AP Calculus BC FRQ Room

Algebraic Manipulation with Radical Functions

Let $$f(x)= \frac{\sqrt{x+5}-3}{x-4}$$, defined for $$x\neq4$$. Answer the following:

Analysis of a Jump Discontinuity

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

Analyzing Continuity on a Closed Interval

Suppose a function $$f(x)$$ is continuous on the closed interval $$[0,5]$$ and differentiable on the

Analyzing Limits of a Combined Exponential‐Log Function

Consider $$f(x)= e^{-x}\,\ln(1+\sqrt{x})$$ for $$x \ge 0$$. Analyze the limits as $$x \to 0^+$$ and

Application of the Squeeze Theorem with Trigonometric Oscillations

Consider the function $$f(x)= x^2*\sin(1/x)$$ for $$x \neq 0$$ and $$f(0)=0$$. Answer the following

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 Analysis of a Piecewise Function

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

Environmental Pollution Modeling

In a lake, a pollutant is added every year at a constant amount of 5 units. However, due to natural

Graphical Analysis of a Removable Discontinuity

Consider the function $$f(x)=\frac{x^2-1}{x-1}$$ for x \neq 1, with a defined value of f(1) = 3. Ans

Inverse Function Analysis and Continuity

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

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

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

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

Limits and Asymptotic Behavior of Rational Functions

Let $$k(x)=\frac{5*x^2-2*x+7}{x^2+4}.$$ Answer the following:

Limits from Table and Graph

A function $$g(x)$$ is studied in a lab experiment. The following table gives sample values of $$g(x

Limits with Composite Logarithmic Functions

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

One-Sided Limits for a Piecewise Function

Consider the piecewise function $$f(x)= \begin{cases} 2*x+1 & \text{if } x< 3 \\ x^2-5*x+8 & \text{i

Oscillatory Behavior and Limits

Consider the function $$f(x)=x\sin(1/x)$$ for x \neq 0, with f(0) defined to be 0. Use the following

Oscillatory Functions and Discontinuity

Consider the function $$f(x)= \begin{cases} \sin\left(\frac{1}{x}\right), & x\neq0 \\ 0, & x=0 \end{

Parameter Determination for Continuity

Let $$h(x)= \begin{cases} \frac{e^{2x} - 1 - a\,\ln(1+bx)}{x} & x \neq 0 \\ c & x = 0 \end{cases}.$$

Population Growth and Limits

The population $$P(t)$$ of a small town is recorded every 10 years as shown in the table below. Assu

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

Seasonal Temperature Curve Analysis

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

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 $$

Squeeze Theorem with an Oscillating Function

Let $$f(x)=x * \cos(\frac{1}{x})$$ for $$x \neq 0$$, and define $$f(0)=0$$. Answer the following:

Squeeze Theorem with Oscillatory Behavior

Examine the function $$s(x)=x^2*\sin(1/x)$$ for x ≠ 0.

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 Concavity and Second Derivative

Let $$f(x)=x^4-4*x^3+6*x^2$$. Analyze the concavity of the function and identify any inflection poin

Analyzing Motion Through Derivatives

A car’s position as a function of time is given by $$s(t) = 4*t^3 - 12*t^2 + 9*t + 5,$$ where $$s

Chain Rule Verification with a Power Function

Let $$f(x)= (3*x+2)^4$$.

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

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

Cost Optimization in Production: Derivative Application

A company's cost function is given by $$K(x)= 0.1x^3 - 2x^2 + 15x + 100$$, where x represents the nu

Differentiating Composite Functions using the Chain Rule

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

Exploration of Derivative Notation and Higher Order Derivatives

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

Market Price Rate Analysis

The market price of a product (in dollars) has been recorded over several days. Use the table below

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

Optimization in Engineering Design

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

Parametric Analysis of a Curve

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

Population Growth Rates

A city’s population (in thousands) was recorded over several years. Use the data provided to analyze

Profit Rate Analysis in Economics

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

Reservoir Management Problem

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

River Flow and Differentiation

The rate of water flow in a river is modeled by $$Q(t)= 5t^2 + 8t + 3$$ in cubic meters per second,

Savings Account Growth: From Discrete Deposits to Continuous Derivatives

An individual deposits $$P$$ dollars at the beginning of each month into an account that earns a con

Secant Line Approximation in an Experimental Context

A temperature sensor records the following data over a short experiment:

Sediment Accumulation in a Dam

Sediment enters a dam reservoir at a rate of $$S_{in}(t)=5\ln(t+1)$$ kg/hour, while sediment is remo

Taylor Expansion of a Polynomial Function Centered at x = 1

Given the polynomial function $$f(x)=3+2*x- x^2+4*x^3$$, analyze its Taylor series expansion centere

Testing Differentiability at a Junction Point

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

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

Urban Population Flow

A city’s population changes due to migration. The inflow of people is modeled by $$M_{in}(t)=8-0.5*t

Using the Limit Definition for a Non-Polynomial Function

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

Chain Rule and Inverse Trigonometric Differentiation

Consider the function $$f(x)= 3*\arccos\left(\frac{x}{4}\right) + \sqrt{1-\frac{x^2}{16}}$$. Answer

Chain Rule with Nested Logarithmic and Exponential Functions

Consider the function $$f(x)= \sqrt{\ln(5*x + e^{x})}$$. Differentiate this function using the chain

Derivative of an Inverse Function with a Quadratic

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

Differentiation in a Logistic Population Model

The population of a species is modeled by the logistic function $$P(t)= \frac{1000}{1+e^{-0.3*(t-5)}

Fuel Tank Dynamics

A fuel storage tank is being filled by a pump at a rate given by the composite function $$P(t)=(4*t+

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 Involving Exponential Functions

Consider the relation defined implicitly by $$e^{x*y} + x^2 - y^2 = 7$$.

Implicit Differentiation with Logarithmic Functions

Consider the equation $$\ln(x+y)= x - y$$.

Inverse Function Derivative Calculation

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

Inverse Function Derivative with Logarithms

Let $$f(x)= \ln(x+2) + x$$ with inverse function $$g(x)$$. Find the derivative $$g'(y)$$ in terms of

Inverse Function Differentiation in a Radical Context

Let $$f(x)= \sqrt{1+ x^3}$$ and let $$g$$ be its inverse function. Answer the following parts:

Inverse Function Differentiation in a Science Experiment

In an experiment, the relationship between an input value $$x$$ and the output is given by $$f(x)= \

Inverse Function Differentiation with a Logarithmic Function

Let $$ f(x)= \ln(x+3) $$. Consider its inverse function $$ f^{-1}(y) $$.

Inverse of a Composite Function

Let $$f(x)=\sqrt{3*x+1}$$ and $$g(x)=x^2-1$$, and define $$h(x)=f(g(x))$$. Analyze the invertibility

Inverse Trigonometric Differentiation

Differentiate the function $$ y= \arctan\left(\frac{2*x}{1-x}\right) $$.

Optimization in Manufacturing Material

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

Particle Motion with Composite Position Function

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

Power Series Representation and Differentiation of a Composite Function

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

Rate of Change in a Biochemical Process Modeled by Composite Functions

The concentration of a biochemical in a cell is modeled by the function $$C(t) = \sin(0.2*t) + 1$$,

Analyzing Concavity through the Second Derivative

A particle’s position is given by $$x(t)=\ln(t^2+1)$$, where $$t$$ is in seconds.

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.

Business Profit Optimization

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

Chemical Concentration Rate Analysis

The concentration of a chemical in a reactor is given by $$C(t)=\frac{5*t}{t+2}$$ M (moles per liter

Comparing Rates: Temperature Change and Coffee Cooling

The temperature of a freshly brewed coffee is modeled by $$T(t)=95*e^{-0.05*t}+25$$ (in °F), where $

Error Propagation in Circular Disk Area Measurement

A circular disk has a measured diameter of 10 cm with a possible error of ±0.05 cm. The area of the

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 $$

Implicit Differentiation: Tangent to a Circle

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

Inflating Spherical Balloon

A spherical balloon is being inflated so that the volume increases at a constant rate of $$dV/dt = 1

L'Hôpital's Rule Application

Evaluate the limit: $$\lim_{t \to \infty} \frac{5*t^3 - 4*t^2 + 7}{7*t^3 + 2*t - 6}$$ using L'Hôpita

L’Hôpital’s Rule for an Exponential Ratio

Analyze the limit of the function $$f(t)=\frac{e^{2*t}-1}{t}$$ as $$t\to 0$$. Answer the following:

Ladder Sliding Down a Wall

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

Linearization Approximation Problem

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

Maclaurin Series for ln(1+x)

Consider the function $$f(x)= \ln(1+x)$$. Its Maclaurin series may be used to approximate values of

Mixed Quadratic Relation

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

Mixing a Saline Solution: Related Rates

A tank contains a saline solution with a constant volume of 50 liters. Salt is added at a rate of 2

Optimization with Material Costs

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

Optimizing Factory Production with Log-Exponential Model

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

Ozone Layer Recovery Simulation

In a simulation of ozone layer dynamics, ozone is produced at a rate of $$I(t)=\frac{25}{t+1}$$ (Dob

Parametric Curve Motion

A particle’s trajectory is given by the parametric equations $$x(t)=t^2-1$$ and $$y(t)=2*t+3$$ for $

Population Growth Analysis

A certain bacterial population in a lab grows according to the model $$P(t)=100\cdot e^{0.03*t}$$, w

Quadratic Function Inversion with Domain Restriction

Let $$f(x)=x^2+4$$. Since quadratic functions are not one-to-one over all real numbers, consider an

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 in a Conical Water Tank

Water is being pumped into a conical tank at a rate of $$2\;m^3/min$$. The tank has a height of 6 m

Revenue Concavity Analysis

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

Seasonal Reservoir Dynamics

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

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

Trigonometric Implicit Relation

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

Water Filtration Plant Analysis

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

Absolute Extrema via Candidate's Test

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

Analysis of a Piecewise Function's Differentiability and Extrema

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

Analyzing Convergence of a Modified Alternating Series

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

Analyzing Extrema for a Rational Function

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

Analyzing Inverses in a Rate of Change Scenario

Consider the function $$f(x)= \ln(x+5) + x$$ defined for $$x > -5$$. This function models a system's

Analyzing The Behavior of a Log-Exponential Function Over a Specified Interval

Consider the function $$h(x)= \ln(x) + e^{-x}$$. A portion of its values is given in the following t

Application in Motion: Approximate Velocity using Taylor Series

A particle’s position is given by $$s(t)=e^{-t}+t^2$$. Using Taylor series approximations near $$t=0

Application of Rolle's Theorem

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

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:

Candidate’s Test for Absolute Extrema in Projectile Motion

A projectile is launched such that its height at time $$t$$ is given by $$h(t)= -16*t^2+32*t+5$$ (in

Chemical Reaction Rate

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

Concavity and Inflection Points Analysis

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

Derivative Sign Chart and Function Behavior

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

Differentiability and Critical Points of a Piecewise Function

Consider the piecewise function $$ f(x)= \begin{cases} x^2 & \text{if } x \le 2, \\ 4*x-4 & \text{i

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

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

Extreme Value Theorem in a Polynomial Function

Consider the function $$h(x)=x^4-8*x^2+16$$ defined on the closed interval $$[-3,3]$$. Answer the fo

Extremum Analysis Using the Extreme Value Theorem

Given the function $$f(x)= \cos(x)-x$$ defined on the interval $$\left[0, \frac{\pi}{2}\right]$$, an

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:

Interpreting a Velocity-Time Graph

A particle’s velocity over the interval $$[0,6]$$ seconds is depicted in the graph provided.

Linear Particle Motion Analysis

A particle moves along a straight line with acceleration given by $$a(t)=6-4*t$$ (in m/s²) for $$t\g

Logarithmic-Quadratic Combination Inverse Analysis

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

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

Optimization in Production Costs

In an economic context, consider the cost function $$C(x)=0.5*x^3-6*x^2+25*x+100$$, where C(x) repre

Optimization Problem: Designing a Box

A company needs to design an open-top box with a square base that has a volume of $$32\,m^3$$. The c

Parameter Estimation in a Log-Linear Model

In a scientific experiment, the data is modeled by $$P(t)= A\,\ln(t+1) + B\,e^{-t}$$. Given that $$P

Pharmaceutical Dosage and Metabolism

A patient is administered a medication with an initial dose of 50 mg. Due to metabolism, the amount

Population Growth vs. Harvest Model

A fish population in a lake grows naturally at a rate given by $$G(t)=\frac{50}{1+t}$$ (in fish/mont

Profit Maximization in Business

A company’s profit function is given by $$P(x) = -2*x^3 + 30*x^2 - 100*x + 150$$, where x represents

Rational Function Discontinuities

Consider the rational function $$ R(x)=\frac{(x-3)(x+2)}{(x-3)(x-1)}.$$ Answer the following parts:

Ski Resort Snow Accumulation and Melting

At a ski resort, snow accumulates naturally at a rate given by $$S(t)=50*\exp(-0.1*t)$$ cm/hour due

Stock Price Analysis

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

Taylor Series for $$\arcsin(x)$$

Derive the Maclaurin series for $$f(x)=\arcsin(x)$$ up to the $$x^5$$ term, determine the radius of

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 $$\ln\left(\frac{1+x}{1-x}\right)$$

Let $$f(x)=\ln\left(\frac{1+x}{1-x}\right)$$. Derive its Taylor series expansion about $$x=0$$, dete

Taylor Series for $$\sqrt{x}$$ Centered at $$x=4$$

For the function $$f(x)=\sqrt{x}$$, find the Taylor series expansion centered at $$x=4$$ including t

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

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

Temperature Variations

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

Travel Distance from Speed Data

A traveler’s speed (in km/h) is recorded at various times during a trip. Use the data to approximate

Trigonometric Function and its Inverse

Consider the function $$f(x)= \sin(x) + x$$ defined on the interval $$[-\pi/2, \pi/2]$$. Answer the

Volume of a Solid of Revolution Using the Washer Method

Find the volume of the solid obtained by revolving the region bounded by $$y=\sqrt{x}$$, $$y=\frac{x

Accumulated Displacement from a Velocity Function

A car’s velocity is given by the function $$v(t)=4 + t$$ (in m/s) over the interval [0, 8] seconds.

Analyzing and Integrating a Function with a Removable Discontinuity

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

Area Estimation Using Trapezoidal Sums from Tabulated Data

The table below provides values of $$h(t)$$ over time for a process: | Time (t) | 0 | 2 | 5 | 8 | |

Average Value and Accumulated Change

For the function $$f(x)= x^2+1$$ defined on the interval [0, 4], find the average value of the funct

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

Comprehensive Integration of a Polynomial Function

Consider the function $$f(x)=(x-3)(x+2)^2$$ on the interval $$[1,5]$$. This problem involves multipl

Cost and Inverse Demand in Economics

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

Cost Function Accumulation

A manufacturer’s marginal cost function is given by $$C'(x)= 0.1*x + 5$$ dollars per unit, where x

Definite Integral Involving an Inverse Function

Evaluate the definite integral $$\int_{1}^{4} \frac{1}{\sqrt{x}}\,dx$$ and explain the significance

Estimating Area Under a Curve from Tabular Data

A function $$f(t)$$ is sampled at discrete time points as given in the table below. Using these data

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:

Fundamental Theorem of Calculus Application

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

Graphical Transformations and Inverse Functions

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

Parameter-Dependent Integral Function Analysis

Define the function $$F(x)=\int_(1)^(x) \frac{\ln(t)}{t} dt$$ for x > 1. This function accumulates t

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

Pollution Accumulation

At a manufacturing plant, the pollutant release rate is given by $$R(t)=5-0.5*t$$ (in mg/hour) for $

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

Population Increase from a Discontinuous Growth Rate

A sudden migration event alters the population growth rate. The growth rate (in individuals per year

Power Series Approximation of an Integral Function

The function $$f(x)=e^{-x^2}$$ does not have an elementary antiderivative. Its definite integral can

Riemann Sum Approximation with Irregular Intervals

A set of experimental data provides the values of a function $$f(x)$$ at irregularly spaced points a

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+

Temperature Change in a Material

A laser heats a material such that its temperature changes at a rate given by $$\frac{dT}{dt} = 2*\s

Total Work Done by a Variable Force

A variable force $$F(x)$$ (in Newtons) is applied along a displacement, and its values are recorded

Autocatalytic Reaction Dynamics

Consider an autocatalytic reaction described by the differential equation $$\frac{dy}{dt} = k*y*\ln|

Capacitor Charging in an RC Circuit

In an RC circuit, when a capacitor is charging, the voltage across the capacitor, $$V(t)$$, satisfie

Capacitor Discharge in an RC Circuit

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

Chemical Reaction Rate Modeling

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

Cooling of a Metal Rod

A metal rod cools according to the differential equation $$\frac{dT}{dt}=-k\,(T-25)$$ with an initia

Disease Spread Model

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

Electrical Circuit Analysis Using an RL Circuit

An RL circuit is described by the differential equation $$L\frac{di}{dt}+R*i=E$$, where $$L$$ is the

Implicit Differentiation and Homogeneous Equation

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

Investment Account Growth with Fees

An investment account with balance $$A(t)$$ grows at a continuously compounded annual rate of $$6\%$

Logistic Differential Equation Analysis

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

Logistic Growth Model

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

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 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(

Particle Motion with Variable Acceleration

A particle moves along a straight line with an acceleration given by $$a(t)=6-4*t$$. At time t = 0,

Relative Motion with Acceleration

A car starts from rest and its velocity $$v(t)$$ (in m/s) satisfies the differential equation $$\fra

Reservoir Contaminant Dilution

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

Second-Order Differential Equation: Oscillations

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

Separable Differential Equation with Initial Condition

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

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 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

Traffic Flow on a Highway

A highway segment experiences an inflow of cars at a rate of $$200+10*t$$ cars per minute and an out

Analyzing Acceleration Data from Discrete Measurements

A vehicle’s acceleration (in m/s²) is recorded at discrete time intervals as given in the table. Use

Analyzing Convergence of a Taylor Series

Consider the function $$g(x)= e^{-x^2}$$. Analyze the Maclaurin series for this function.

Arc Length of the Logarithmic Curve

For the function $$f(x)=\ln(x)$$ defined on the interval $$[1,e]$$, determine the arc length of the

Area Between Curves in a Business Context

A company’s revenue and cost (in dollars) for producing items (in hundreds) are modeled by the funct

Area Between Curves: Supply and Demand Analysis

In an economic model, the supply and demand functions for a product (in hundreds of units) are given

Area Between Two Curves in a Water Channel

A channel cross‐section is defined by two curves: the upper boundary is given by $$f(x)=12-0.8*x$$ a

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 Analysis

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

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

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

Cyclist's Journey: Displacement versus Total Distance

A cyclist's velocity is given by $$v(t)=\sin(t)$$ (in m/s) for $$t\in[0,2\pi]$$. Answer the followin

Determining the Length of a Curve

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

Drone Motion Analysis

A drone’s vertical acceleration is modeled by $$a(t) = 6 - 2*t$$ (in m/s²) for time $$t$$ in seconds

Integration in Cost Analysis

In a manufacturing process, the cost per minute is modeled by $$C(t)=t^2 - 4*t + 7$$ (in dollars per

Inverse Function Analysis

Consider the function $$f(x)=3*x^3+2$$ defined for all real numbers.

Motion Analysis of a Car

A car has an acceleration given by $$a(t)=2-0.5*t$$ for $$0\le t\le8$$ seconds. The initial velocity

Motion Analysis on a Particle with Variable Acceleration

A particle moves along a straight line with acceleration given by $$a(t)=6-4*t$$ (in m/s²). The init

Net Change and Direction of Motion

A particle’s velocity is given by $$v(t)=\sin(t)-\frac{1}{2}*t$$ for $$0 \le t \le 6$$.

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

Oval Path Implicit Differentiation

A particle moves along a path described by the equation $$x^2 + 2 * x * y + y^2 = 10$$. Answer the f

Particle Motion from Acceleration

A particle has an acceleration given by $$a(t)=3*t-6$$ (m/s²). With initial conditions $$v(0)=2$$ m/

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

Piecewise Velocity Analysis

A particle moves along a straight line with velocity given by the following piecewise function: $$v

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

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.

Sand Pile Dynamics

Sand is being added to a pile at a rate given by $$A(t)=8-0.5*t$$ (kg/min) for $$0\le t\le12$$ minut

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 by the Shell Method: Rotating a Region

Consider the region bounded by the curve $$y=\sqrt{x}$$, the line $$y=0$$, and the vertical line $$x

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

Volume Using Washer Method

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

Water Tank Dynamics: Inflow and Outflow

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

Work Done by a Variable Force

A variable force is applied along a frictionless surface and is given by $$F(x)=6-0.5*x$$ (in Newton

Work Done by a Variable Force

A variable force applied to move an object along a straight line is given by $$F(x)=3*x^2$$ (in newt

Work Done by a Variable Force

A force acting on an object along a displacement is given by $$F(x)=3*x^2 -2*x+1$$ (in Newtons), whe

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

Acceleration in Polar Coordinates

An object moves in the plane with its position expressed in polar coordinates by $$r(t)= 4+\sin(t)$$

Arc Length of a Parametrically Defined Curve

A curve is defined by the parametric equations $$x(t)=t^2$$ and $$y(t)=\frac{t^3}{3}$$ for $$0 \leq

Arc Length of a Polar Curve

Consider the polar curve $$r(\theta)= 1+\cos(\theta)$$ for \(0 \le \theta \le \pi\).

Arc Length of a Polar Curve

Consider the polar curve given by $$r(θ)= 1+\sin(θ)$$ for $$0 \le θ \le \pi$$. Answer the following:

Area Between Two Polar Curves

Consider the two polar curves $$r_1(θ)= 3+\cos(θ)$$ and $$r_2(θ)= 1+\cos(θ)$$. Answer the following:

Curvature of a Space Curve

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

Cycloid and Its Arc Length

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

Distance Traveled in a Turning Curve

A curve is defined by the parametric equations $$x(t)=4*\sin(t)$$ and $$y(t)=4*\cos(t)$$ for $$0\le

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

Integrating a Vector-Valued Function

A particle has a velocity given by $$\vec{v}(t)= \langle e^t, \cos(t) \rangle$$. Its initial positio

Integration of Speed in a Parametric Motion

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

Intersection of Parametric Curves

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

Kinematics in Polar Coordinates

A particle’s position in polar coordinates is given by $$r(t)= \frac{5*t}{1+t}$$ and $$\theta(t)= \f

Lissajous Figures and Their Properties

A Lissajous curve is defined by the parametric equations $$x(t)=5*\sin(3*t)$$ and $$y(t)=5*\cos(2*t)

Multi-Step Problem Involving Polar Integration and Conversion

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

Optimization in Parametric Projectile Motion

A projectile is launched from the ground with an initial speed of $$20\,m/s$$ at an angle $$\alpha$$

Parametric Slope and Arc Length

Consider the parametric curve defined by $$x(t)= t-\ln(t)$$ and $$y(t)= t\cdot\ln(t)$$ for $$t > 1$$

Particle Trajectory in Parametric Motion

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

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 with Parametric Equations

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

Relative Motion of Two Objects

Two objects A and B move in the plane with positions given by the vector functions $$\vec{r}_A(t)= \

Spiral Motion with a Damped Vector Function

An object moves according to the spiral vector function $$\mathbf{r}(t)=\langle e^{-t}\cos(t),\; e^{

Tangent Lines to Polar Curves

Consider the polar curve $$r(\theta)= 3\sin(\theta)$$. Analyze the tangent line at a point correspo

Vector-Valued Function Integration

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

Vector-Valued Functions: Velocity and Acceleration

A particle's position is given by the vector-valued function $$\vec{r}(t) = \langle \sin(t), \ln(t+1

Vector-Valued Kinematics

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