AP Calculus BC FRQ Room

Absolute Value Function Limits

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

Algebraic Manipulation in Limit Evaluation

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

Algebraic 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 a Composite Function Involving a Limit

Let $$f(x)=\sin(x)$$ and define the function $$g(x)=\frac{f(x)}{x}$$ for $$x\neq0$$, with the conven

Analyzing a Function with a Removable Discontinuity

Consider the function $$r(x)=\frac{x^2-9}{x-3}$$ for $$x\neq3$$ and $$r(3)=2.$$ Answer the follow

Analyzing Limits Using Tabular Data

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

Application of the Squeeze Theorem

Let $$f(x)=x^2 * \sin(\frac{1}{x})$$ for $$x \neq 0$$. Answer the following:

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

Continuity Analysis in Road Ramp Modeling

A highway ramp is modeled by the function $$y(x)= \frac{(x-3)(x+2)}{x-3}$$ for $$x\neq3$$, where x i

Continuity Analysis Involving Logarithmic and Polynomial Expressions

Consider the function $$f(x)= \begin{cases} \frac{\ln(x+2)}{x} & \text{if } x<0 \\ (x+1)^2 & \text{i

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

Continuity and Asymptotes of a Log‐Exponential Function

Examine the function $$f(x)= \ln(e^x + e^{-x})$$.

Continuity in Piecewise Defined Functions

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

Economic Growth and Continuity

The function $$E(t)$$ represents an economy's output index over time (in years). A table provides th

Evaluating Limits via Rationalizing Techniques

Let $$f(x)=\frac{\sqrt{2*x+9}-3}{x}.$$ Answer the following parts.

Exponential Inflow with a Shift in Outflow Rate

A water tank receives water at a rate given by $$R_{in}(t)=20\,e^{-t}$$ liters per minute. The water

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 $$x=2$$.

Infinite Limits and Vertical Asymptotes

Let $$g(x)=\frac{1}{(x-2)^2}$$. Answer the following:

L'Hôpital's Rule for Indeterminate Forms

Evaluate the limit $$h(x)=\frac{e^{2*x}-1}{\sin(3*x)}$$ as x approaches 0.

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 Involving Trigonometric Functions

Consider the function $$q(x)=\frac{1-\cos(2*x)}{x^2}$$.

Limits with Infinite Discontinuities

Consider the function $$k(x)=\frac{1}{x-2}$$.

Logarithmic Function Limits

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

Modeling with a Removable Discontinuity

A water reservoir’s inflow rate is modeled by the function $$R(t)=\frac{t^2-16}{t-4}$$ liters per mi

Physical Applications: Temperature Continuity

A temperature sensor records temperature (in °C) over time according to the function $$T(t)=\frac{t^

Rational Function Analysis of a Drainage Rate

A drain’s outflow rate is given by $$R_{out}(t)=\frac{3\,t^2-12\,t}{t-4}$$ for \(t\neq4\). Answer th

Related Rates with an Expanding Spherical Balloon

A spherical balloon is being inflated so that its volume increases at a rate of $$\frac{dV}{dt}=100\

Trigonometric Rate Function Analysis

A pump’s output is modified by a trigonometric factor. The outflow rate is recorded as $$R(t)=\frac{

Advanced Analysis of a Composite Piecewise Function

Consider the function $$g(x)= \begin{cases} \frac{2*x^2-8}{x-2} & x \neq 2 \\ 5 & x=2 \end{cases}$$

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

Derivative of a Function Involving an Absolute Value

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

Determining Rates of Change with Secant and Tangent Lines

A function is graphed by the equation $$f(t)=t^3-3*t$$ and its graph is provided. Use the graph to a

Differentiating a Series Representing a Function

Consider the function defined by the infinite series $$S(x)= \sum_{n=0}^\infty \frac{(-1)^n * x^{2*

Differentiating Composite Functions

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

Economic Model Rate Analysis

A company models its cost variations with respect to price $$p$$ using the function $$C(p)=e^{-p}+\l

Estimating Temperature Change

A scientist recorded the temperature of a liquid at different times (in minutes) as it was heated. U

Exploration of Derivative Notation and Higher Order Derivatives

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

Graph Behavior of a Log-Exponential Function

Let $$f(x)=e^{-x}+\ln(x)$$, where the domain is $$x>0$$.

Graphical Estimation of Tangent Slopes

Using the provided graph of a function g(t), analyze its rate of change at various points.

Heat Transfer in a Rod: Modeling and Differentiation

The temperature distribution along a rod is given by $$T(x)= 100 - 2x^2 + 0.5x^3$$, where x is in me

Implicit Differentiation: Elliptic Curve

Consider the curve defined by $$2*x^2 + 3*x*y + y^2 = 20$$.

Irrigation Reservoir Analysis

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

Maclaurin Series and Convergence for 1/(1-x)

An economist is using the function $$f(x)=\frac{1}{1-x}$$ to model economic behavior. Analyze the Ma

Population Growth Rate

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

Product of Exponential and Trigonometric Functions

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

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

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,

River Flow Dynamics

A river experiences seasonal variations. Its inflow is modeled by $$F_{in}(t)=30+10\cos((\pi*t)/12)$

Satellite Orbit Altitude Modeling

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

Secant Line Approximations and Instantaneous Slopes

The function $$g(x)=e^{x} - 2*x$$ models the mass (in grams) of a chemical in a reaction over time,

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

Temperature Change with Provided Data

The temperature at different times after midnight is modeled by $$T(t)=5*\ln(t+1)+20$$, with $$t$$ i

Widget Production Rate

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

Analyzing a Composite Function from a Changing Systems Model

The displacement of an object is given by $$s(t) = \sqrt{t^2 + 4*t + 5}$$ (in meters), where $$t$$ i

Chain Rule with Exponential Function

Consider the function $$h(x)= e^{\sin(4*x)}$$ which models a process with exponential growth modulat

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 Chain Rule with Exponential and Trigonometric Functions

Consider the function $$f(x) = e^{\cos(x)}$$. Analyze its derivative and explain the role of the cha

Composite Function Rates in a Chemical Reaction

In a chemical reaction, the concentration of a substance at time $$t$$ is given by $$C(t)= e^{-k*(t+

Composite Functions in a Biological Growth Model

A biologist models the substrate concentration by the function $$ g(t)= \frac{1}{1+e^{-0.5*t}} $$ an

Composite Functions in a Biological Model

In a biological model, the concentration of a substance is given by $$P(x)=e^{-\sqrt{x^2+1}}$$, wher

Composite Functions in Population Growth

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

Differentiation of a Nested Trigonometric Function

Let $$h(x)= \sin(\arctan(2*x))$$.

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\

Differentiation of an Inverse Exponential Function

Let $$f(x)=e^{2*x}-7$$, and let g denote its inverse function. Answer the following parts.

Differentiation of an Inverse Function

Let f be a differentiable and one-to-one function with inverse $$f^{-1}$$. Suppose that $$f(3)=7$$ a

Engine Air-Fuel Mixture

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

Ice Cream Storage Dynamics

An ice cream storage facility receives ice cream at a rate given by the composite function $$I(t)=d(

Implicit Differentiation Involving Product and Logarithm

Consider the curve defined by $$x*y + \ln(y) = x^2$$. Answer the following parts:

Implicit Differentiation with an Exponential Function

Given the equation $$ e^{x*y}= x+y $$, use implicit differentiation.

Implicit Differentiation with Exponential and Trigonometric Components

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

Implicit Differentiation with Trigonometric Functions

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

Indoor Air Quality Control

In a controlled laboratory environment, the rate of fresh air introduction is modeled by the composi

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 with a Logarithmic Function

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

Inverse Function Differentiation with Combined Logarithmic and Exponential Terms

Let $$f(x)=e^{x}+\ln(x)$$ for $$x>1$$ and let g be its inverse function. Answer the following.

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 of a Piecewise Function

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

Inverse Trigonometric Differentiation

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

Inverse Trigonometric Differentiation in Navigation

A ship's course angle is given by $$ \theta= \arcsin\left(\frac{3*x}{5}\right) $$, where x is the ho

Inverse Trigonometric Function Differentiation

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

Logarithmic and Composite Differentiation

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

Pipeline Pressure and Oil Flow

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

Rainwater Harvesting System

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

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:

Air Pressure Change in a Sealed Container

The air pressure in a sealed container is modeled by $$P(t)=\frac{100}{1+e^{-0.5*(t-10)}}$$, where $

Analysis of Particle Motion

A particle’s velocity is given by $$v(t)= 4t^3 - 3t^2 + 2$$. Analyze the particle’s motion by invest

Analyzing Pollutant Concentration in a River

The concentration of a pollutant in a river is modeled by $$C(t)=50-5*t+0.5*t^2$$, where C is in mg/

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.

Applying L'Hospital's Rule to a Transcendental Limit

Evaluate the limit $$\lim_{x\to 0}\frac{e^{2*x}-1}{\sin(3*x)}$$.

Area Under a Curve: Definite Integral Setup

Consider the function $$f(x) = x^3 - 4x + 1$$ on the interval $$[0, 3]$$. Explore the area between t

Bacterial Population Growth Analysis

A laboratory culture of bacteria has an initial population of $$P_0=1000$$ and grows according to th

Biology: Logistic Population Growth Analysis

A population is modeled by the logistic function $$P(t)= \frac{100}{1+ 9e^{-0.5*t}}$$, where $$t$$ i

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

Conical Tank Water Flow

Water is pumped into a conical tank at a rate of $$\frac{dV}{dt}=9\text{ ft}^3/\text{min}$$. The tan

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),

Economic Marginal Cost Analysis

A manufacturer’s total cost for producing $$x$$ units is given by $$C(x)= 0.01*x^3 - 0.5*x^2 + 10*x

Graphical Data and Derivatives

A set of experimental data is provided below, showing the concentration (in moles per liter) of a ch

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.

Interpreting Derivatives from Experimental Concentration Data

An experiment records the concentration (in moles per liter) of a substance over time (in minutes).

Linearization of Implicit Equation

Consider the implicit equation $$x^2 + y^2 - 2*x*y = 1$$, which defines $$y$$ as a function of $$x$$

Maclaurin Series for ln(1+x)

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

Marginal Cost and Revenue Analysis

A company’s cost function is given by $$C(x)=100+25*x+4*x^2$$, where $$C(x)$$ is measured in dollars

Optimization of Material Cost for a Pen

A rectangular pen is to be built against a wall, requiring fencing on only three sides. The area of

Series Integration in Fluid Flow Modeling

The flow rate of a fluid is modeled by $$Q(t)= \sum_{n=0}^{\infty} (-1)^n \frac{(0.1t)^{n+1}}{n+1}$$

Temperature Change of Coffee: Exponential Cooling

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

Temperature Conversion Model Inversion

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

Vector Function: Particle Motion in the Plane

A particle moves in the plane with a position vector given by $$\mathbf{r}(t)=\langle t, t^2 \rangle

Absolute Extrema and the Candidate’s Test

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

Analysis of an Exponential Function

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

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

Area and Volume of Region Bounded by Exponential and Linear Functions

Consider the functions $$f(x)=e^{x}$$ and $$g(x)=x+2$$. The region enclosed by these curves will be

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

Derivatives and Inverses

Consider the function $$f(x)=\ln(x)+x$$ for x > 0, and let g(x) denote its inverse function. Answer

Error Approximation using Linearization

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

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 for a Piecewise Function

Let $$h(x)$$ be defined on $$[-2,4]$$ as $$ h(x)= \begin{cases} -x^2+4 & \text{if } x \le 1, \\ 2x-

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

Implicit Differentiation and Tangent Slope

Consider the curve defined implicitly by $$x^2 + x*y + y^2 = 7$$. Answer the following parts:

Integration of a Series Representing an Economic Model

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

Inverse Analysis for a Function with Multiple Transformations

Consider the function $$f(x)= 2*(x-1)^3 + 3$$ for all real numbers. Answer the following parts.

Logarithmic-Quadratic Combination Inverse Analysis

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

Manufacturing Optimization in Production

A company’s profit (in thousands of dollars) from producing x (in thousands of units) is given by $$

Mean Value Theorem on a Quadratic Function

Consider the function $$f(x)=x^2-4*x+3$$ defined on the closed interval $$[1, 5]$$. Verify that the

MVT Application: Rate of Temperature Change

The temperature in a room is modeled by $$T(t)= -2*t^2+12*t+5$$, where $$t$$ is in hours. Analyze th

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 Rectangular Garden

A homeowner plans to build a rectangular garden adjacent to a river (so the side along the river nee

Projectile Motion and Maximum Height

A projectile is launched with its height (in meters) given by the function $$h(t) = -5*t^2 + 20*t +

Projectile Trajectory: Parametric Analysis

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

Rate of Reaction: Concentration Change

In a chemical reaction, the concentration (in mM) of a reactant is modeled by $$C(t) = 50*e^{-0.3*t}

Sign Chart Construction from the Derivative

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

Stock Price Analysis

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

Stress Analysis in Engineering Structures

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

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

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

Taylor Series in Differential Equations: $$y'(x)=y(x)\cos(x)$$

Consider the initial value problem $$y'(x)= y(x)\cos(x)$$ with $$y(0)=1$$. Assume a power series sol

Temperature Change in a Weather Balloon

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

Water Tank Dynamics

A water tank receives water from a pipe at a rate of $$R(t)=3*t+5$$ liters/min and loses water throu

Water Tank Volume Analysis

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

Accumulated Displacement from a Piecewise Velocity Function

A particle moves along a line with velocity given by $$ v(t)= \begin{cases} t^2-1, & 0 \le t < 2, \\

Accumulated Population Change from a Growth Rate Function

A population changes at a rate given by $$P'(t)= 0.2*t^2 - 1$$ (in thousands per year) for t between

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

Arc Length of an Architectural Arch

An architectural arch is described by the curve $$y=4 - 0.5*(x-2)^2$$ for $$0 \le x \le 4$$. The len

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

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

Area Under a Parametric Curve

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

Car Motion: From Acceleration to Distance

A car has an acceleration given by $$a(t)= 3 - 0.5*t$$ m/s² for time t in seconds. The initial velo

Center of Mass of a Rod with Variable Density

A thin rod of length 10 m has a linear density given by $$\rho(x)= 2 + 0.3*x$$ (in kg/m), where x is

Charging a Battery

An electric battery is charged with a variable current given by $$I(t)=4+2*\sin\left(\frac{\pi*t}{6}

Chemical Reaction Rates

A chemical reaction in a vessel occurs at a rate given by $$R(t)= 8*e^{-t/2}$$ mmol/min. Determine t

Comparing Riemann Sum Approximations for an Increasing Function

A function f(x) is given in the table below: | x | 0 | 2 | 4 | 6 | |---|---|---|---|---| | f(x) | 3

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

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

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.

Graphical Analysis of Riemann Sums

A graph titled 'Graph of Experimental Data' shows a curve representing the height function $$h(t)$$

Integration of a Complex Trigonometric Function

Evaluate the integral $$\int_{0}^{\pi/2} 4*\cos^3(t)*\sin(t) dt$$.

Interpreting the Constant of Integration in Cooling

An object cools according to the differential equation $$\frac{dT}{dt}=-k*(T-20)$$ where $$T(t)$$

Net Change in Drug Concentration

The rate of change of a drug's concentration in the bloodstream is given by $$R(t)=8*e^{-0.5*t}$$ (i

Non-Uniform Subinterval Riemann Sum

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

Partial Fractions Integration

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

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

Region Bounded by a Parabola and a Line: Area and Volume

Consider the region bounded by the curves $$y=x^{2}$$ and $$y=2*x+3$$. Answer the following:

Riemann Sum Approximations: Midpoint vs. Trapezoidal

Consider the function $$f(x)=e^(-x)$$ on the interval [0, 3]. Compare the approximations for the def

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

Volume of a Solid: Cross-Sectional Area

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

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

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,

Bacterial Growth with Predation

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

Bank Account Growth with Continuous Compounding

A bank account balance $$A(t)$$ grows according to the differential equation $$\frac{dA}{dt}= r*A$$,

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

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

Compound Interest and Investment Growth

An investment account grows according to the differential equation $$\frac{dA}{dt}=r*A$$, where the

Cooling of an Electronic Component

An electronic component overheats and its temperature $$T(t)$$ (in $$^\circ C$$) follows Newton's La

Differential Equation Involving Logarithms

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

Differential Equation with Exponential Growth and Logistic Correction

Consider the modified differential equation $$\frac{dy}{dt} = a*y - b*y^2$$, where $$a$$ and $$b$$ a

Economic Investment Growth Model with Regular Deposits

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

Estimating Total Change from a Rate Table

A car's velocity (in m/s) is recorded at various times according to the table below:

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

Growth and Decay in a Bioreactor

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

Logistic Growth: Time to Half-Capacity

Consider a logistic population model governed by the differential equation $$\frac{dP}{dt}=kP\left(1

Logistic Model with Harvesting

A fish population is modeled by a modified logistic differential equation that includes harvesting.

Mixing Problem in a Saltwater Tank

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

Motion Under Gravity with Air Resistance

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

Oscillatory Behavior in Differential Equations

Consider the second-order differential equation $$\frac{d^2y}{dx^2}+y=0$$, which describes simple ha

Population Dynamics with Harvesting

A species population is modeled by the differential equation $$\frac{dP}{dt} = 0.2*P\left(1-\frac{P}

Power Series Solutions for a Differential Equation

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

Second-Order Differential Equation in a Mass-Spring System

A mass-spring system without damping is modeled by the differential equation $$m\frac{d^2x}{dt^2}+kx

Series Convergence and Error Analysis

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

Slope Field Analysis for $$\frac{dy}{dx}=x$$

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

Tank Draining Problem

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

Accumulated Rainfall

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

Accumulation of Rainwater in a Reservoir

During a storm lasting 6 hours, rain falls on a reservoir at a rate given by $$R(t)=3+2\sin(t)$$ (cm

Arc Length and Average Speed for a Parametric Curve

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

Area Under a Curve with a Discontinuity

Consider the function $$f(x)=\frac{1}{x+2}$$ defined on $$[0,3]$$.

Average Temperature Over a Day

A research team studies the variation in water temperature in a lake over a 24‐hour period. The temp

Average Velocity of a Car

A car's velocity is given by $$v(t)=20-4*\ln(t+1)$$ (in m/min) for $$t$$ in minutes on the interval

Car Braking and Stopping Distance

A car decelerates with an acceleration given by $$a(t)=-2*t$$ (in m/s²) and has an initial velocity

Chemical Mixing in a Tank

A tank initially contains 100 liters of water. A chemical solution with a concentration of 0.5 g/l f

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}{

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

Displacement vs. Distance: Analysis of Piecewise Velocity

A particle moves along a line with velocity given by $$v(t)=\begin{cases} t^2, & 0 \le t < 2,\\ 8-t^

Drone Motion Analysis

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

Environmental Contaminant Spread Analysis

A contaminant enters a lake at a rate given by $$r(t)=4e^{-0.5*t}$$ kilograms per day, where $$t$$ i

Force on a Submerged Plate

A vertical rectangular plate is submerged in water. The plate is 3 m wide and extends from a depth o

Integral Approximation Using Taylor Series

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

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 Cash Flow Analysis

A company’s net cash flow is modeled by $$N(t)=50*\ln(t+1) - 2*t$$ (in thousands of dollars per mont

Particle Motion with Variable Acceleration

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

Pollution Concentration in a Lake

A lake has a pollution concentration modeled by $$C(x) = 16 - x^2$$ (in mg/L), where $$x$$ (in meter

Projectile Motion under Gravity

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

River Crossing: Average Depth and Flow Calculation

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

Series and Integration Combined: Error Bound in Integration

Consider the integral $$\int_{0}^{0.5} \frac{1}{1+x^2} dx$$. Use the Taylor series expansion of the

Surface Area of a Solid of Revolution

Consider the curve $$y=\sqrt{x}$$ on the interval $$[0,9]$$. When this curve is rotated about the x-

Surface Area of a Solid of Revolution

Consider the curve $$y= \sqrt{x}$$ over the interval $$0 \le x \le 4$$. When this curve is rotated a

Total Charge in an Electrical Circuit

In an electrical circuit, the current is given by $$I(t)=5*\cos(0.5*t)$$ (in amperes), where \(t\) i

Total Distance Traveled with Changing Velocity

A runner’s velocity is given by $$v(t)=3*(t-1)*(t-4)$$ m/s for $$0 \le t \le 5$$ seconds. Note that

Volume of a Solid via Shell Method

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

Volume of a Solid via the Disc Method

The region bounded by the curve $$y=\sqrt{x}$$ and the x-axis for $$x\in[0,4]$$ is revolved about th

Work Done by a Variable Force

A variable force is applied along a straight line and is given by $$F(x)=3*\ln(x+1)$$ (in Newtons),

Work Done by a Variable Force

A variable force acting along the x-axis is given by $$F(x) = 2 * x + 3$$ (in Newtons). An object mo

Analyzing a Cycloid

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

Arc Length and Surface Area of Revolution from a Parametric Curve

Consider the curve defined by $$x(t)=\cos(t)$$ and $$y(t)=\ln(\sec(t)+\tan(t))$$ for $$0 \le t < \fr

Arc Length of a Cycloid

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

Arc Length of a Parametric Curve

Consider the parametric curve defined by $$x(t)= t^2$$ and $$y(t)= t^3$$ for $$0 \le t \le 1$$. Anal

Arc Length of a Parametric Curve with Logarithms

Consider the curve defined parametrically by $$x(t)=\ln(t+1)$$ and $$y(t)=\sqrt{t}$$ for $$0 \le t \

Area Between Polar Curves

In the polar coordinate plane, consider the region bounded by the curves $$r = 2 + \cos(\theta)$$ (t

Comprehensive Motion Analysis Using Parametric and Vector-Valued Functions

A particle moves with position given by $$ r(t)=\langle t*e^{-t},\;\ln(1+t) \rangle $$ for $$ t\ge0

Concavity and Inflection Points of a Parametric Curve

For the curve defined by $$x(t)=e^{t}-t$$ and $$y(t)=\ln(1+t^2)$$ for $$t \ge 0$$, answer the follow

Conversion and Differentiation of a Polar Curve

Consider the polar curve defined by $$ r=2+\sin(\theta) $$. Study its conversion to Cartesian coordi

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

Differentiation and Integration of a Vector-Valued Function

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

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

Implicit Differentiation with Implicitly Defined Function

Consider the equation $$x^2+xy+y^2=7$$, which defines $$y$$ implicitly as a function of $$x$$.

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

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)

Motion Along a Helix

A particle moves along a helix described by the vector-valued function $$\vec{r}(t)=<\cos(t),\, \sin

Motion on a Circle in Polar Coordinates

A particle moves along a circular path of constant radius $$r = 4$$, with its angle given by $$θ(t)=

Optimization on a Parametric Curve

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

Parametric Representation of Circular Motion

An object moves along a circle of radius $$5$$, with its position given by $$x(t)=5\cos(t)$$ and $$y

Particle Motion in the Plane

A particle moves in the plane with its position described by the parametric equations $$x(t)=3*\cos(

Polar Coordinates: Area of a Leaf-Shaped Curve

Consider the polar curve $$r(\theta)=2*\cos(\theta)$$ for $$-\frac{\pi}{2}\le\theta\le\frac{\pi}{2}$

Polar to Cartesian Conversion

Consider the polar curve defined by $$r = 4*\cos(\theta)$$.

Satellite Orbit: Vector-Valued Functions

A satellite’s orbit is modeled by the vector function $$\mathbf{r}(t)=\langle \cos(t)+0.1*\cos(6*t),

Slope of a Tangent Line for a Polar Curve

For the polar curve defined by \(r=3+\sin(\theta)\), determine the slope of the tangent line at \(\t

Tangent Line to a Parametric Curve

Consider the circle parametrized by $$x(t)=3\sin(t)$$ and $$y(t)=3\cos(t)$$ for $$0\le t\le 2\pi$$.

Time of Nearest Approach on a Parametric Path

An object travels along a path defined by $$x(t)=5*t-t^2$$ and $$y(t)=t^3-6*t$$ for $$t\ge0$$. Answe

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