AP Calculus BC FRQ Room

Algebraic Method for Evaluating Limits

Consider the function $$h(x)=\frac{x^2-9}{x-3}.$$ Answer the following parts.

Analyzing Discontinuities in a Piecewise Function

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

Analyzing Limits Using Tabular Data

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

Application of the Intermediate Value Theorem in Temperature Change

A laboratory experiment records the temperature $$T(t)$$ in a reaction over time $$t$$ (in minutes).

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

Calculating Tangent Line from Data

The table below gives a function $$f(x)$$ representing the distance (in meters) of a moving object f

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 for a Logarithmic Transform Function

Consider the function $$f(x)= \ln\Bigl(\frac{e^{3x}-1}{x}\Bigr)$$ for $$x \neq 0$$ and define $$f(0)

Continuity of a Trigonometric Function Near Zero

Consider the function defined by $$ f(x)= \begin{cases} \frac{\sin(5*x)}{x}, & x \neq 0 \\ L, & x =

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

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

Identifying and Removing Discontinuities

The function $$f(x)=\frac{x^2-9}{x-3}$$ is undefined at x = 3.

Identifying and Removing Discontinuities in a Traffic Flow Model

A model for traffic flow during rush hour is given by $$C(t)= \frac{t^2-9}{t-3}$$ for $$t \neq 3$$.

Indeterminate Limit with Exponential and Log Functions

Examine the limit $$\lim_{x \to 0} \frac{e^{2x} - \cos(x) - 1}{\ln(1+x^2)}.$$

Inverse Function Analysis and Continuity

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

Investigating Limits Involving Nested Rational Expressions

Evaluate the limit $$\lim_{x\to3} \frac{\frac{x^2-9}{x-3}}{x-2}$$. (a) Simplify the expression and e

Limits Involving Exponential Functions

Consider the function $$p(x)=\frac{e^x}{e^x+5}$$.

Non-Existence of a Limit due to Oscillation

Consider the function $$h(x)= \sin(\frac{1}{x})$$. Answer the following regarding its limit as x app

Oscillatory Behavior and Squeeze Theorem

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

Piecewise Inflow Function and Continuity Check

A water tank's inflow is measured by a piecewise function due to a change in sensor calibration at \

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

Series Representation and Convergence Analysis

Consider the power series $$S(x)=\sum_{n=1}^{\infty} \frac{(-1)^{n+1}*(x-2)^n}{n}.$$ (Calculator per

Squeeze Theorem in Oscillatory Functions

Consider the function $$f(x)= x\sin(1/x)$$ for $$x \neq 0$$ and define $$f(0)=0$$.

Vertical Asymptote Analysis in a Rational Function

Consider the function $$g(x)=\frac{x+1}{x-3}$$, which is undefined at $$x=3$$. Answer the following:

Water Filling a Leaky Tank

A water tank is initially empty. Every minute, 10 liters of water is added to the tank, but due to a

Car Motion and Critical Velocity

The position of a car is described by $$s(t)=t^3 - 12*t^2 + 36*t$$ (with $$s$$ in meters and $$t$$ i

Chemical Reaction Rate Control

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

Composite Exponential-Log Function Analysis

Consider the function $$f(x)=e^{x}*\ln(x+3)$$, which arises in the study of compound reactions in ch

Comprehensive Analysis of $$e^{-x^2}$$

The function $$f(x)=e^{-x^2}$$ is used to model temperature distribution in a material. Provide a co

Derivative of Inverse Functions

Let $$f(x)=3*x+\sin(x)$$, which is assumed to be one-to-one with an inverse function $$f^{-1}(x)$$.

Differentiating Composite Functions

Let $$f(x)=\sqrt{2*x^2+3*x+1}$$. (a) Differentiate $$f(x)$$ with respect to $$x$$ using the appropr

Differentiating Composite Functions using the Chain Rule

Consider the function $$S(x)=\sin(3*x^2+2)$$ which might model the stress on a structure as a functi

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

Electricity Consumption: Series and Differentiation

A household's monthly electricity consumption increases geometrically due to seasonal variations. Th

Finding the Derivative of a Logarithmic Function

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

Implicit Differentiation for a Rational Equation

Consider the curve defined by $$\frac{x*y}{x+y} = 3$$.

Implicit Differentiation with Exponential and Trigonometric Functions

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

Implicit Differentiation: Conic with Mixed Terms

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

Instantaneous vs. Average Rate of Change

Consider the trigonometric function $$f(x)= \sin(x)$$.

Interpreting Derivative Notation in a Real-World Experiment

A reservoir's water level (in meters) is measured at different times (in minutes) as shown in the ta

Irrigation Reservoir Analysis

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

Limit Definition of the Derivative for a Quadratic Function

Let $$f(x)= 5*x^2 - 4$$. Use the limit definition of the derivative to compute $$f'(x)$$.

Plant Growth Rate Analysis

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

Product of Exponential and Trigonometric Functions

Let $$f(x)=e^(2*x)*\sin(x)$$. This function models oscillating growth. Answer the following:

Radius of Convergence of a Power Series for e^x

Consider the power series representation $$\sum_{n=0}^{\infty} \frac{x^n}{n!}$$, known to represent

Related Rates: Changing Shadow Length

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

Related Rates: Constant Area Rectangle

A rectangle maintains a constant area of $$A = l*w = 50$$ m², where the length l and width w vary wi

Related Rates: Two Moving Vehicles

A car is traveling east at 60 km/h and a truck is traveling north at 80 km/h. Let $$x$$ and $$y$$ be

Second Derivative and Concavity Analysis

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

Tangent and Normal Lines to a Curve

Given the function $$p(x)=\ln(x)$$ defined for $$x > 0$$, analyze its rate of change at a specific p

Testing Differentiability at a Junction Point

Consider the function $$f(x)= \begin{cases} x+2 & x < 3 \\ 5 & x = 3 \\ 2*x-1 & x > 3 \end{cases}$$.

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

Using the Limit Definition for a Non-Polynomial Function

Consider the function $$f(x)= \sqrt{x+4}$$ for $$x\ge -4$$. Answer the following:

Water Tank: Inflow-Outflow Dynamics

A water tank initially contains $$1000$$ liters of water. Water enters the tank at a rate of $$R_{in

Water Treatment Plant Simulator

A water treatment plant receives contaminated water at a rate of $$R_{in}(t)=50e^{-0.1*t}$$ liters p

Chain Rule Application: Differentiating a Nested Trigonometric Function

Consider the function $$f(x) = \sin(\cos(2*x))$$. Analyze its derivative using the chain rule.

Chain Rule with Trigonometric Composite Function

Examine the function $$ h(x)= \sin((2*x^2+1)^2) $$.

Chain, Product, and Implicit: A Motion Problem

A particle moves along a curve defined by the parametric equations $$x(t)=e^{-t}\cos(t)$$ and $$y(t)

Composite Function with Hyperbolic Sine

A cable's displacement over time is modeled by $$s(t)= \sinh(\ln(t+1))$$, where $$t$$ is in seconds.

Composite Population Growth Function

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

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))

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 +

Derivative of an Inverse Function with a Quadratic

Consider the function $$f(x) = x^2 + 6*x + 9$$ defined on $$x \ge -3$$. Let $$g$$ be the inverse of

Differentiation Involving Absolute Values and Composite Functions

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

Graphical Analysis of a Composite Function

Let $$f(x)=\ln(4*e^{x}+1)$$. A graph of f is provided. Answer the following parts.

Higher-Order Derivatives via Implicit Differentiation

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

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: Circle and Tangent Line

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

Implicit Differentiation: Second Derivatives of a Circle

Given the circle $$x^2+y^2=10$$, answer the following parts:

Infinite Series in a Financial Deposit Model

An investor makes monthly deposits that follow a geometric sequence, with the deposit in the nth mon

Inverse Analysis of a Radical Function

Consider the function $$f(x)=\sqrt{2*x+3}$$ defined for $$x \ge -\frac{3}{2}$$. Analyze its invertib

Inverse Function Differentiation for a Cubic Function

Let $$ f(x)= x^3+x $$. This function is invertible over all real numbers.

Inverse Function Differentiation for a Quadratic Function

Let $$ f(x)= (x+1)^2 $$ with the domain $$ x\ge -1 $$. Consider its inverse function $$ f^{-1}(y) $$

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 Function Differentiation

Let $$y=\arctan(\sqrt{3*x+1})$$. Answer the following parts:

Investment Growth and Rate of Change

An investor makes monthly deposits that increase according to an arithmetic sequence. The deposit am

Logarithmic and Composite Differentiation

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

Maximizing the Garden Area

A rectangular garden is to be built alongside a river, so that no fence is needed along the river. T

Rainwater Harvesting System

A rainwater harvesting system collects water in a reservoir. The inflow rate is given by the composi

Reservoir Levels and Evaporation Rates

A reservoir is being filled with water from an inflow while losing water through controlled release

Rocket Fuel Consumption Analysis

A rocket’s fuel consumption rate is modeled by the composite function $$C(t)=n(m(t))$$, where $$m(t)

Analyzing a Motion Graph

A car's velocity over time is modeled by the piecewise function given in the graph. For $$0 \le t <

Cost Function Analysis in Production

A company's cost for producing $$x$$ items is given by $$C(x)=0.5*x^3-4*x^2+10*x+500$$ dollars.

Expanding Circular Pool

A circular pool is being designed such that water flows in uniformly, expanding its surface area. Th

Filling a Conical Tank

A conical water tank has its radius related to its height by $$r=\frac{h}{2}$$, and its volume is gi

Hyperbolic Motion

A particle moves along a path given by the hyperbola $$x*y = 16$$. The particle's position changes w

Implicit Differentiation on an Ellipse

Consider the ellipse defined by $$\frac{x^2}{16}+\frac{y^2}{9}=1$$, which represents a track. A runn

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'Hôpital's Analysis

Evaluate the limit $$\lim_{x\to\infty}\frac{5*x^3-4*x^2+1}{7*x^3+2*x-6}$$. Answer the following part

L'Hôpital's Rule in Context

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

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

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

Pool Water Volume Change

The volume of water in a pool is described by the function $$V(t)=8*t^2-32*t+4$$, where $$V$$ is mea

Population Decline Modeled by Exponential Decay

A bacteria population is modeled by $$P(t)=200e^{-0.3t}$$, where t is measured in hours. Answer the

Population Growth Differential

Consider an implicit relationship between a population $$N$$ and time $$t$$ given by $$\ln(N) + t =

Population Growth Rate

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

Related Rates: Circular Oil Spill

An oil spill on a lake forms a circular patch whose area is given by $$A= \pi*r^2$$, where $$r$$ is

Revenue Concavity Analysis

A company's revenue over time is modeled by $$R(t)=100\ln(t+1)-2t$$. Answer the following:

Tangent Line and Rate of Change Analysis

A scientist collected experimental data on the concentration of a chemical, and the graph below repr

Vehicle Motion on a Curved Path

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

Absolute Extrema and the Candidate’s Test

Let $$f(x)=x^3-3x^2-9x+5$$ be defined on the closed interval $$[-2,5]$$. Answer the following parts:

Analysis of Total Distance Traveled

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

Application of the Extreme Value Theorem in Economics

A company's revenue is modeled by $$R(x)= -2*x^2+40*x+100$$, where $$x$$ is the number of units sold

Bouncing Ball with Energy Loss

A ball is dropped from a height of 100 meters. Each time it bounces, it reaches 60% of the height fr

Chemical Reaction Rate

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

Concavity and Inflection Points

Examine the function $$p(x)= x^3-3*x^2-9*x+30$$ to determine its concavity and any inflection points

Elasticity Analysis of a Demand Function

The demand function for a product is given by $$Q(p) = 100 - 5*p + 0.2*p^2$$, where p (in dollars) i

Epidemic Infection Model

In a community experiencing an epidemic, the rate of new infections is modeled by $$I(t)=\frac{200}{

Finding Local Extrema for an Exponential-Logarithmic Function

The function $$g(x)= e^x\ln(x)$$ is defined for $$x > 0$$. Analyze the function as follows:

Fuel Consumption in a Generator

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

Integration of a Series Representing an Economic Model

An economist models the marginal cost by the power series $$MC(q)=\sum_{n=0}^\infty (-1)^n * \frac{q

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.

Inverse Analysis for a Logarithmic Function

Let $$f(x)= \ln(2*x+5)$$ for $$x > -2.5$$. Answer the following parts.

Mean Value Theorem with a Trigonometric Function

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

Modeling Real World with the Mean Value Theorem

A car travels along a straight road with its position at time $$t$$ (in seconds) given by $$ s(t)=0.

Optimization in a Geometric Setting: Garden Design

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

Optimization in a Log-Exponential Model

A firm's profit is given by the function $$P(x)= x\,e^{-x} + \ln(1+x)$$, where x (in thousands) repr

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

Parameter Identification in a Log-Exponential Function

The function $$f(t)= a\,\ln(t+1) + b\,e^{-t}$$ models a decay process with t \(\geq 0\). Given that

Population Growth Modeling

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

Projectile Trajectory: Parametric Analysis

A projectile is launched with its trajectory given in parametric form by $$x(t) = 10*t$$ and $$y(t)

Rate of Change in a Chemical Reaction

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

Related Rates: Draining Conical Tank

Water is draining from a conical tank with a height of \(10\,m\) and a top diameter of \(8\,m\). Wat

Related Rates: Expanding Balloon

A spherical balloon is being inflated so that its volume $$V$$ increases at a constant rate of $$\fr

River Pollution Analysis

A river receives a pollutant through industrial discharge at a rate of $$P_{in}(t)=10*\exp(-0.1*t)$$

Road Trip Analysis

A car's speed (in mph) during a road trip is recorded at various times. Use the table provided to an

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

Square Root Function Inverse Analysis

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

Stock Price Analysis

The daily closing price of a stock (in dollars) is recorded at various days. Use the stock price dat

Stress Analysis in Engineering Structures

A beam experiences stress along its length given by $$S(x)=5*x^3-30*x^2+45*x$$ where x is the distan

Taylor Series for $$e^{\sin(x)}$$

Let $$f(x)=e^{\sin(x)}$$. First, obtain the Maclaurin series for $$\sin(x)$$ up to the $$x^3$$ term,

Temperature Variations

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

Antiderivatives and the Constant of Integration

The function $$f(x)=3*x^{2}$$ has an antiderivative $$F(x)$$.

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

Area Estimation Using Riemann Sums for $$f(x)=x^2$$

Consider the function $$f(x)=x^2$$ on the interval $$[1,4]$$. A table of computed values for the lef

Area Estimation with Riemann Sums

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

Area Under a Parametric Curve

Consider the parametric curve defined by $$x(t)=t$$ and $$y(t)=t^2$$ for $$t \in [0,3]$$. The area u

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

Average Value of an Exponential Function

For the function $$f(x)= x*e^{-x}$$, determine the average value on the interval $$[0,2]$$. Answer t

Calculating Work Using Integration

A variable force is given by $$F(x)=5*x^2-2*x$$ (in Newtons) and is applied along the direction of m

Convergence of an Improper Integral

Consider the function $$f(x)=\frac{1}{x*(\ln(x))^2}$$ for $$x > 1$$.

Cross-Sectional Area of a River Using Trapezoidal Rule

The depth $$h(x)$$ (in meters) of a river’s cross-section is measured at various points along a hori

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 Complex Integral

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

Finding the Area Between Curves

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

Fundamental Theorem of Calculus Application

Let $$F(x)=\int_{2}^{x} (t^{2} - 4*t + 3) dt$$. Answer the following:

Graphical Transformations and Inverse Functions

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

Implicit Differentiation Involving an Integral

Consider the relationship $$y^2 + \int_{1}^{x} \cos(t)\, dt = 4$$. Answer the following parts.

Improper Integral Evaluation

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

Integrated Growth in Economic Modeling

A company experiences revenue growth at an instantaneous rate given by $$r(t)=0.5*t+2$$ (in millions

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)=

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.

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

Partial Fractions Integration

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

Rate of Production in a Factory

A factory has a production rate given by $$R(t)=100+20*\cos\left(\frac{\pi*t}{4}\right)$$ units per

Reservoir Water Level

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

Revenue Accumulation and Constant of Integration

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

Taylor/Maclaurin Series Approximation and Error Analysis

Consider the function $$f(x)=\ln(1+x)$$. This function is infinitely differentiable at x = 0 and has

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

Volume of a Solid: Cross-Sectional Area

A solid has cross-sectional area perpendicular to the x-axis given by $$A(x)= (4-x)^2$$ for $$0 \le

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,

City Population with Migration

The population $$P(t)$$ of a city changes as individuals migrate in at a constant rate of $$500$$ pe

Combined Differential Equations and Function Analysis

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

Cooling Coffee Data Analysis

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

Depreciation Model of a Vehicle

A vehicle's value depreciates continuously over time according to the differential equation $$\frac{

Economic Growth Model

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

Economic Investment Growth Model with Regular Deposits

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

Estimating Instantaneous Rate from a Table

A function $$f(x)$$ is defined by the following table of values:

FRQ 8: RC Circuit Analysis

In an RC circuit, the voltage across the capacitor, $$V(t)$$, satisfies the differential equation $$

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 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

Growth and Decay in a Bioreactor

In a bioreactor, the concentration of a chemical P (in mg/L) evolves according to the differential e

Logistic Growth Population Model

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

Mixing in a Chemical Reaction

A solution in a tank is undergoing a chemical reaction described by the differential equation $$\fra

Mixing Problem in a Saltwater Tank

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

Motion Under Gravity with Air Resistance

An object falling under gravity experiences air resistance proportional to its velocity. Its motion

Newton's Law of Cooling: Temperature Change

A hot object is cooling in a room with an ambient temperature of 20°C. Measurements of the object's

Particle Motion with Damping

A particle moving along a straight line is subject to damping and its motion is modeled by the secon

Population Growth with Logistic Differential Equation

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

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 Drag

A projectile is launched horizontally with an initial velocity $$v_0$$. Due to air resistance, the h

RC Circuit: Voltage Decay

In an RC circuit, the voltage across a capacitor satisfies $$\frac{dV}{dt} = -\frac{1}{R*C} * V$$. G

Salt Tank Mixing Problem

A tank contains $$100$$ L of water with $$10$$ kg of salt. Brine containing $$0.5$$ kg of salt per l

Second-Order Differential Equation: Oscillations

Consider the second-order differential equation $$\frac{d^2y}{dx^2}= -9*y$$ with initial conditions

Separable Differential Equation and Slope Field Analysis

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

Separable Differential Equation with Initial Condition

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

Separation of Variables with Trigonometric Functions

Solve the differential equation $$\frac{dy}{dx} = \frac{\sin(x)}{1+y^2}$$ by using separation of var

Sketching a Solution Curve from a Slope Field

A slope field for the differential equation $$\frac{dy}{dt}=y(1-y)$$ is provided. Use the slope fiel

Variable Carrying Capacity in Population Dynamics

In a modified logistic model, the carrying capacity of a population is time-dependent and given by $

Accumulated Rainfall

The rainfall intensity in a region is given by $$R(t)=0.2*t^2+1$$ (in cm/hour), where $$t$$ is measu

Accumulated Rainfall

The rate of rainfall over a 12-hour storm is modeled by $$r(t)=4*\sin\left(\frac{\pi}{12}*t\right) +

Arc Length of a Curve

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

Arc Length of a Logarithmic Curve

Consider the curve defined by $$y = \ln(\sec(t))$$ for $$t$$ in the interval $$[0,\pi/4]$$. Determin

Area Between a Rational Function and Its Asymptote

Consider the function $$f(x)=\frac{2*x+3}{x+1}$$ and its horizontal asymptote $$y=2$$ over the inter

Area Between Curves: Enclosed Region

The curves $$f(x)=x^2$$ and $$g(x)=x+2$$ enclose a region. Answer the following:

Area Under a Parametric Curve

Consider the parametric equations $$x= t^2$$ and $$y= t^3 + t$$ for $$t \in [0,2]$$. Find the area u

Average Population Density

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

Average Temperature Computation

Consider a scenario in which the temperature (in °C) in a region is modeled by the function $$T(t)=

Average Temperature in a City

An urban planner recorded the temperature variation over a 24‐hour period modeled by $$T(t)=10+5*\si

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.

Draining a Conical Tank Related Rates

Water is draining from a conical tank that has a height of $$8$$ meters and a top radius of $$3$$ me

Implicit Function Differentiation

Consider the implicitly defined function $$\sin(x * y) + x^2 = \ln(y)$$. Answer the following:

Medical Imaging: Reconstruction of a Cross-Section

In a medical imaging technique, the cross-sectional area of a tumor is modeled by $$A(x)=5*e^{-0.5*x

Moment of Inertia of a Thin Plate

A thin plate occupies the region bounded by the curves $$y= x$$ and $$y= x^2$$ for $$0 \le x \le 1$$

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 Change via Rate Integration

A small town’s population changes at a rate given by $$P'(t)=100*e^{-0.3*t}$$ (persons per year) wit

Projectile Motion Analysis

A projectile is launched vertically upward with an initial velocity of $$20$$ m/s. The only accelera

Rainfall Accumulation Analysis

The rainfall rate (in cm/hour) at a location is modeled by $$r(t)=0.5+0.1*\sin(t)$$ for $$0 \le t \l

River Crossing: Average Depth and Flow Calculation

The depth of a river along a 100-meter cross-section is modeled by $$d(x)=2+\cos\left(\frac{\pi}{50}

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

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 via Shell Method

Consider the region bounded by $$y=x^2$$ and $$y=4$$ in the first quadrant. This region is revolved

Volume of a Solid with Variable Cross Sections

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

Water Tank Dynamics: Inflow and Outflow

A water tank receives water through an inflow at a rate given by $$I(t)=10+2*t$$ (liters per minute)

Work Done by a Variable Force

A force acting on an object is given by the function $$F(x)=3*x^2$$ (in Newtons). The object moves a

Work Done in Pumping Water from a Tank

A cylindrical tank has a radius of $$3$$ meters and a height of $$10$$ meters. The tank is completel

Analysis of a Cycloid

A cycloid is defined by the parametric equations $$x(t)=2*(t-\sin(t))$$ and $$y(t)=2*(1-\cos(t))$$ f

Analysis of a Polar Rose

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

Component-Wise Integration of a Vector-Valued Function

Given the acceleration vector $$\textbf{a}(t)= \langle 3\cos(t), -3\sin(t) \rangle$$, answer the fol

Converting and Analyzing a Polar Equation

Examine the polar equation $$r=2+3\cos(\theta)$$.

Converting Polar to Cartesian and Computing Slope

The polar curve is given by the equation $$r=4\cos(\theta)$$. Answer the following:

Curvature of a Parametric Curve

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

Curvature of a Space Curve

Consider the vector-valued function $$\mathbf{r}(t)=\langle \sin(t), \cos(t), t \rangle$$ for $$t \i

Designing a Race Track with Parametric Equations

An engineer designs a race track whose left and right boundaries are described by the curves $$C_1:

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

Determining Curvature from a Vector-Valued Function

Consider the curve defined by $$\mathbf{r}(t)=\langle t, t^2, t^3 \rangle$$ for $$t \ge 0$$. Analyze

Differentiation and Integration of a Vector-Valued Function

Let $$\mathbf{r}(t)=\langle e^{-t}, \sin(t), \cos(t) \rangle$$ for $$t \in [0,\pi]$$.

Discontinuities in a Piecewise-Defined Function

Consider the function $$f(x)= \begin{cases} \frac{x^2-4}{x-2} & x < 2 \\ 3 & x = 2 \\ x+1 & x > 2 \e

Exploring Polar Curves: Spirals and Loops

Consider the polar curve $$r=θ$$ for $$0 \le θ \le 4\pi$$, which forms a spiral. Analyze the spiral

Integration of Speed in a Parametric Motion

For the parametric curve defined by $$x(t)=\cos(t)$$ and $$y(t)=\sin(t)$$ for $$0 \le t \le 2\pi$$,

Integration of Vector-Valued Acceleration

A particle's acceleration is given by the vector function $$\mathbf{a}(t)=\langle 2*t,\; 6-3*t \rang

Intersection Points of Polar Curves

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

Modeling Periodic Motion with a Vector Function

A particle moves in the plane according to the vector function $$\mathbf{r}(t)=\langle \cos(2*t),\;

Modeling with Polar Data

A researcher collects the following polar coordinate data for a phenomenon.

Motion Along a Helix

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

Optimization in Garden Design using Polar Coordinates

A garden is to be designed in the shape of a circular sector with radius $$r$$ and central angle $$\

Parameter Elimination in Logarithmic and Quadratic Relationships

Given the parametric equations $$x(t)= \ln(t)$$ and $$y(t)= t^2 - 4*t + 3$$ for $$t > 0$$, eliminate

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 Intersection of Curves

Consider the curves $$C_1: x(t)=\cos(t),\, y(t)=\sin(t)$$ for $$0 \le t \le 2\pi$$ and $$C_2: x(s)=1

Polar Curve Sketching and Area Estimation

A polar curve is described by sample data given in the table below.

Projectile Motion in Parametric Form

A projectile is launched with an initial speed of $$20\,m/s$$ at an angle of $$30^\circ$$ above the

Projectile Motion via Vector-Valued Functions

A projectile is launched from the origin with an initial velocity given by \(\mathbf{v}(0)=\langle 5

Self-Intersection in a Parametric Curve

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

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 Functions and Curvature

Let the vector-valued function be $$\vec{r}(t)= \langle t, t^2, t^3 \rangle$$.

Vector-Valued Functions in Motion

A particle's position is given by the vector-valued function $$\mathbf{r}(t)=\langle \sin(t), \cos(t