AP Calculus BC FRQ Room

Applying Algebraic Techniques to Evaluate Limits

Examine the limit $$\lim_{x\to4} \frac{\sqrt{x+5}-3}{x-4}$$. Answer the following: (a) Evaluate the

Asymptotic Behavior of a Water Flow Function

In a reservoir, the net water flow rate is modeled by the rational function $$R(t)=\frac{6\,t^2+5\,t

Car Braking Distance and Continuity

A car decelerates to a stop, and its velocity $$v(t)$$ in m/s is recorded in the following table, wh

Computing a Limit Using Algebraic Manipulation

Evaluate the limit $$\lim_{x\to2} \frac{x^2-4}{x-2}$$ using algebraic manipulation.

Continuity Analysis of an Integral Function

Let $$F(x)=\int_{0}^{x} f(t)\,dt,$$ where $$ f(t)= \begin{cases} t+1, & t < 2 \\ 3, & t \ge 2 \end{

Continuity Assessment of a Rational Function with a Redefined Value

Consider the function $$r(x)= \begin{cases}\frac{x^2-9}{x-3}, & x \neq 3 \\ 7, & x=3\end{cases}$$.

Evaluating Limits Involving Absolute Value Functions

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

Exploring Removable and Nonremovable Discontinuities

Consider the function $$f(x)=\frac{(x-2)(x+3)}{(x-2)}$$ for $$x\neq2$$ and $$f(2)=7$$. Answer the fo

Factorization and Limits

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

Higher‐Order Continuity in a Log‐Exponential Function

Define $$ f(x)= \begin{cases} \frac{e^x - 1 - \ln(1+x)}{x^3}, & x \neq 0 \\ D, & x = 0, \end{cases}

Identifying and Removing a Discontinuity

Consider the function $$g(x)=\frac{x^2-4}{x-2}$$ for $$x\neq2$$, which is undefined at $$x=2$$.

Investment Portfolio Rebalancing

An investment portfolio is rebalanced periodically, yielding profits that form a geometric sequence.

Jump Discontinuity Analysis using Table Data

A function f is defined by experimental measurements near $$x=2$$. Use the table provided to answer

Limits Involving Exponential Functions

Consider the function $$f(x)= \frac{e^{2*x}-1}{x}$$ defined for $$x\neq0$$.

Limits Involving Radical Functions

Examine the function $$m(x)=\frac{\sqrt{x}-2}{x-4}$$.

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

Radical Function Limit via Conjugate Multiplication

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

Rational Function Analysis with Removable Discontinuities

Consider the function $$f(x)=\frac{(x+3) * (x-1)}{(x-1)}$$ for $$x \neq 1$$. This function exhibits

Squeeze Theorem with a Log Function

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

Trigonometric Limits

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

Water Flow Measurement Analysis

A water flow sensor measures the flow rate $$Q(t)$$ (in cubic meters per second) of a stream at vari

Analysis of a Piecewise Function

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

Analyzing a Polynomial with Higher Order Terms

Consider the function $$f(x)=4*x^5 - 2*x^3 + x - 7$$. Answer the following:

Average vs Instantaneous Rate of Change in Temperature Data

The table below shows the temperature (in °C) recorded at several times during an experiment. Use t

Circular Motion Analysis

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

Composite Function Behavior

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

Composite Function Differentiation and Taylor Series for $$e^{\sin(x)}$$

Consider the composite function $$f(x)=e^{\sin(x)}$$. A physicist needs to approximate this function

Cooling Tank System

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

Cost Minimization in Packaging

A company's packaging box has a cost function given by $$C(x)=0.05*x^2 - 4*x + 200$$, where $$x$$ is

Derivative from First Principles

Let $$f(x)=\sqrt{x}$$. Use the limit definition of the derivative to find $$f'(x)$$.

Differentiation of Inverse Functions

Let $$f(x)=3*x+2$$ and let $$f^{-1}(x)$$ denote its inverse function. Answer the following:

Electricity Consumption: Series and Differentiation

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

Implicit Differentiation with Inverse Functions

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

Implicit Differentiation with Trigonometric Functions

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

Implicit Differentiation: Conic with Mixed Terms

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

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

Limit Definition of the Derivative for a Trigonometric Function

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

Maclaurin Series for ln(1+x)

A scientist modeling logarithmic growth wishes to approximate the function $$\ln(1+x)$$ around $$x=0

Optimization in a Chemical Reaction

The rate of a chemical reaction is modeled by the function $$R(x)=x*e^{-x}+\ln(x+2)$$, where $$x$$ r

Parametric Analysis of a Curve

A particle moves along a path given by the parametric equations $$x(t)=t^2+1$$ and $$y(t)=t^3-3*t$$,

Pharmacokinetics: Drug Concentration Analysis

The concentration of a drug in the bloodstream is modeled by the function $$C(t)=10*\ln(t+2)*e^{-0.3

Product of Exponential and Trigonometric Functions

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

Product Rule in Differentiation

Suppose the cost function is given by $$Q(x)=(3*x^2 - x)*e^{x}$$, which represents a cost (in dollar

Rainwater Harvesting System

A rainwater harvesting system collects water with an inflow rate of $$R_{in}(t)=100+25\sin((\pi*t)/1

Revenue Change Analysis via the Product Rule

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

Satellite Orbit Altitude Modeling

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

Secant to Tangent Convergence

Consider the natural logarithm function $$f(x)=\ln(x)$$ for \(x>0\). Answer the following:

Tangent and Normal Lines

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

Tangent Line to a Logarithmic Function

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

Taylor Series for sin(x) Approximation

A researcher studying oscillatory phenomena wishes to approximate the function $$f(x)=\sin(x)$$ for

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 Treatment Plant Simulator

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

Widget Production Rate

A widget manufacturing plant produces widgets according to the function $$P(t)=4*t^2 - 3*t + 10$$ wh

Analysis of a Composite Chemical Concentration Model

The concentration of a chemical in a reaction is modeled by the composite function $$C(t)= \ln(0.5*t

Chain Rule with Trigonometric Composite Function

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

Combined Differentiation: Inverse and Composite Function

Let $$f(x)= \ln(2*x+1)$$ and let $$g$$ be the inverse function of $$f$$. Answer the following parts:

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 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

Dam Water Release and River Flow

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

Differentiation of a Product Involving Inverse Trigonometric Components

Let $$m(x)=\arctan(x)+x*\arcsin\left(\frac{x}{2}\right)$$, where the domain of arcsin requires $$-2\

Geometric Context: Sun Angle and Shadow Length Inverse Function

Consider the function $$f(\theta)=\tan(\theta)+\theta$$ for $$0<\theta<\frac{\pi}{2}$$, which models

Implicit Differentiation in Circular Motion

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

Implicit Differentiation with Exponential and Trigonometric Components

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

Implicit Differentiation with Trigonometric Components

Consider the equation $$x*\sqrt{y} + \cos(y) = x^2$$, where $$y$$ is a function of $$x$$. Differenti

Implicit Differentiation: Conic Section Analysis

Consider the conic section defined by $$x^2 + 3*x*y + y^2 = 5$$. Answer the following:

Implicit Equation with Logarithmic and Exponential Terms

The relation $$\ln(x+y)+e^{x-y}=3$$ defines y implicitly as a function of x. Answer the following pa

Inverse Function Derivative Calculation

Let $$f$$ be a one-to-one differentiable function for which the table below summarizes selected info

Inverse Function Derivative for the Natural Logarithm

Consider the function $$f(x) = \ln(x+1)$$ for $$x > -1$$ and let $$g$$ be its inverse function. Anal

Inverse Function Differentiation for a Trigonometric-Polynomial Function

Let $$f(x)= \sin(x) + x^2$$ be defined on the interval $$[0, \pi/2]$$ so that it is invertible, with

Inverse Trigonometric Differentiation

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

Logarithmic and Composite Differentiation

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

Optimization with Composite Functions - Minimizing Fuel Consumption

A car's fuel consumption (in liters per 100 km) is modeled by $$F(v)= v^2 * e^{-0.1*v}$$, where $$v$

Projectile Motion and Composite Exponential Functions

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

Tangent Line to a Circle via Implicit Differentiation

Consider the circle defined by $$x^2 + y^2 = 25$$. At the point $$(3, -4)$$, determine the slope of

Analyzing a Motion Graph

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

Analyzing Motion on an Inclined Plane

A sled moves along an inclined plane with displacement given by $$s(t)=5*t^2-0.5*t^3$$, where $$s$$

Application of L’Hospital’s Rule

Evaluate the limit $$\lim_{x \to \infty} \frac{5x^3 - 2x + 1}{3x^3 + 7}$$ using L’Hospital’s Rule.

Approximating Function Values Using Linearization

Consider the function $$f(x)=x^4$$. Use linearization at x = 4 to approximate the value of $$f(3.98)

Bacterial Growth and Linearization

A bacterial population is modeled by $$P(t)=100e^{0.3*t}$$, where $$t$$ is in hours. Answer the foll

Comparison of Series Convergence and Error Analysis

Consider the series $$S(x)= \sum_{n=0}^{\infty} \frac{(-1)^n x^n}{2^n}$$ and $$T(x)= \sum_{n=0}^{\in

Conical Tank Filling

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

Conical Tank Filling - Related Rates

A conical water tank has its volume given by $$V= \frac{1}{3}\pi*r^2*h$$, where \(r\) is the radius

Cooling Temperature Model

The temperature of a heated object cooling in a room is modeled by $$T(t)= 80 + 120*e^{-0.25*t}$$, w

Derivative of Concentration in a Chemical Reaction

The concentration of a reactant in a chemical reaction is modeled by $$C(t)=\frac{8e^{-0.5t}}{1+e^{-

Economic Optimization: Profit Maximization

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

Expanding Rectangle: Related Rates

A rectangle has a length $$l$$ and width $$w$$ that are changing with time. At a certain moment, the

Exponential and Trigonometric Bounded Regions

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

Forensic Gas Leakage Analysis

A gas tank under investigation shows leakage at a rate of $$O(t)=3t$$ (liters per minute) while it i

Fuel Consumption Rate Analysis

The fuel consumption of a car (in gallons per 100 miles) is modeled by $$C(v)=0.05*v^2+1$$, where $$

Graphical Analysis of an Inverse Function

Consider a continuously differentiable and strictly increasing function $$f(x)$$ as depicted in the

Horizontal Tangents on Cubic Curve

Consider the curve defined by $$x^3 + y^3 - 6*x*y = 0$$.

L'Hôpital’s Rule in Chemical Reaction Rates

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

Linearization Approximation Problem

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

Motion Model Inversion

Suppose that the position of a particle moving along a line is given by $$f(t)=t^3+t$$. Analyze the

Parametric Motion Analysis

A particle moves such that its position is described by the parametric equations $$x(t)= t^2 - 4*t$$

Parametric Motion with Logarithmic and Radical Components

A particle’s motion is described by the vector function $$\mathbf{r}(t)=\langle \ln(t+1),\sqrt{t} \r

Particle Motion Analysis

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

Related Rates in a Circular Pool

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

Related Rates: Expanding Circular Oil Spill

In a coastal region, an oil spill is spreading uniformly and forms a circular region. The area of th

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

Revenue and Marginal Analysis

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

Savings Account Dynamics

A bank account receives deposits at a rate of $$I(t)=50+10t$$ (dollars per month) and experiences wi

Seasonal Reservoir Dynamics

A reservoir receives water inflow influenced by seasonal variations, modeled by $$I(t)=50+30\sin\Big

Series Solution of a Drug Concentration Model

The drug concentration in the bloodstream is modeled by $$C(t)= \sum_{n=0}^{\infty} \frac{(-t)^n}{n!

Spherical Balloon Inflation

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

Temperature Conversion Model Inversion

The temperature conversion function is given by $$f(x)=\frac{9}{5}*x+32$$, which converts Celsius to

Vehicle Motion on a Curved Path

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

Amusement Park Ride Braking Distance

An amusement park ride uses a sequence of friction pads to stop a roller coaster. The first pad diss

Analysis of an Absolute Value Function

Consider the function $$f(x)=|x^2-4|$$. Answer the following parts:

Analysis of Critical Points for Increasing/Decreasing Intervals

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

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

Application of the Mean Value Theorem

Let $$f(x)=\frac{x}{x^2+1}$$ be defined on the interval $$[0,2]$$. Answer the following questions us

Asymptotic Behavior and Limits of a Logarithmic Model

Examine the function $$f(x)= \ln(1+e^{-x})$$ and its long-term behavior.

Bacterial Culture with Periodic Removal

A laboratory experiment involves a bacterial culture that, at the beginning of an hour, has 200 bact

Concavity and Inflection Points Analysis

Consider the function \( f(x)=\ln(x) - x \) where \( x > 0 \). Answer the following parts:

Derivative Analysis of a Rational Function

Consider the function $$s(x)=\frac{x}{x^2+1}$$. Answer the following parts:

Epidemic Infection Model

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

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

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

Garden Design Optimization

A gardener wants to design a rectangular garden adjacent to a river, so that fencing is required for

Graph Analysis of a Logarithmic Function

Consider the function $$g(x)= \ln(x) - \frac{1}{x}$$ defined for $$x>0$$. Analyze its behavior and g

Implicit Differentiation and Inverse Function Analysis

Consider the implicit equation $$x^2 + y^2 + x*y = 7$$, where y is a function of x near the point wh

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

Logarithmic-Exponential Function Analysis

Consider the function $$f(x)= e^(x) + x$$ defined for all real numbers. Answer the following questio

Logarithmic-Quadratic Combination Inverse Analysis

Consider the function $$f(x)= \ln(x^2+1)$$ for $$x \ge 0$$. Answer the following parts.

Motion Analysis: Particle’s Position Function

A particle moves along a straight line and its position is given by $$s(t)=t^4 - 8*t^2 + 16$$ (in me

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

Optimizing Fencing for a Field

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

Piecewise Function with Absolute Value

Consider the function defined by $$ g(x)=\begin{cases} |x-1| & \text{if } x<2, \\ 3x-5 & \text{if }

Real-World Modeling: Radioactive Decay with Logarithmic Adjustment

A radioactive substance decays according to $$N(t)= N_0\,e^{-0.03t}$$. In an experiment, the recorde

Related Rates: Changing Shadow Length

A 2-meter tall lamppost casts a shadow of a 1.6-meter tall person who is walking away from the lampp

Relative Extrema Using the First Derivative Test

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

River Sediment Transport

Sediment enters a river from a landslide at a rate of $$S_{in}(t)=4*\exp(0.2*t)$$ tonnes/day and is

Sign Chart Construction from the Derivative

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

Accumulated Change Prediction

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

Accumulation Function and the Fundamental Theorem of Calculus

Let $$F(x) = \int_{2}^{x} \sqrt{1 + t^3}\, dt$$. Answer the following parts regarding this accumulat

Accumulation Function from a Rate Function

The rate at which water flows into a tank is given by $$r(t)=3\sqrt{t}$$ (in liters per minute) for

Antiderivatives and the Fundamental Theorem of Calculus

Given the function $$f(x)= 2*x+3$$, use the Fundamental Theorem of Calculus to evaluate the definite

Area and Volume for an Exponential Function Region

Consider the curve $$y=e^{-x}$$ for $$x\ge0$$. Answer the following:

Area Between a Curve and Its Tangent

For the function $$f(x)=x^3-3*x^2+2*x$$, analyze the area between the curve and its tangent line at

Area Between Curves

Consider the curves given by $$f(x)=x^2$$ and $$g(x)=2*x$$. A graph of these curves is provided. Det

Area Estimation with Riemann Sums

A water flow rate function f(x) (in m³/s) is measured at various times. The table below shows the me

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

Composite Functions and Inverses

Consider \(f(x)= x^2+1\) for \(x \ge 0\). Answer the following questions regarding \(f\) and its inv

Convergence of an Improper Integral

Consider the improper integral $$\int_{1}^{\infty} \frac{1}{x^{p}}\,dx$$, where $$p$$ is a positive

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

Cost and Inverse Demand in Economics

Consider the cost function representing market demand: $$f(x)= x^2 + 4$$ for $$x\ge0$$. Answer the f

Definite Integral using U-Substitution

Evaluate the integral $$\int_{1}^{5} (2*x - 3)^4\,dx$$ using the method of u-substitution.

Determining Velocity and Displacement from Acceleration

A particle's acceleration is given by $$a(t)=4*t-8$$ (in m/s²) for $$0 \le t \le 3$$ seconds. The in

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

Estimating Area Under a Curve Using Riemann Sums

Consider the function $$f(x)$$ whose values on the interval $$[0,10]$$ are given in the table below.

Evaluating a Complex Integral

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

Evaluation of an Improper Integral

Consider the integral $$\int_{1}^{\infty} \frac{1}{x^{2}} dx$$. Answer the following:

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:

Investigating Partition Sizes

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

Limit of a Riemann Sum as a Definite Integral

Consider the limit of the Riemann sum given by $$\lim_{n \to \infty} \sum_{i=1}^{n} \left( \frac{6}{

Modeling Bacterial Growth Through Accumulated Change

A bacteria population's growth rate is given by $$r(t)=\frac{2*t}{1+t^{2}}$$ (in thousands per hour)

Parametric Integral and Its Derivative

Let $$I(a)= \int_{0}^{a} \frac{t}{1+t^2}dt$$ where a > 0. This integral is considered as a function

Population Growth: Rate to Accumulation

A population's growth rate (in thousands of individuals per year) is modeled by $$P'(t)=2*t - 1$$ fo

Power Series Analysis and Applications

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

Reservoir Water Level

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

Series Convergence and Integration with Power Series

Consider the power series $$\sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{(2n+1)!}$$, which represents $$

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 of Water Flow in a River

The water flow rate through a river, given in cubic meters per second, is measured at different time

Bacterial Growth with Predation

A bacterial culture grows according to the differential equation $$\frac{dB}{dt}= r*B - P$$, where $

Chemical Reaction Rate

In a chemical reaction, the concentration $$C(t)$$ of a reactant decreases according to the first-or

Cooling Model Using Newton's Law

Newton's law of cooling states that the temperature T of an object changes at a rate proportional to

Differential Equations in Economic Modeling

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

Euler's Method and Differential Equations

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

Euler's Method Approximation

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

Existence and Uniqueness in an Implicit Differential Equation

Consider the implicit initial value problem given by $$y\,e^{y}+x=0$$ with the initial condition $$y

Exponential Growth via Slope Field Analysis

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

Flow Rate in River Pollution Modeling

A river system is modeled to study pollutant concentration $$C(t)$$ (in mg/L). Polluted water with c

Infectious Disease Spread Model

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

Logistic Model in Product Adoption

A company models the adoption rate of a new product using the logistic equation $$\frac{dP}{dt} = 0.

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

Mixing Problem with Differential Equations

A tank initially contains $$S(0)=S_0$$ grams of salt dissolved in a volume $$V$$ liters of water. Br

Modeling Exponential Growth

A population follows the differential equation $$\frac{dP}{dt} = k*P$$. Given that the population do

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

An object with an initial temperature of $$80^\circ C$$ is placed in a room at a constant temperatur

Nonlinear Differential Equation with Implicit Solution

Consider the nonlinear differential equation $$\frac{dy}{dx} = \frac{x^2+y^2}{2xy}$$. An implicit so

Particle Motion with Damping

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

Power Series Solutions for a Differential Equation

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

Radioactive Decay with Constant Source

A radioactive material is produced at a constant rate S while simultaneously decaying. This process

Separable DE: Basic SIPPY Problem

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

Separation of Variables with Trigonometric Functions

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

Series Convergence and Error Analysis

Consider the power series representation $$f(x)=\sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}$

Series Solution for a Differential Equation

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

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 Sketching a Solution Curve

The differential equation $$\frac{dy}{dx}=x-y$$ has been represented by a slope field. Answer the fo

Slope Field and Solution Curve Sketching

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

Tank Draining Problem

A tank with a variable cross-sectional area is being emptied. The height \(h(t)\) of the water satis

Temperature Change and Differential Equations

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

Viral Spread on Social Media

Let $$V(t)$$ denote the number of viral posts on a social media platform. Posts go viral at a consta

Area Between Two Curves: Parabola and Line

Consider the functions $$f(x)= 5*x - x^2$$ and $$g(x)= x$$. These curves enclose a region in the pla

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 Temperature Analysis

A weather station records the temperature throughout a day. The temperature, in degrees Celsius, is

Average Value of a Piecewise Function

Consider the piecewise function defined on $$[0,4]$$ by $$ f(x)= \begin{cases} x^2 & \text{for } 0

Average Value of a Population Growth Rate

The instantaneous growth rate of a bacterial population is modeled by the function $$r(t)=0.5*\cos(0

Chemical Reaction Rate Analysis

During a chemical reaction, the concentration of a reactant is modeled by $$C(t)=20e^{-0.3*t}$$ (in

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 versus Displacement

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

Drug Concentration Profile Analysis

The functions $$f(t)=5*t+10$$ and $$g(t)=2*t^2+3$$ (where t is in hours and concentration in mg/L) r

Economic Analysis: Consumer and Producer Surplus

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

Electric Charge Accumulation

A circuit has a current given by $$I(t)=4e^{-t/3}$$ A for $$t$$ in seconds. Analyze the charge accum

Electrical Charge Distribution

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

Finding the Centroid of a Planar Region

Consider the region bounded by the curves $$y=x$$ and $$y=x^2$$ between the vertical lines $$x=0$$ a

Logarithmic and Exponential Equations in Integration

Let $$f(x)=\ln(x+2)$$. Consider the expression $$\frac{1}{6}\int_0^6 [f(x)]^2dx=k$$, where it is giv

Particle Motion: Position, Velocity, and Acceleration

A particle moves along a straight line with acceleration $$a(t)=4-2*t$$ (in m/s²), initial velocity

Profit-Cost Area Analysis

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

Projectile Maximum Height

A ball is thrown upward with an acceleration of $$a(t)=-9.8$$ m/s², an initial velocity of $$v(0)=20

Solid of Revolution via Disc Method

Consider the region bounded by the curve $$y = x^2$$ and the x-axis for $$0 \le x \le 3$$. This regi

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

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 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

Volume of an Irregular Tank

A water tank has a varying cross-sectional profile described by $$y(x)=\sqrt{25 - (x-5)^2}$$, for $$

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 by a Variable Force

A variable force given by $$F(x)= 2*x + 3$$ (in Newtons) is applied to an object as it moves along a

Work Done by a Variable Force

A variable force acting along a straight line is given by $$F(x) = \frac{10}{x+2}$$ (in Newtons). Fi

Work Done by a Variable Force

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

Work Done in Lifting a Cable

A cable of length 10 m with a uniform mass density of 5 kg/m hangs vertically from a winch. The winc

Work Done on a Non-linear Spring

A non-linear spring exerts a force given by $$F(x) = 3 * x^2 + 2 * x$$ (in Newtons), where $$x$$ (in

Work Done with a Discontinuous Force Function

A force acting on an object is defined piecewise by $$F(x)=\begin{cases} x^2, & 0 \le x < 4,\\ 16, &

Acceleration Analysis in a Vector-Valued Function

Consider the vector function describing an object's motion: $$\textbf{r}(t)= \langle \ln(t+2), \sqrt

Analyzing Oscillatory Motion in Parametric Form

The motion of an oscillating particle is given by $$x(t)=e^{-t}\cos(2t)$$ and $$y(t)=e^{-t}\sin(2t)$

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$

Arc Length of a Parametric Curve

Consider the curve defined by $$x(t)=t^3-3*t$$ and $$y(t)=t^2+2$$ for $$t \in [0,2]$$.

Arc Length of a Polar Curve

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

Average Position from a Vector-Valued Function

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

Catching a Thief: A Parametric Pursuit Problem

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

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

Conversion of Polar to Parametric Form

A particle’s motion is given in polar form by the equations $$r = 4$$ and $$\theta = \sqrt{t}$$ wher

Drone Altitude Measurement from Experimental Data

A drone’s altitude (in meters) is recorded at various times (in seconds) as shown in the table below

Exponential Decay in Vector-Valued Functions

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

Exponential Growth in Parametric Representation

A model for population growth is given by the parametric equations $$x(t)=t$$ and $$y(t)=e^{0.3t}$$,

Intersection of Parametric Curves

Two curves are given by the parametric equations $$x_1(t)=t^2,\; y_1(t)=t^3$$ and $$x_2(s)=1-s^2,\;

Intersection of Parametric Curves

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

Motion Along a Helix

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

Optimization on a Parametric Curve

A curve is described by the parametric equations $$x(t)= e^{t}$$ and $$y(t)= t - e^{t}$$.

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 Coordinate Area Calculation

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

Polar Coordinates: Area Between Curves

Consider two polar curves: the outer curve given by $$R(\theta)=4$$ and the inner curve by $$r(\thet

Spiral Intersection on the X-Axis

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

Vector-Valued Function Analysis

Let the vector-valued function be given by $$\vec{r}(t)=<e^{t},\, \sin(t),\, \cos(t)>$$ for $$0\leq

Vector-Valued Integration

Let the vector-valued function $$r(t) = \langle t, t^2, t^3 \rangle$$ represent the position of a pa