AP Calculus BC FRQ Room

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 of a Composite Function

Let $$f(x)=\frac{\sin(\sqrt{4+x}-2)}{x}$$ for $$x \neq 0$$. Answer the following:

Calculating Tangent Line from Data

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

Continuity in Piecewise Functions with Parameters

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

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 a Limit with Algebraic Manipulation

Examine the function $$g(x)= \frac{\sqrt{x+9}-3}{x}$$ for $$x \neq 0$$.

Experimental Data Limit Estimation from a Table

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

Fuel Efficiency and Speed Graph Analysis

A car manufacturer records data on fuel efficiency (in mpg) as a function of speed (in mph). A graph

Graphical Analysis of a Continuous Polynomial Function

Consider the function $$f(x)=2*x^3-5*x^2+x-7$$ and its graph depicted below. The graph provided accu

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

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 at Infinity and Asymptotic Behavior

Given the rational function $$f(x)=\frac{5*x^2-3*x+2}{2*x^2+x-1}$$, answer the following: (a) Evalua

Limits Involving Absolute Value Functions

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

Limits via Improper Integration Representation

Consider the function defined by the integral $$f(x)= \int_{1}^{x} \frac{1}{t^2} dt$$ for x > 1. Add

Limits with a Parameter in a Trigonometric Function

Consider the function $$f(x)= \begin{cases} \frac{\sin(a*x)}{x} & x \neq 0 \\ b & x=0 \end{cases}$$,

Logarithmic Function Limits

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

Piecewise Function Continuity

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

Seasonal Temperature Curve Analysis

A graph represents the average daily temperature (in $$^\circ C$$) as a function of the day of the y

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

Water Tank Flow Analysis

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

Analyzing a Function with an Oscillatory Component

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

Applying Product and Quotient Rules

For the function $$h(x)=\frac{(3*x^2+2)*(x-4)}{x+1}$$, determine its derivative by appropriately app

Car Acceleration: Secant and Tangent Slope

A car's position along a straight road is given by $$s(t)= 2t^3 - 9t^2 + 12t$$, where s is in meters

Cooling Model Rate Analysis

The temperature of a cooling object is modeled by $$T(t)=e^{-2*t}+\ln(t+3)$$, where $$t$$ is time in

Derivative Using Limit Definition

Let $$f(x)=\frac{1}{x+2}$$. Using the definition of the derivative, find $$f'(x)$$.

Derivatives of Inverse Functions

Let $$f(x)=\ln(x)$$ with inverse function $$f^{-1}(x)=e^x$$. Answer the following parts.

Differentiation in Biological Growth Models

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

Differentiation of a Rational Function

Consider the function $$f(x) = \frac{2*x^2+3*x}{x-1}$$, which is defined on its domain. Analyze the

Engineering Analysis of Log-Exponential Function

In an engineering system, the output voltage is given by $$V(x)=\ln(4*x+1)*e^{-0.5*x}$$, where $$x$$

Epidemiological Rate Change Analysis

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

Error Bound Analysis for $$e^{2x}$$

In a study of reaction rates, the function $$f(x)=e^{2*x}$$ is used. Analyze the error in approximat

Graphical Derivative Analysis

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

Implicit Differentiation for a Rational Equation

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

Implicit Differentiation with Inverse Functions

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

Instantaneous Rate of Change in Motion

A particle moves along a straight line with position given by $$s(t)=t^3-6*t^2+9*t+2$$, where $$t$$

Limit Definition of Derivative for a Rational Function

For the function $$f(x)=\frac{1}{x+1}$$, use the limit definition of the derivative to answer the fo

Logarithmic Differentiation

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

Optimization in Engineering Design

A manufacturer designs a cylindrical can with a fixed volume of $$1000\,cm^3$$. The surface area of

Optimization Using Derivatives

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

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

Renewable Energy Storage

A battery storage system experiences charging at a rate of $$C(t)=50+10\sin(0.5*t)$$ kWh and dischar

Satellite Orbit Altitude Modeling

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

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

Tangent Line to a Logarithmic Function

Consider the function $$f(x)= \ln(x+1)$$.

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

Chain Rule and Implicit Differentiation in a Pendulum Oscillation Experiment

In a pendulum experiment, the angle \(\theta(t)\) in radians satisfies the relation $$\cos(\theta(t)

Chain Rule Combined with Inverse Trigonometric Differentiation

Let $$h(x)= \arccos((2*x-1)^2)$$. Answer the following:

Composite and Inverse Differentiation in an Electrical Circuit

In an electrical circuit, the current is modeled by $$ I(t)= \sqrt{20*t+5} $$ and the voltage is giv

Composite Differentiation in Biological Growth

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

Composite Function Analysis

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

Composite Function Modeling with Chain Rule

A chemical reaction rate is modeled by the composite function $$R(x)=f(g(x))$$ where $$f(u)=\sin(u)$

Composite Function with Implicitly Defined Inner Function

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

Composite Functions in Biological Growth

Let a model for bacteria growth be represented by $$f(t)=e^{2*t}$$, and let the effect of nutrient c

Differentiation of an Inverse Trigonometric Function

Define $$h(x)= \arctan(\sqrt{x})$$. 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 Implicit Differentiation in a Nonlinear Model

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

Implicit Differentiation in a Conic Section

Consider the curve defined by $$x^2 + x*y + y^2 = 9$$.

Implicit Differentiation in a Non-Standard Function

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

Implicit Differentiation Involving Logarithms

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

Implicit Differentiation with Exponential and Trigonometric Components

Consider the relation $$ (x^2 + y^2) * e^{y} = x $$. Answer the following:

Inverse Analysis via Implicit Differentiation for a Transcendental Equation

Consider the equation $$e^{x*y}+x-y=0$$ defining y implicitly as a function of x near a point where

Inverse Differentiation of a Trigonometric Function

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

Inverse Function Analysis for Exponential Functions

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

Inverse Function Derivative in an Exponential Model

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

Inverse Function Differentiation for a Cubic Function

Let $$f(x)= x^3 + x$$ be an invertible function with inverse $$g(x)$$. Use the inverse function deri

Inverse Function Differentiation in a Trigonometric Context

Let $$f(x)= \sin(x) + x$$, defined on the interval $$[0, \frac{\pi}{2}]$$, and let $$g$$ be its inve

Inverse Function Differentiation in Thermodynamics

In a thermodynamics experiment, a differentiable one-to-one function $$f$$ describes the temperature

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 of a Piecewise Function

Consider the piecewise function $$f(x)=\begin{cases} x^2 & x \ge 0 \\ -x & x < 0 \end{cases}$$. Anal

Inverse of a Shifted Logarithmic Function

Analyze the function $$f(x)=\ln(x-1)+2$$ defined for $$x>1$$ and its inverse.

Inverse Trigonometric Functions in Navigation

A ship navigates such that its angular position relative to a fixed reference is given by $$\theta =

Navigation on a Curved Path: Boat's Eastward Velocity

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

Projectile Motion and Composite Exponential Functions

A projectile’s height at time $$t$$ (in seconds) is modeled by the function $$h(t)= e^{- (t-2)^2}$$.

Reservoir Level: Inverse Function Application

A reservoir's water level $$h$$ (in feet) is related to time $$t$$ (in minutes) through an invertibl

Second Derivative via Implicit Differentiation

Given the relation $$x^2 + x*y + y^2 = 7$$, answer the following:

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

Temperature Control: Heating Element Dynamics

A room's temperature is controlled by a heater whose output is given by the composite function $$H(t

Biological Growth Rate

A bacterial culture grows according to the model $$P(t)= 500*e^{0.8*t}$$, where \(P(t)\) is the popu

Concavity and Acceleration in Motion

A car’s position is modeled by $$s(t)= t^3 - 6*t^2 + 9*t+5$$ with time $$t$$ in seconds. Analyze the

Cooling Coffee: Temperature Change

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

Drug Concentration Dynamics

The concentration of a drug in the bloodstream is modeled by $$C(t)= 50*e^{-0.2*t} + 10$$ (in mg/L),

Estimation Error with Differentials

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

Implicit Differentiation: Tangent to a Circle

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

Inflating Balloon: Radius and Surface Area

A spherical balloon is being inflated such that its volume increases at a constant rate of 12 cm³/s.

Instantaneous vs. Average Speed in a Race

An athlete’s displacement during a 100 m race is modeled by $$s(t)=2*t^3-t^2+1$$, where $$s(t)$$ is

L'Hospital's Rule in Indeterminate Form Computation

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

Linearization to Estimate Change in Electrical Resistance

The resistance of a wire is modeled by $$R(T)=R_0(1+\alpha*T)$$, where $$R_0=100$$ ohms and $$\alpha

Logarithmic Function Series Analysis

The function $$L(x)= \sum_{n=1}^{\infty} \frac{(-1)^{n+1}(x-1)^n}{n}$$ represents $$\ln(x)$$ centere

Optimization with Material Costs

A company plans to design an open-top rectangular box with a square base that must have a volume of

Piecewise Velocity and Acceleration Analysis

A particle moves along a straight line with its velocity given by $$ v(t)= \begin{cases} t^2-4*t+3,

Pollution Decay and Inversion

A model for pollution decay is given by the function $$f(t)=\frac{100}{1+t}$$ where $$t\ge0$$ repres

Projectile Motion Analysis

A projectile is launched such that its horizontal and vertical positions are modeled by the parametr

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: Inflating Spherical Balloon with Exponential Volume Rate

A spherical balloon is being inflated so that its volume changes at a rate of $$\frac{dV}{dt}=8e^{0.

Security Camera Angle Change

A security camera is mounted on a 15 m tall tower. Let $$x$$ denote the horizontal distance from the

Series Analysis in Profit Optimization

A company's profit function near a break-even point is approximated by $$\pi(x)= 1000 + \sum_{n=1}^{

Series Integration for Work Calculation

A force along a displacement is given by $$F(x)= \sum_{n=0}^{\infty} \frac{(-1)^n x^n}{n+2}$$ (in Ne

Water Filtration Plant Analysis

A water filtration plant processes water entering at a rate of $$I(t)=60-2t$$ (liters per minute) an

Analysis of Relative Extrema and Increasing/Decreasing Intervals

A particle moves along a line with position given by $$s(x)=x^3-6*x^2+9*x+4$$, where $$x$$ represent

Average and Instantaneous Velocity Analysis

A bird’s position is given by $$s(t)=2*t^2-t+1$$ (in meters) for $$t\in[0,3]$$ seconds.

Car Depreciation Analysis

A new car is purchased for $$30000$$ dollars. Its value depreciates by 15% each year. Analyze the de

Chemical Reaction Rate

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

Concavity Analysis of a Population Growth Model

A biologist models a species’ population (in thousands) with the function $$f(x) = x^3 - 9*x^2 + 24*

Concavity and Inflection Points

Let $$f(x)=x^3-6x^2+9x+2.$$ Answer the following parts:

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

Determining Absolute Extrema for a Trigonometric-Polynomial Function

Consider the function $$f(x)= x+\cos(x)$$ defined on the closed interval $$[0, 2\pi]$$. Determine th

Determining Absolute Extrema in a Motion Context

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

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

Exponential Decay in Velocity

A particle’s velocity is modeled by the function $$v(t)=10e^{-0.5*t}-3$$ (in m/s) for $$t\ge0$$.

Extreme Value Analysis of a Logarithmic-Exponential-Polynomial Function

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

Inverse Function and Critical Points in a Business Context

A company models its profit (in thousands of dollars) by $$f(x)= \ln(4*x+7)$$ for $$x \ge 0$$, where

Investigating a Composite Function Involving Logarithms and Exponentials

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

Loan Amortization with Increasing Payments

A loan of $$20000$$ dollars is to be repaid in equal installments over 10 years. However, the repaym

Mean Value Theorem Application

Let \( f(x) = x^3 - 3*x^2 + 2 \) be defined on the closed interval \([0,3]\). Answer the following p

Mean Value Theorem with Trigonometric Function

Consider the function $$f(x)= \sin(x)$$ on the interval $$[0,\pi]$$.

Optimizing Fencing for a Field

A farmer has $$100$$ meters of fencing to construct a rectangular field that borders a river (no fen

Rolle's Theorem on a Cubic Function

Consider the cubic function $$f(x)= x^3-3*x^2+2*x$$ defined on the interval $$[0,2]$$. Verify that t

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

Tangent Line and Linearization

Consider the function $$ f(x)=\sqrt{x+5}.$$ Answer the following parts:

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

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

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 Change in a Weather Balloon

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

Vector Analysis of Particle Motion

A particle moves in the plane with its position given by the vector function $$\mathbf{r}(t) = \lang

Accumulation Function in an Investment Model

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

Antiderivative with Initial Condition

Find the general antiderivative of the function $$f(x)=5*x^3-2*x+6$$ and determine the particular an

Antiderivatives and the Constant of Integration

Consider the rate function $$ r(t)= 2*t + 3 $$ where t represents time in seconds.

Applying the Fundamental Theorem of Calculus

Consider the function $$f(x)=2*x$$. Use the Fundamental Theorem of Calculus to evaluate the definite

Area and Volume of a Region Bounded by Trigonometric Functions

Consider the curves $$y=\sin(x)$$ and $$y=\cos(x)$$ for $$0 \le x \le \frac{\pi}{4}$$. Answer the fo

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

Biomedical Modeling: Drug Concentration Dynamics

A drug concentration in the bloodstream is modeled by $$f(t)= 5\left(1 - e^{-0.3*t}\right)$$ for $$t

Continuity and Integration of a Sinc-like Function

Consider the function $$ f(x)= \begin{cases} \frac{\sin(x)}{x} & \text{if } x \neq 0, \\ 1 & \text{i

Drug Concentration in a Bloodstream

A patient receives an intravenous drug infusion at a rate $$R_{in}(t)=3e^{-0.1*t}$$ mg/min for $$0 \

Economic Applications: Consumer and Producer Surplus

In a market, the demand function is given by $$D(p)=100-2p$$ and the supply function by $$S(p)=20+3p

Error Analysis in Riemann Sum Approximations

Consider approximating the integral $$\int_{0}^{2} x^3\,dx$$ using a left-hand Riemann sum with $$n$

Error Estimation in Riemann Sum Approximations

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

Estimating Area Under a Curve Using Riemann Sums

A function $$f(x)$$ is defined on the interval $$[0,6]$$. The following table provides the values of

Evaluating an Integral Involving an Exponential Function

Evaluate the definite integral $$\int_{0}^{\ln(4)} e^{x}\,dx$$.

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

Fundamental Theorem and Total Accumulated Growth

A bacteria culture grows according to the logistic model $$\frac{dN}{dt}=N\left(1-\frac{N}{10000}\r

Heat Energy Accumulation

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

Integration by U-Substitution and Evaluation of a Definite Integral

Evaluate the definite integral $$\int_{0}^{1} \frac{2*t}{(t^2+1)^2}\, dt$$ by applying U-substitut

Non-Uniform Subinterval Riemann Sum

A function $$f(t)$$ is measured at non-uniform time intervals as recorded in the table below: | t (

Population Model Inversion and Accumulation

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

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 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

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

Rocket Height Determination via U-Substitution

A rocket’s velocity is modeled by the function $$v(t)=t * e^(t^(2))$$ (in m/s) for $$t \ge 0$$. With

Scooter Motion with Variable Acceleration

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

Series Representation and Term Operations

Consider the power series for $$\arctan(x)$$ given by $$\sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{2n+

Tank Filling Problem

Water flows into a tank at a rate given by $$R(t)=8e^{-0.5*t}+2$$ (in liters per minute) for $$t\geq

Trapezoidal and Riemann Sums from Tabular Data

A scientist collects data on the concentration of a chemical over time as given in the table below.

Trapezoidal Approximation of a Definite Integral from Tabular Data

The table below shows the height H(t) (in meters) of a liquid in a tank at specific times. Use a tra

Volume by Cross-Section: Squares on a Parabolic Base

A solid has a base in the xy-plane bounded by the curves $$y=x^2$$ and $$y=4$$. Cross-sections perpe

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 Tank Inflow and Outflow

A water tank begins operation at t = 0 with an initial volume of 0 liters. Water flows in through an

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 Piecewise Function with Potential Discontinuities

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

Capacitor Discharge in an RC Circuit

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

Complex Related Rates Problem Involving a Moving Ladder

A 10-meter ladder leans against a vertical wall. The bottom of the ladder slides away from the wall

Cooling with Time-Varying Ambient Temperature

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

Differential Equations in Economic Modeling

An economist models the rate of change of a commodity price $$P(t)$$ with the differential equation

Direction Fields and Phase Line Analysis

Consider the autonomous differential equation $$\frac{dy}{dt}=(y-2)(3-y)$$. Answer the following par

Disease Spread Model

In a simplified epidemiological model, the number of infected individuals \(I(t)\) evolves according

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

In a simplified epidemic model, the number of infected individuals $$I(t)$$ is modeled by the logist

Euler's Method Approximation

Consider the initial value problem $$\frac{dy}{dt}=t\sqrt{y}$$ with $$y(0)=1$$. Use Euler's method w

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 8: RC Circuit Analysis

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

FRQ 18: Enzyme Reaction Rates

A chemical concentration $$C(t)$$ in a reaction decreases according to the differential equation $$\

Growth and Decay in a Bioreactor

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

Homogeneous Differential Equation

Solve the homogeneous differential equation $$\frac{dy}{dx}= \frac{x^2+y^2}{x*y}$$ using the substit

Implicit Differentiation and Homogeneous Equation

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

Infectious Disease Spread Model

In a closed population of N individuals, the number of infected individuals $$I(t)$$ is modeled by t

Interpreting Slope Fields for a Differential Equation

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

Investment Growth with Nonlinear Dynamics

The rate of change of an investment amount $$I$$ is modeled by the nonlinear differential equation $

Logistic Equation with Harvesting

A fish population in a lake follows a logistic growth model with the addition of a constant harvesti

Logistic Population Model

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

Medicine Infusion and Elimination Model

A patient receives an intravenous infusion of a drug such that the infusion rate is $$R(t)=e^{0.2*t}

Mixing Problem in a Tank

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

Modeling Cooling and Heating: Temperature Differential Equation

Suppose the temperature of an object changes according to the differential equation $$\frac{dT}{dt}

Modeling Disease Spread with Differential Equations

In a simple model for disease spread, the number of infected individuals, $$I(t)$$, evolves accordin

Modeling Free Fall with Air Resistance

An object falls under gravity while experiencing air resistance proportional to its velocity. The mo

Newton's Law of Cooling

A hot liquid cools in a room maintained at a constant temperature $$T_{room}$$. The temperature $$T(

Population Dynamics with Harvesting

Consider a population model that includes constant harvesting, given by the differential equation $$

Population Growth with Logistic Differential Equation

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

Projectile Motion with Air Resistance

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

RC Circuit: Voltage Decay

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

Reservoir Contaminant Dilution

A reservoir has a constant volume of 10,000 L and contains a pollutant with amount $$Q(t)$$ (in kg)

Saltwater Mixing Problem

A tank initially contains 1000 L of a salt solution with a concentration of 0.2 kg/L (thus 200 kg of

Separable Differential Equation and Maclaurin Series Approximation

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

Separable Differential Equation and Slope Field Analysis

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

Temperature Change with Variable Ambient Temperature

A heated object is cooling in an environment where the ambient temperature changes over time. For $$

Arc Length of a Logarithmic Curve

Determine the arc length of the curve $$f(x)= \ln(x)$$ on the interval $$[1,e]$$.

Area Between Curves in Window Design

An architect is designing a decorative window whose outline is bounded by the curves $$y=5*x-x^2$$ a

Area Between Curves: Parabolic and Linear Functions

Consider the functions $$f(x)=5*x-x^2$$ and $$g(x)=x$$. Determine the area enclosed between these cu

Area Calculation: Region Under a Parabolic Curve

Let $$f(x)=4-x^2$$. Consider the region bounded by the curve $$f(x)$$ and the x-axis.

Average and Instantaneous Acceleration

For a particle, the acceleration is given by $$a(t)=4*\sin(t)-t$$ (in m/s²) for $$t\in[0,\pi]$$. Giv

Average Chemical Concentration Analysis

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

Average Daily Temperature

The temperature during a day is modeled by $$T(t)=10+5*\sin((\pi/12)*t)$$ (in °C), where $$t$$ is th

Average Temperature of a Day

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

Average Value of a Temperature Function

A region’s temperature throughout a day is modeled by the function $$T(t)=10+5*\sin(\frac{\pi}{12}*t

Average Value of a Trigonometric Function

A function representing sound intensity is given by $$I(t)= 4*\cos(2*t) + 10$$ over the time interva

Center of Mass of a Non-uniform Rod

A thin rod of length 10 m has a linear density given by $$\lambda(x)= 3 + 0.5*x$$ (in kg/m) for $$0

Center of Mass of a Rod with Variable Density

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

Cost Analysis: Area Between Production Cost Curves

Suppose two cost functions for producing goods are given by $$f(x)=20+2*x$$ and $$g(x)=5*x-\frac{1}{

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.

Economic Analysis: Consumer and Producer Surplus

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

Electrical Charge Distribution

A wire of length 3 meters carries a charge distribution given by $$\rho(x)=\frac{5}{1+x^2}$$ (in cou

Kinematics: Motion with Variable Acceleration

A particle moves along a straight line with acceleration $$a(t)=4-2*t$$ (in m/s²). The particle has

Optimization of Material Usage in a Container

A container's volume is given by $$V(h)=\int_0^h \pi*(3-0.5*\ln(1+x))^2dx$$, where $$h$$ is the heig

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

Position and Velocity from Tabulated Data

A particle’s velocity (in m/s) is measured at discrete time intervals as shown in the table. Use the

Projectile Motion under Gravity

An object is projected vertically upward with an initial velocity of $$20$$ m/s and from an initial

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.

Surface Area of a Rotated Curve

Consider the curve $$y=x^3$$ on the interval $$[0,2]$$. This curve is rotated about the x-axis, form

Surface Area of a Rotated Parabolic Curve

The curve $$y = x^2$$ is rotated about the x-axis for $$x$$ in the interval $$[0,3]$$ to form a surf

Volume of a Solid by the Disc Method

Consider the region bounded by the curve $$y=\sqrt{x}$$ and the x-axis for $$x\in[0,4]$$. This regio

Volume of a Solid with Equilateral Triangle Cross Sections

Consider the region bounded by $$y= \sqrt{x}$$ and $$y=0$$ for $$x \in [0,1]$$. A solid is formed by

Volume of a Solid with Square Cross Sections

The base of a solid is the region in the plane bounded by $$y=x$$ and $$y=x^2$$ (with $$x$$ between

Volume of a Solid with Square Cross Sections

Consider the region bounded by the curve $$f(x)= 4 - x^2$$ and the x-axis for $$x \in [-2,2]$$. A so

Volume of a Wavy Dome

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

Volume of an Arch Bridge Support

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

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)

Arc Length of a Decaying Spiral

Consider the parametric equations $$x(t)= e^{-t}\cos(t)$$ and $$y(t)= e^{-t}\sin(t)$$ for $$t \ge 0$

Circular Motion: Speed and Acceleration Components

A car travels around a circle of radius 5, described by the parametric equations $$x(t)=5\cos(t)$$ a

Converting Polar to Cartesian and Computing Slope

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

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

Dynamics in Polar Coordinates

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

Implicit Differentiation and Curves in the Plane

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

Inner Loop of a Limaçon in Polar Coordinates

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

Intersection of Polar Curves

Consider the polar curves given by $$r=2\sin(\theta)$$ and $$r=1+\cos(\theta)$$. Answer the followin

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),\;

Parametric Motion and Change of Direction

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

Parametric Representation of an Ellipse

An ellipse is represented by the parametric equations $$x(t)=4\cos(t)$$ and $$y(t)=3\sin(t)$$ for $$

Particle Trajectory in Parametric Motion

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

Periodic Motion: A Vector-Valued Function

A point moves on a circle with position given by $$\vec{r}(t)= \langle \cos(2t), \sin(2t) \rangle$$

Polar Coordinate Area Calculation

Consider the polar curve $$r=4*\sin(θ)$$ for $$0 \le θ \le \pi$$. This equation represents a circle.

Polar Differentiation and Tangent Lines

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

Polar Equations and Slope Analysis

Given the polar curve $$r = 4\sin(\theta)$$, analyze its Cartesian form and tangent properties.

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 Modeled by Vector-Valued Functions

A projectile is launched with an initial velocity vector $$\vec{v}_0=\langle 10, 20 \rangle$$ (in m/

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 via Vector-Valued Functions

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

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 $$

Synthesis of Parametric, Polar, and Vector Concepts

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

Tangent Line to a 3D Vector-Valued Curve

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

Vector-Valued Functions: Motion in the Plane

The position of a particle in space is given by $$\vec{r}(t)=\langle e^t, \ln(1+t), t^2 \rangle$$ fo

Work Done by a Force along a Vector Path

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