the function $$y(x)$$ defined by the differential equation $$\frac{dy}{dx} = \frac{2*x}{1+y

Compound Interest and Investment Growth

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

Compound Interest with Continuous Payment

An investment account grows with a continuous compound interest rate $$r$$ and also receives continu

Cooling of a Smartphone Battery

A smartphone battery cools according to Newton’s law: $$\frac{dT}{dt} = -k*(T-T_{room})$$. Initially

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 Investment Growth Model with Regular Deposits

An investment account grows with continuously compounded interest at a rate $$r$$ and receives conti

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 and Differential Equations

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

Exact Differential Equation

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

Exponential Growth with Shifted Dependent Variable

The differential equation $$\frac{dy}{dx} = e^{x}*(y+2)$$ is used to model a growth process where th

FRQ 6: Radioactive Decay

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

FRQ 7: Projectile Motion with Air Resistance

A projectile is launched vertically upward with an initial velocity of 50 m/s. Its vertical motion i

FRQ 9: Epidemiological Model Differential Equation

An epidemic evolves according to the differential equation $$\frac{dI}{dt}=r*I*(M-I)$$, where $$I(t)

Implicit Differential Equations and Slope Fields

Consider the implicit differential equation $$x\frac{dy}{dx}+ y = e^x$$. Answer the following parts.

Integrating Factor Method

Solve the differential equation $$\frac{dy}{dx} + \frac{2}{x} y = \frac{\sin(x)}{x}$$ for $$x>0$$.

Investment Growth Model

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

Mixing Problem in a Saltwater Tank

A tank initially contains $$100$$ liters of water with a salt concentration of $$2\,g/l$$. Brine wit

Modeling Currency Exchange Rates with Differential Equations

Suppose the exchange rate $$E(t)$$ (domestic currency per foreign unit) evolves according to the dif

Non-linear Differential Equation using Separation of Variables

Consider the differential equation $$\frac{dy}{dx}= \frac{x*y}{x^2+1}$$. Answer the following questi

Power Series Solutions for a Differential Equation

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

Projectile Motion with Air Resistance

A projectile is fired vertically upward with an initial velocity of $$50\,m/s$$. The projectile expe

Projectile Motion with Air Resistance

A projectile is launched with an initial speed $$v_0$$ at an angle $$\theta$$ relative to the horizo

Separable Differential Equation and Slope Field Analysis

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

Separable Differential Equation and Slope Field Analysis

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

Tank Mixing Problem

A tank contains 1000 L of a well‐mixed salt solution. Brine containing 0.5 kg/L of salt flows into t

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 a Function with a Removable Discontinuity

Consider the function $$f(x)=\frac{x^2-4}{x-2}$$ for $$x\neq2$$, with an assigned value of $$f(2)=3$

Arc Length of a Parabolic Curve

Find the arc length of the curve defined by $$y = x^2$$ for $$x$$ in the interval $$[0,3]$$.

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 $

Average Concentration of a Drug in Bloodstream

The concentration of a drug in the bloodstream is modeled by $$C(t)=3e^{-0.9*t}+2$$ mg/L, where $$t$

Average Population Density

In an urban study, the population density (in thousands per km²) of a city is modeled by the functio

Average Temperature Analysis

A research team models the ambient temperature in a region over a 24‐hour period with the function $

Average Value of a Piecewise Function

Consider the function $$g(x)$$ defined piecewise on the interval $$[0,6]$$ by $$g(x)=\begin{cases} x

Average Value of a Polynomial Function

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

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

Average Velocity of a Runner

A runner's velocity is modeled by $$v(t)=5+3\cos(0.5*t)$$ (m/s) for $$0\le t\le10$$ seconds. Answer

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

Complex Integral Evaluation with Exponential Function

Evaluate the integral $$I=\int_1^e \frac{2*\ln(t)}{t}dt$$ by applying a suitable substitution.

Distance Traveled versus Displacement

A particle moves along a line with velocity given by $$v(t)=t^3 - 6*t^2 + 9*t$$ (in m/s) for $$t\in[

Economic Analysis: Consumer and Producer Surplus

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

Optimizing the Thickness of a Cooling Plate

The local heat conduction efficiency at a point on a cooling plate is modeled by the function $$A(x)

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

Profit-Cost Area Analysis

A company’s profit (in thousands of dollars) is modeled by $$P(x) = -x^2 + 10*x$$ and its cost by $$

Pumping Water from a Conical Tank

An inverted right circular conical tank has a height of $$10$$ meters and a top radius of $$4$$ mete

River Cross Section Area

The cross-sectional boundaries of a river are modeled by the curves $$y = 5 * x - x^2$$ and $$y = x$

Surface Area of a Solid of Revolution

Consider the curve $$y=\sqrt{x}$$ on the interval $$[0,9]$$. When this curve is rotated about the x-

Total Change in Temperature Over Time (Improper Integral)

An object cools according to the function $$\Delta T(t) = e^{-0.5*t}$$, where $$t\ge 0$$ is time in

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 Hollow Cylinder Using the Shell Method

A hollow cylindrical tube of height 5 m is formed by rotating the rectangular region bounded by $$x

Volume of a Solid Obtained by Rotation

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

Volume of a Solid of Revolution Between Curves

Consider the region bounded by the curves $$y=\sqrt{x}$$ and $$y=\frac{x}{2}$$ for $$x \in [0,4]$$.

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

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 Done by a Variable Force

A force acting along a straight line is given by $$F(x)=10 - 0.5*x$$ newtons for $$0 \le x \le 12$$

Work Done by a Variable Force

A variable force acting along a straight line is given by $$F(x)=5*x$$ (in Newtons), where $$x$$ is

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

Analysis of a Polar Rose

Examine the polar curve given by $$ r=3*\cos(3\theta) $$.

Analysis of a Vector-Valued Function

Consider the vector-valued function $$\mathbf{r}(t)= \langle t^2+1,\; t^3-3*t \rangle$$, where $$t$$

Arc Length Calculation of a Cycloid

Consider a cycloid described by the parametric equations $$x(t)=r*(t-\sin(t))$$ and $$y(t)=r*(1-\cos

Arc Length of a Cycloid

A cycloid is generated by a circle of radius \(r=1\) rolling along a straight line. The cycloid is g

Arc Length of a Polar Curve

Consider the polar curve given by $$r = 2 + 2*\sin(\theta)$$ for $$0 \le \theta \le \pi$$.

Area Between Polar Curves

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

Catching a Thief: A Parametric Pursuit Problem

A police car and a thief are moving along a straight road. Initially, both are on the same road with

Combined Motion Analysis

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

Designing a Roller Coaster: Parametric Equations

The path of a roller coaster is modeled by the equations $$x(t)=t-\sin(t)$$ and $$y(t)=1-\cos(t)$$ f

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,

Dynamics in Polar Coordinates

A particle moves such that its polar coordinates are given by $$ r(\theta)=1+\theta $$, where $$ \th

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

Intersection Analysis with the Line y = x

Given the parametric equations $$x(t)=\ln(t+2)$$ and $$y(t)=t^2-1$$ for $$t \ge 0$$, answer the foll

Intersection of Parametric Curves

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

Intersection of Polar and Parametric Curves

Consider the polar curve given by $$r = 2\cos(θ)$$ and the parametric curve defined by $$x(t)= 1+t^2

Kinematics in the Plane: Parametric Motion

A particle moves in the plane with its position given by the parametric equations $$ x(t)=t^2-2*t $$

Motion on a Circle in Polar Coordinates

A particle moves along a circular path of constant radius $$r = 4$$, with its angle given by $$θ(t)=

Parametric Curve Intersection

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

Parametric Equations and Tangent Slopes

Consider the parametric equations $$x(t)= t^3 - 3*t$$ and $$y(t)= t^2$$, for $$t \in [-2, 2]$$. Anal

Parametric Equations from Real-World Data

A moving vehicle’s position is modeled by the parametric equations $$ x(t)=3*t+1 $$ and $$ y(t)=t^2-

Polar Differentiation and Tangent Lines

Consider the polar curve $$r(\theta)= 1+\cos(\theta)$$.

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

Polar to Cartesian Conversion and Tangent Slope

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

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 with Parametric Equations

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

Reparameterization between Parametric and Polar Forms

A curve is described by the parametric equations $$x(t)=a*t$$ and $$y(t)=b*t^2$$, where $$a$$ and $$

Satellite Orbit: Vector-Valued Functions

A satellite’s orbit is modeled by the vector function $$\mathbf{r}(t)=\langle \cos(t)+0.1*\cos(6*t),

Spiral Intersection on the X-Axis

Consider the spiral defined by the parametric equations $$x(t) = e^{-t}\cos(2*t)$$ and $$y(t)= e^{-t

Spiral Motion in Polar Coordinates

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

Vector-Valued Fourier Series Representation

The vector function $$\mathbf{r}(t)=\langle \cos(t), \sin(t), 0 \rangle$$ for $$t \in [-\pi,\pi]$$ c

Vector-Valued Function and Particle Motion

Consider the vector-valued function $$\vec{r}(t)= \langle e^t, \sin(t), \cos(t) \rangle$$ representi

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

Vector-Valued Functions and Kinematics

A particle moves in space with its position given by the vector-valued function $$\vec{r}(t)= \langl

Vector-Valued Functions: Tangent and Normal Components

A car’s motion on a plane is described by the vector-valued function $$\mathbf{r}(t)=\langle t^2, 4*

Vector-Valued Kinematics

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