AP Calculus BC FRQ Room

Algebraic Manipulation in Limit Evaluation

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

Algebraic Method for Evaluating Limits

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

Continuity Across Piecewise‐Defined Functions with Mixed Components

Let $$ f(x)= \begin{cases} e^{\sin(x)} - \cos(x), & x < 0, \\ \ln(1+x) + x^2, & 0 \le x < 2, \\

Continuity in Piecewise Defined Functions

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

Continuity in Piecewise Functions with Parameters

A function is defined piecewise by $$f(x)=\begin{cases}a*x^2+3,& x<1,\\ b*x+1,& x\ge 1.\end{cases}$$

Continuous Extension of a Log‐Ratio Function

Define $$g(x)= \frac{\ln(1+e^x)}{x}$$ for $$x \neq 0$$ and let $$g(0)=m$$ be chosen for continuity.

Estimating Limits from Tabulated Data

A function $$g(x)$$ is experimentally measured near $$x=2$$. Use the following data to estimate $$\l

Examining Continuity with an Absolute Value Function

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

Exploring the Squeeze Theorem

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

Exponential Function Limit and Continuity

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

Graphical Analysis of Removable Discontinuity

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

Horizontal Asymptote of a Rational Function

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

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 $$t \neq 3$$.

Investigating Limits and Areas Under Curves

Consider the region bounded by the curve $$y=\frac{1}{x}$$, the vertical line $$x=1$$, and the verti

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 Absolute Value Functions

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

Limits Involving Trigonometric Functions and the Squeeze Theorem

Examine the following trigonometric limits: (a) Evaluate $$\lim_{x\to0} \frac{\sin(4*x)}{x}$$. (b) E

Maclaurin Polynomial Approximation and Error Analysis for $$\ln(1+x)$$

Consider the function $$f(x)=\ln(1+x)$$. You are asked to approximate $$f(0.5)$$ using its Maclaurin

Removable Discontinuity in a Rational Function

Consider the function given by $$f(x)= \frac{(x+3)*(x-1)}{(x-1)}$$ for $$x \neq 1$$. Answer the foll

Squeeze Theorem in Oscillatory Functions

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

Taylor Series Expansion for $$\arctan(x)$$

Consider the function $$f(x)=\arctan(x)$$ and its Taylor series about $$x=0$$.

Water Tank Flow Analysis

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

Water Tank Inflow with Oscillatory Variation

A water tank is equipped with a sensor that records the inflow rate with a slight oscillatory error.

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

Composite Function and Chain Rule Application

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

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

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

Derivative via the Limit Definition: A Rational Function

Consider the function $$f(x)=\frac{1}{x+2}$$. Use the limit definition of the derivative to find $$f

Differentiation from First Principles

Let $$h(x)=3*x^2+2*x-1$$. Use the limit definition of the derivative to analyze this function.

Evaluating the Derivative Using the Limit Definition

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

Exploration of the Definition of the Derivative as a Limit

Consider the function $$f(x)=\frac{1}{x}$$ for $$x\neq0$$. Answer the following:

Hot Air Balloon Altitude Analysis

A hot air balloon’s altitude is modeled by the function $$h(t)=5*\sqrt{t+1}$$, where $$h$$ is in met

Implicit Differentiation in Circular Motion

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

Implicit Differentiation: Conic with Mixed Terms

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

Implicit Differentiation: Square Root Equation

Consider the curve defined by $$\sqrt{x} + \sqrt{y} = \sqrt{10}$$, where $$x, y \ge 0$$.

Logarithmic Differentiation

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

Maclaurin Series for e^x Approximation

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

Optimization and Tangent Lines

A rectangular garden is to be constructed along a river with 100 meters of fencing available for thr

Population Growth Rate

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

Position Recovery from a Velocity Function

A particle moving along a straight line has a velocity function given by $$v(t)=6-3*t$$ (in m/s) for

Radioactive Decay and Derivative

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

Second Derivative of a Composite Function

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

Tangent Line Approximation

Consider the function $$g(t)=t^2 - 4*t + 7$$. Answer the following parts to find the equation of the

Tangent Line Approximation

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

Tangent Line Estimation in Transportation Modeling

A vehicle's displacement along a highway is modeled by $$s(t)=\ln(3*t+1)*e^{t}$$, where $$t$$ denote

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

Traffic Flow Analysis

A highway on-ramp has vehicles entering at a rate of $$V_{in}(t)=30+2*t$$ vehicles per minute and ve

Vibration Analysis: Rate of Change in Oscillatory Motion

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

Bacterial Growth and Nutrient Concentration

A bacterial culture grows such that the number of bacteria at the end of each hour is given by the g

Composite and Implicit Differentiation with Trigonometric Functions

Consider the equation $$\sin(x*y)+x^2=y^2$$ which relates x and y. Answer the following parts:

Composite Function Analysis

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

Composite Function with a Radical in a Shadow Length Model

The length of a shadow cast by an object is modeled by the function $$s(t)= \sqrt{100+4*t^2}$$, wher

Composite Functions in Population Growth

Consider a population $$P(t) = f(g(t))$$ modeled by the functions $$g(t) = 2 + t^2$$ and $$f(u) = 10

Differentiation of Inverse Trigonometric Functions

Consider the function $$f(x)= \sin(x)$$ for $$x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$$ and

Engine Air-Fuel Mixture

In an engine, the fuel injection rate is modeled by the composite function $$F(t)=w(z(t))$$, where $

Exponential Composite Function Differentiation

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

Implicit Differentiation in a Nonlinear Trigonometric Equation

Consider the equation $$ x^2+\sin(x*y)= y^2 $$.

Implicit Differentiation Involving Logarithms

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

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 in Logistic Population Growth

A population model is given by $$P(t)=\frac{100}{1+4e^{-0.5*t}}$$ for t \ge 0. Analyze the inverse f

Inverse Trigonometric Function in a Navigation Problem

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

Parameter Dependent Composite Function

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

Pipeline Pressure and Oil Flow

In an oil pipeline, the driving pressure is modeled by the composite function $$P(t)=r(s(t))$$, wher

Power Series Representation and Differentiation of a Composite Function

Let $$f(x)= \sin(x^2)$$ and consider its Maclaurin series expansion.

Taylor Polynomial and Error Bound for a Trigonometric Function

Let $$f(x) = \cos(2*x)$$. Develop a second-degree Taylor polynomial centered at 0, and analyze the a

Analyzing Motion on a Curved Path

A particle moves along a path defined by $$x(t)=\cos(t)$$ and $$y(t)=\sin(t)$$ for $$t \in [0,2\pi]$

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

Bacterial Culture Dynamics

In a bioreactor, bacteria are introduced at a rate given by $$I(t)=200e^{-0.1t}$$ (cells per minute)

Business Profit Optimization

A firm's profit is modeled by $$P(x)= -4*x^2 + 240*x - 1000$$, where $$x$$ (in hundreds) represents

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

Financial Model Inversion

Consider the function $$f(x)=\ln(x+2)+x$$ which models a certain financial indicator. Although an ex

Inflating Spherical Balloon

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

Integration Region: Exponential and Polynomial Functions

Let the region be bounded by the curves $$y = x^2$$ and $$y = e^x$$. Analyze the area of the region

Inverse Trigonometric Composition

Consider the function $$f(x)=2*\sin(x)-1$$ defined on $$\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$$.

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.

Ladder Sliding Down a Wall

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

Linear Account Growth in Finance

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

Logarithmic Differentiation and Asymptotic Behavior

Let $$f(x)=\frac{(x^2+1)^3}{e^{2x}(x-1)^2}$$ for x > 1. Answer the following:

Maximizing Revenue in a Business Model

A company’s revenue as a function of price is given by $$R(p)= p*(100 - 2*p)$$, where $$p$$ is the p

Particle Motion Analysis

A particle moves along a straight line and its position at time $$t$$ seconds is given by $$s(t)= t^

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

Popcorn Sales Growth Analysis

A movie theater observes that the number of popcorn servings sold increases by 15% each week. Let $$

Population Growth: Rate of Change Analysis

A town's population is modeled by the function $$P(t)=500\, e^{0.03t}$$, where $$t$$ is measured in

Rate of Change in Logarithmic Brightness

The brightness of a star, measured on a logarithmic scale, is given by $$B(t)=\ln(100+t^2)$$, where

Related Rates: Expanding Circular Ripple

A circular ripple in a pond expands such that its area, given by $$A=\pi r^2$$, is increasing at a c

Related Rates: Inflating Spherical Balloon

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

Related Rates: Pool Water Level

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

Revenue Concavity Analysis

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

Series Representation of a CDF

A cumulative distribution function (CDF) is given by $$F(x)= \sum_{n=0}^{\infty} \frac{(-1)^n (x-2)^

Spherical Balloon Inflation

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

Trigonometric Implicit Relation

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

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

Vertical Projectile Motion

An object is thrown vertically upward with an initial velocity of 20 m/s and experiences a constant

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 a Decay Model with Constant Input

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

Analysis of a Function with Oscillatory Behavior

Consider the function $$f(x)=x+\sin(x)$$. Answer the following parts:

Analysis of a Quartic Function as a Perfect Power

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

Application of Rolle's Theorem to a Trigonometric Function

Consider the function $$f(x)=\cos(x)$$ on the interval [0,π]. Answer the following parts:

Application of the Mean Value Theorem in Motion

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

Composite Function and Inverse Analysis

Let $$f(x)= e^(x) - x$$ defined for all real numbers, and consider its behavior.

Derivative Sign Chart and Function Behavior

Given the function $$ f(x)=\frac{x}{x^2+1},$$ answer the following parts:

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

Dynamic Analysis Under Time-Varying Acceleration in Two Dimensions

A particle moves in the plane with acceleration given by $$\vec{a}(t)=\langle3\cos(t),-2\sin(t)\rang

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}{

Error Approximation using Linearization

Consider the function $$f(x) = \sqrt{4*x + 1}$$.

Error Estimation in Approximating $$e^x$$

For the function $$f(x)=e^x$$, use the Maclaurin series to approximate $$e^{0.3}$$. Then, determine

Graphical Analysis of a Differentiable Function

A function $$f(x)$$ is given, and its graph appears as shown in the stimulus. Answer the following p

Inverse Function in a Physical Context

Suppose $$f(t)= t^3 + 2*t + 1$$ represents the displacement (in meters) of an object over time t (in

Investigation of a Fifth-Degree Polynomial

Consider the function $$f(x)=x^5-5*x^4+10*x^3-10*x^2+5*x-1$$. Answer the following parts:

Light Reflection Between Mirrors

A beam of light is directed between two parallel mirrors. With each reflection, 70% of the light’s i

Mean Value Theorem in Motion

A car travels along a straight road and its position is modeled by $$s(x) = x^2$$ (in kilometers), w

Numerical Integration using Taylor Series for $$\cos(x)$$

Approximate the integral $$\int_{0}^{0.5} \cos(x)\,dx$$ by using the Maclaurin series for $$\cos(x)$

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

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

Taylor Series for $$\frac{1}{1-3*x}$$

Consider the function $$f(x)=\frac{1}{1-3*x}$$. Derive its Taylor series expansion about $$x=0$$, de

Taylor Series for $$\sqrt{1+x}$$

Consider the function $$f(x)=\sqrt{1+x}$$. In this problem, compute its 3rd degree Maclaurin polynom

Temperature Change in a Weather Balloon

A weather balloon’s temperature and altitude are related by the implicit equation $$T*e^{z} + z = 50

Volume by Cross Sections Using Squares

A region in the xy-plane is bounded by $$y=x$$, $$y=0$$, and $$x=3$$. Perpendicular to the x-axis, c

Water Tank Volume Analysis

Water is being added to a tank at a varying rate given by $$r(t) = 3*t^2 - 12*t + 15$$ (in liters/mi

Accumulated Change Prediction

A population grows continuously at a rate proportional to its size. Specifically, the growth rate is

Accumulated Change via U-Substitution

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

Accumulation Function Analysis

A function $$A(x) = \int_{0}^{x} (e^{-t} + 2)\,dt$$ represents the accumulated amount of a substance

Arc Length of a Power Function

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

Area Between Curves

Consider the curves $$f(x)=x^2$$ and $$g(x)=4*x - 3$$. These curves intersect and enclose a region.

Area Under a Parametric Curve

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

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 \

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

Consumer Surplus in an Economic Model

For a particular product, the demand function is given by $$D(p)=100 - 5p$$ and the supply function

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

Distance from Acceleration Data

A car's acceleration is recorded in the table below. Given that the initial velocity is $$v(0)= 10$$

Economic Surplus: Area between Supply and Demand Curves

In an economic model, the demand function is given by $$D(x)=10 - x^2$$ and the supply function by $

Economics: Accumulated Earnings

A company’s instantaneous revenue rate (in dollars per day) is modeled by the function $$R(t)=1000\s

Evaluating an Integral via U-Substitution

Evaluate the integral $$\int_{1}^{5} (x-4)^{10}\,dx$$ using u-substitution.

Graphical Transformations and Inverse Functions

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

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 Rational Function via Partial Fractions

Evaluate the indefinite integral $$\int \frac{2*x+3}{x^2+x-2}\,dx$$ by using partial fractions.

Integration of a Trigonometric Function by Two Methods

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

Marginal Cost and Production

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

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

Power Series Analysis and Applications

Consider the function with the power series representation $$f(x)=\sum_{n=0}^\infty \frac{(-1)^n x^{

Riemann Sum Approximation of Area

Given the following table of values for the function $$f(x)$$ on the interval $$[0,4]$$, use Riemann

Riemann Sum from a Table: Plant Growth Data

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

Trapezoidal Rule Error Estimation

Given the function $$f(x)=\ln(x)$$ on the interval $$[1,4]$$, answer the following:

U-Substitution Integration Challenge

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

Variable Interest Rate and Continuous Growth

An investment grows continuously with a variable interest rate given by $$r(t)=0.05+0.01*t$$. The in

Volume of a Solid with Square Cross-Sections

Consider the region bounded by the curve $$y=x^{2}$$ and the line $$y=4$$. Cross-sections taken perp

Work Done by a Variable Force

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

Analysis of a Nonlinear Differential Equation

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

Analysis of an Inverse Function from a Differential Equation Solution

Suppose a differential equation is solved to give an increasing function $$f(x)=\ln(2*x+3)$$ defined

Area and Volume from a Differential Equation-derived Family

Consider the family of curves that are solutions to the differential equation $$\frac{dy}{dx} = 2*x$

Autonomous Differential Equations and Stability Analysis

An autonomous differential equation has the form $$\frac{dy}{dt} = f(y)$$ with critical points at $$

Basic Separation of Variables: Solving $$\frac{dy}{dx}=\frac{x}{y}$$

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

Braking of a Car

A car decelerates according to the differential equation $$\frac{dv}{dt} = -k*v$$, where k is a posi

Capacitor Discharge in an RC Circuit

In an RC circuit, the voltage $$V(t)$$ across a discharging capacitor obeys the differential equatio

Chemical Reaction in a Closed System

The concentration $$C(t)$$ of a reactant in a closed system decreases according to the differential

Chemical Reaction Rate Modeling

In a chemical reaction, the concentration $$C(t)$$ (in moles per liter) of a reactant decreases acco

City Population with Migration

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

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

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

Epidemic Spread Modeling

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

Euler's Method Approximation

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

Free-Fall with Air Resistance

An object falling under gravity experiences air resistance proportional to the square of its velocit

FRQ 3: Population Growth and Logistic Model

A population $$P(t)$$ grows according to the logistic differential equation $$\frac{dP}{dt} = k*P\le

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 12: Bacterial Growth with Limiting Resources

A bacterial culture grows according to the differential equation $$\frac{dP}{dt}=r*P-c*P^2$$, where

Logistic Growth in Populations

A population of organisms is modeled by the logistic differential equation $$\frac{dP}{dt} = rP \lef

Logistic Growth Model

A population P grows according to the logistic differential equation $$\frac{dP}{dt} = rP\left(1-\fr

Mixing Problem with Constant Flow Rate

A tank holds 500 L of water and initially contains 10 kg of dissolved salt. Brine with a salt concen

Mixing Problem with Constant Rates

A tank contains $$200\,L$$ of a well-mixed saline solution with $$5\,kg$$ of salt initially. Brine w

Motion along a Line with a Separable Differential Equation

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

Newton's Law of Cooling

Newton's Law of Cooling is given by the differential equation $$\frac{dT}{dt} = -k*(T-T_a)$$, where

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

Population Dynamics with Harvesting

A fish population in a lake is modeled by the logistic equation with harvesting: $$\frac{dP}{dt}=r\,

Population Growth with Harvesting

A fish population in a lake is modeled by the differential equation $$\frac{dP}{dt}= rP - H$$, where

Projectile Motion with Air Resistance

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

Rainfall in a Basin: Differential Equation Model

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

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

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

Temperature Change and Differential Equations

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

Tumor Growth Under Chemotherapy

A tumor's size $$S(t)$$ (in cm³) grows at a rate proportional to its size, at $$0.08*S(t)$$, but che

Variable Carrying Capacity in Population Dynamics

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

Verification of a Candidate Solution

Consider the candidate solution $$y(x)= \sqrt{4*x^2+3}$$ proposed for the differential equation $$\f

Arc Length in Polar Coordinates

Find the length of the curve defined in polar coordinates by $$r(θ)= 1+ \cos(θ)$$ for $$θ \in [0, 2\

Area Between a Parabola and a Line

Consider the region bounded by the curves $$y=5*x - x^2$$ and $$y=x$$ where they intersect. Answer t

Area Between Curves: Park Design

A park layout is bounded by two curves: $$f(x)=10-x^2$$ and $$g(x)=2*x+2$$. Answer the following par

Area Between Economic Curves

In an economic model, two cost functions are given by $$C_1(x)=100-2*x$$ and $$C_2(x)=60-x$$, where

Area of One Petal of a Polar Curve

The polar curve defined by $$r = \cos(2\theta)$$ forms a rose with four petals. Find the area of one

Average Temperature in a City

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

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

Averaging Chemical Concentration in a Reactor

In a chemical reactor, the concentration of a substance is given by $$C(t)=100*e^{-0.5*t}+20$$ (mg/L

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

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 Length of a Curve

Find the arc length of the curve given by $$y=\sqrt{4*x}$$ for $$x\in[0,9]$$.

Displacement and Distance from a Variable Velocity Function

A particle moves along a straight line with velocity function $$v(t)= \sin(t) - 0.5$$ for $$t \in [0

Distance Traveled from a Velocity Function

A car has a velocity function given by $$v(t)=t^2-4*t+3$$ (in m/s) for $$t$$ in seconds from 0 to 5.

Electric Charge Distribution Along a Rod

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

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

Integral Approximation Using Taylor Series

Approximate the integral $$\int_{0}^{0.2} \sin(2*x)\,dx$$ by using the Taylor series expansion of $$

Mass of a Wire with Variable Density

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

Optimizing the Shape of a Parabolic Container

A container is formed by rotating the region under the curve $$y=8 - x^2$$ for $$0 \le x \le \sqrt{8

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)

Particle Motion with Variable Acceleration

A particle's acceleration is given by $$a(t)=4*e^{-t} - 2$$ for $$t$$ in seconds over the interval $

Particle Motion with Velocity Reversal

A particle moves along a straight line with an acceleration given by $$a(t)=12-6*t$$ (in m/s²) for $

Projectile Motion with Constant Acceleration

A ball is thrown upward and moves under the constant acceleration due to gravity $$a(t)=-9.8$$ (in m

Rainfall Accumulation Analysis

A local weather station records the rainfall intensity (in mm/h) over a 6-hour period. Use integrati

Salt Concentration in a Mixing Tank

A tank initially contains 50 L of water with 5 g of salt. A salt solution with a concentration of 0.

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 Hollow Cylinder Using the Washer Method

A manufacturer designs a hollow cylindrical container. The outer surface is modeled by $$y=10-\sqrt{

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 a Solid with the Washer Method

Consider the region bounded by $$y=x^2$$ and $$y=0$$ between $$x=0$$ and $$x=1$$. This region is rot

Volume with Square Cross-Sections

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

Work Done 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

Analyzing a Cycloid

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

Arc Length of a Parametric Curve

The curve defined by $$x(t)=\ln(t+1)$$ and $$y(t)=\sqrt{t}$$ for $$t \in [0,4]$$ is given.

Arc Length of a Vector-Valued Curve

A vector-valued function is given by $$\mathbf{r}(t)=\langle e^t,\, \sin(t),\, \cos(t) \rangle$$ for

Area and Tangent for a Polar Curve

The polar curve is defined by $$r = 2+\cos(\theta)$$.

Computing the Area Between Two Polar Curves

Consider the polar curves given by $$R(\theta)=3+2*\cos(\theta)$$ (outer curve) and $$r(\theta)=1+\c

Converting Polar to Cartesian and Computing Slope

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

Curvature Analysis of a Space Curve

Consider the vector function $$\mathbf{r}(t)=\langle t,\; t^2,\; \ln(t+1)\rangle$$ for $$t\geq0$$. A

Curvature and Oscillation in Vector-Valued Functions

Given the vector function $$\vec{r}(t)=\langle \ln(t+1), \cos(t) \rangle$$ for $$t \ge 0$$, answer t

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:

Finding the Slope of a Tangent to a Parametric Curve

Consider the parametric equations $$x(t)=e^t$$ and $$y(t)=e^{-t}$$, where $$t \in \mathbb{R}$$.

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 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)=

Oscillatory Motion in a Vector-Valued Function

Consider the vector-valued function $$\vec{r}(t)= \langle \sin(2*t), \cos(3*t) \rangle$$ for $$t \in

Parameter Values from Tangent Slopes

A curve is defined parametrically by $$x(t)=t^2-4$$ and $$y(t)=t^3-3t$$. Answer the following:

Parametric Curve with a Loop and Tangent Analysis

Consider the parametric curve defined by $$x(t)=t^3-3*t$$ and $$y(t)=t^2$$, where $$t\in[-2,2]$$. An

Parametric Curve: Intersection with a Line

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

Parametric Equations and 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 and Tangency

Two curves are given in parametric form by: Curve 1: $$x_1(t)=t^2,\, y_1(t)=2t$$; Curve 2: $$x_2(s

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

Parametric Particle with Acceleration and Jerk

A particle's motion is given by the parametric equations $$x(t)=t^4-6*t^2$$ and $$y(t)=2*t^3-9*t$$ f

Parametric Spiral Curve Analysis

The curve defined by $$x(t)=t\cos(t)$$ and $$y(t)=t\sin(t)$$ for $$t \in [0,4\pi]$$ represents a spi

Particle Motion in Circular Motion

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

Particle Trajectory in Parametric Motion

A particle moves along a curve with parametric equations $$x(t)= t^2 - 4*t$$ and $$y(t)= t^3 - 3*t$$

Polar and Parametric Form Conversion

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

Polar to Parametric Conversion and Arc Length

A curve is defined in polar coordinates by $$r= 1+\sin(\theta)$$. Convert and analyze the curve.

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

Real-World Data Analysis from Tabular Measurements

A vehicle's distance (in meters) along a straight road is recorded at various times (in seconds) as

Related Rates: Expanding Balloon

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

Roller Coaster Design: Parametric Path

A roller coaster is modeled by the parametric equations $$x(t)=t-\cos(t)$$ and $$y(t)=t-\sin(t)$$ fo

Vector-Valued Function Integration

A particle moves along a straight line with constant acceleration given by $$ a(t)=\langle 6,\;-4 \r

Vector-Valued Function with Constant Acceleration

A particle moves in the plane with its position given by $$\vec{r}(t)=\langle 5*t, 3*t+2*t^2 \rangle

Velocity and Acceleration of a Particle

A particle’s position in three-dimensional space is given by the vector-valued function $$\mathbf{r}