AP Calculus BC FRQ Room

Analysis of a Piecewise Function with Multiple Definitions

Consider the function $$h(x)=\begin{cases} \frac{x^2-9}{x-3} & \text{if } x<3, \\ 2*x-1 & \text{if

Analysis of a Rational Inflow Function with a Discontinuity

A water tank is monitored by an instrument that records the inflow rate as $$R(t)=\frac{t^2-9}{t-3}$

Analyzing Limits of a Composite Function

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

Asymptotic Behavior and Horizontal Limits

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

Bacterial Growth Experiment

A laboratory experiment involves a bacterial culture whose population at hour $$n$$ is modeled by a

Composite Functions: Limits and Continuity

Let $$f(x)=x^2-1$$, which is continuous for all $$x$$, and let $$g(x)=f(\sqrt{x+1})$$.

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{

Determining Continuity via Series Expansion

Consider the function $$f(x)= \frac{e^x - \ln(1+x) - x - 1}{x^2}$$ for $$x \neq 0$$ with $$f(0)=L$$.

End Behavior and Horizontal Asymptote Analysis

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

End Behavior of an Exponential‐Log Function

Consider the function $$f(x)= e^{-x} \ln(1+x)$$. Analyze its behavior by investigating the limit as

Endpoint Behavior of a Continuous Function

Let $$m(x)=\sqrt{x+4}$$ be defined on the interval $$[-4,5]$$. Answer the following:

Establishing Continuity in a Piecewise Function

Consider the piecewise-defined function $$p(x)= \begin{cases} \frac{x^2-4}{x-2} & x \neq 2, \\ k & x

Evaluating a Limit with Algebraic Manipulation

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

Fuel Efficiency and Speed Graph Analysis

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

Graphical Analysis of Limits and Asymptotic Behavior

A graphical study titled 'Graph of Experimental Data' shows the measured concentration of a chemical

Horizontal Asymptote of a Rational Function

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

Indeterminate Limit with Exponential and Log Functions

Examine the limit $$\lim_{x \to 0} \frac{e^{2x} - \cos(x) - 1}{\ln(1+x^2)}.$$

Intermediate Value Theorem in a Continuous Function

Consider the continuous function $$p(x)=x^3-3*x+1$$ on the interval $$[-2,2]$$. Answer the followi

Limits at a Point: Removable Discontinuity Analysis

Consider the function $$f(x)=\frac{(x+3)*(x-2)}{(x+3)*(x+5)}$$ which is not defined at $$x=-3$$. Ans

Logarithmic Function Limits

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

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

One-Sided Infinite Limits in Rational Functions

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

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 $$f(x)= \frac{x^2-1}{x-1}$$ for $$x \neq 1$$, and suppose that $$f(1)=m$$. Answer the following:

Physical Applications: Temperature Continuity

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

Piecewise Function Critical Analysis

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

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

Rational Function and Removable Discontinuity

Consider the function $$g(x) = \frac{(x+3)(x-2)}{(x-2)}$$, defined for $$x \neq 2$$, and suppose tha

Related Rates: Changing Shadow Length

A streetlight is mounted at the top of a 12 m tall pole. A person 1.8 m tall walks away from the pol

Removable Discontinuity in a Rational Function

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

Trigonometric Rate Function Analysis

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

Acceleration and Jerk in Motion

The position of a car is given by $$s(t)=t^3-6*t^2+9*t+2$$, where $$t$$ is time in seconds and $$s(t

Analyzing a Removable Discontinuity in a Rational Function

Consider the function defined by $$f(x)= \begin{cases} \frac{x^2-1}{x-1} & x \neq 1 \\ 3 & x = 1 \e

Car Motion and Critical Velocity

The position of a car is described by $$s(t)=t^3 - 12*t^2 + 36*t$$ (with $$s$$ in meters and $$t$$ i

Chemical Mixing Tank

In an industrial process, a mixing tank receives a chemical solution at a rate of $$C_{in}(t)=25+5*t

Chemical Reaction Rate

In a chemical reaction, the concentration of a reactant is modeled by $$C(t)=10 - 2*\ln(t+1)$$, wher

Composite Exponential-Log Function Analysis

Consider the function $$f(x)=e^{x}*\ln(x+3)$$, which arises in the study of compound reactions in ch

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

Derivative of a Composite Function Using the Limit Definition

Consider the function $$h(x)=(2*x+3)^3$$. Use the limit definition of the derivative to answer the f

Derivatives of a Rational Function

Consider the function $$g(x)= \frac{2*x^3 - 1}{x^2+4}$$. Use differentiation rules to answer the fol

Differentiability of an Absolute Value Function

Consider the function $$f(x) = |x|$$.

Differentiation of a Trigonometric Function

Let $$f(x)=\sin(x)+x*\cos(x)$$. Differentiate the function using the sum and product rules.

Epidemiological Rate Change Analysis

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

Exponential Growth and Its Derivative

A culture of bacteria grows according to the model $$P(t)= 100*e^{0.03*t},$$ where \(P(t)\) is th

Graphical Derivative Analysis

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

Implicit Differentiation in Circular Motion

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

Implicit Differentiation: Exponential-Polynomial Equation

Consider the curve defined by $$e^(x*y) + x^2 = y^2$$.

Instantaneous Rate of Change of a Polynomial Function

Consider the function $$f(x)=2*x^3 - 5*x^2 + 3*x - 7$$ which represents the position (in meters) of

Instantaneous Rate of Change of a Trigonometric Function

Consider the function $$h(t)=\sin(2*t) + \cos(t)$$ which models the displacement (in centimeters) of

Logarithmic Differentiation Simplification

Consider the function $$h(x)=\ln\left( \frac{(x^2+1)^{3}*e^{2*x}}{\sqrt{x+2}} \right)$$.

Maclaurin Series for ln(1+x)

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

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 Problem via Derivatives

A manufacturer’s cost in dollars for producing $$x$$ units is modeled by the function $$C(x)= x^3 -

Product of Exponential and Trigonometric Functions

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

Radioactive Decay with Logarithmic Correction

A radioactive substance decays following the model $$A(t)=A_0*e^{-k*t}+\ln(t+1)$$, where $$t$$ is th

Related Rates: Draining Conical Tank

Water is draining from an inverted conical tank with a height of 6 m and a top radius of 3 m. The vo

Satellite Orbit Altitude Modeling

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

Tangent Line Estimation in Transportation Modeling

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

Taylor Series Expansion of ln(x) About x = 2

For a financial model, the function $$f(x)=\ln(x)$$ is expanded about $$x=2$$. Use this expansion to

Taylor Series for Cos(x) in Temperature Modeling

An engineer uses the cosine function to model periodic temperature variations. Approximate $$\cos(x)

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

Temperature Function Analysis

Suppose the temperature over time is modeled by $$T(t)=e^(2*t)*\sin(t)$$, where $$t$$ is measured in

Traffic Flow and Instantaneous Rate

The number of cars passing through an intersection per minute is modeled by $$F(t)= 3t^2 + 2t + 10$$

Water Treatment Plant Simulator

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

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

Bacterial Culture: Nutrient Inflow vs Waste Outflow

In a bioreactor, the nutrient inflow rate is given by $$N(t)=\ln(2*t+1)$$ (in units/min). The waste

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

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

Composite Implicit Differentiation Involving Trigonometric and Polynomial Terms

Consider the relation $$\sin(x*y) + y^3 = x$$.

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)}

Differentiation of Inverse Trigonometric Functions

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

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+

Implicit Differentiation and Inverse Challenges

Consider the implicit relation $$x^2+ x*y+ y^2=10$$ near the point (2,2).

Implicit Differentiation in Economic Equilibrium

In a market, the relationship between the price $$x$$ (in dollars) and the demand $$y$$ (in thousand

Implicit Differentiation in Trigonometric Equations

For the equation $$\cos(x*y) + x^2 - y^2 = 0$$, y is defined implicitly as a function of x.

Implicit Differentiation of a Composite Equation

Given the implicit relation $$x^2*y + \sin(y) = x$$, answer the following:

Implicit Differentiation of an Ellipse

The ellipse is given by $$4*x^2 + 9*y^2 = 36$$.

Implicit Differentiation on an Elliptical Curve

Consider the ellipse $$\frac{x^2}{9}+\frac{y^2}{4}=1$$ representing an object’s cross-section. Answe

Implicit Differentiation with an Exponential Function

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

Implicit Differentiation with Logarithmic Functions

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

Implicit Differentiation with Trigonometric Equation

Consider the curve defined implicitly by $$\sin(x*y) + x^2 = y^3$$. Answer the following parts:

Indoor Air Quality Control

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

Inverse Function Analysis for Exponential Functions

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

Inverse Function Differentiation with Composite Trigonometric Functions

Let $$f(x)= \sin(2*x) + x$$, which is differentiable and one-to-one. It is given that $$f(\pi/6)= 1$

Inverse Trigonometric Function Differentiation

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

Inverse Trigonometric Functions in Navigation

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

Modeling with Composite Functions: Pollution Concentration

The pollutant concentration in a lake is modeled by $$C(t) = \sqrt{100 - 2*e^{-0.1*t}}$$, where $$t$

Pipeline Pressure and Oil Flow

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

Tangent Line for a Parametric Curve

A curve is given parametrically by $$x(t)= t^2 + 1$$ and $$y(t)= t^3 - t$$.

Approximating Changes with Differentials

Given that the surface area of a sphere is $$A = 4\pi r^2$$, use differentials to approximate the ch

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

Cooling Analysis using Newton’s Law of Cooling

An object cools in a room according to Newton's Law of Cooling, given by $$T(t)=T_{env}+ (T(0)-T_{en

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^{-

Differentials in Engineering: Beam Stress Analysis

The stress S (in Pascals) experienced by an engineering beam under load is modeled by $$S(x)=0.02*x^

Implicit Differentiation on an Ellipse

An ellipse representing a racetrack is given by $$\frac{x^2}{25}+\frac{y^2}{9}=1$$. A runner's x-coo

Implicit Differentiation: Tangent to a Circle

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

Integration Region: Exponential and Polynomial Functions

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

Interpreting the Derivative in Straight Line Motion

A particle moves along a straight line with velocity given by $$ v(t)= 3*t^2 - 4*t + 2 $$. Analyze a

L'Hospital's Rule for Indeterminate Limits

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

Linearization in Finance

The value of an investment is modeled by $$V(x)=1000x^{0.5}$$ dollars, where x represents a market i

Linearization of Trigonometric Implicit Function

Consider the implicit equation $$\tan(x + y) = x - y$$, which implicitly defines $$y$$ as a function

Maximizing Revenue in a Business Model

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

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 of a Rectangular Enclosure

A rectangular enclosure is to be built adjacent to a river. Only three sides of the enclosure requir

Particle Motion Along a Line with Polynomial Velocity

A particle moves along the x-axis with velocity $$v(t)=4*t^3-9*t^2+6*t-1$$ (m/s). Given that $$s(0)=

Population Decline Modeled by Exponential Decay

A bacteria population is modeled by $$P(t)=200e^{-0.3t}$$, where t is measured in hours. Answer the

Production Cost Analysis

A company’s production cost $$C$$ (in dollars) and production level $$x$$ (in thousands of units) sa

Reactant Flow in a Chemical Reactor

In a chemical reactor, a reactant is introduced at a rate $$I(t)=15\sin(\frac{t}{2})$$ (grams per mi

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

Related Rates: Inflating Spherical Balloon with Exponential Volume Rate

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

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

Series Analysis in Acoustics

The sound intensity at a distance is modeled by $$I(x)= I_0 \sum_{n=0}^{\infty} \frac{(-1)^n (x-10)^

Series Approximation of a Temperature Function

The temperature in a chemical reaction is modeled by $$T(t)= 100 + \sum_{n=1}^{\infty} \frac{(-1)^n

Series Integration for Work Calculation

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

Series Integration in Fluid Flow Modeling

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

Analyzing a Function with Implicit Logarithmic Differentiation

Consider the implicit equation $$x\,\ln(y) + y\,e^x = 10$$. Analyze this function by differentiating

Area Between a Curve and Its Tangent

Consider the curve $$f(x)=x^2$$ and its tangent line at \(x=1\). Investigate the region bounded by t

Area Enclosed by a Polar Curve

Consider the polar curve defined by $$r(\theta) = 2 + 2*\sin(\theta)$$. This curve represents a lima

Average and Instantaneous Velocity Analysis

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

Curve Sketching Using Derivatives

For the function $$f(x)=\frac{x}{x^2+1}$$, use its derivatives to sketch a rough graph of the functi

Determining Convergence and Error Analysis in a Logarithmic Series

Investigate the series $$L(x)=\sum_{n=1}^\infty (-1)^{n+1} * \frac{(x-1)^n}{n}$$, which represents a

Echoes in an Auditorium

In an auditorium, an audio signal produces echoes. The first echo has an intensity that is 70% of th

Extreme Value Theorem in Temperature Variation

A metal rod’s temperature (in °C) along its length is modeled by the function $$T(x) = -2*x^3 + 12*x

Finding and Interpreting Inflection Points in a Complex Function

Analyze the function $$f(x)= e^{-x}\,\ln(x)$$ for $$x > 0$$. Investigate the points of inflection an

Investigation of Extreme Values on a Closed Interval

For a particle moving along a path given by $$f(x)=x^3-6*x^2+9*x+5$$ where $$x\in[0,5]$$, analyze it

Linear Approximation of a Radical Function

For the function $$f(x)= \sqrt{x+1}+x$$, find its linear approximation at $$x=3$$ and use it to appr

Logistic Growth Model Analysis

Consider the logistic growth model given by $$P(t)=\frac{100}{1+9e^{-0.5*t}}$$. Answer the following

Optimization in a Log-Exponential Model

A firm's profit is given by the function $$P(x)= x\,e^{-x} + \ln(1+x)$$, where x (in thousands) repr

Optimizing Material for a Container

An open-top rectangular container with a square base must have a fixed volume of $$32$$ cubic feet.

Population Growth Modeling

A region's population (in thousands) is recorded over a span of years. Use the data provided to anal

Rolle's Theorem on a Cubic Function

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

Second Derivative Test for Critical Points

Consider the function $$f(x)=x^3-9*x^2+24*x-16$$.

Sign Chart Construction from the Derivative

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

Staircase Design for a Building

A staircase is being designed for a building. The first step has a height of 7 inches, and each subs

Tangent Line to a Parametric Curve

A curve is defined by the parametric equations $$x(t) = \cos(t)$$ and $$y(t) = \sin(t) + \frac{t}{2}

Taylor Polynomial for $$\cos(x)$$ Centered at $$x=\pi/4$$

Consider the function $$f(x)=\cos(x)$$. You will generate the second degree Taylor polynomial for f(

Taylor Polynomial for $$\ln(x)$$ about $$x=1$$

For the function $$f(x)=\ln(x)$$, find the third degree Taylor polynomial centered at $$x=1$$. Then,

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

Taylor Series in Economics: Cost Function

An economic cost function is modeled by $$C(x)=1000\,e^{-0.05*x}+50\,x$$, where x represents the pro

Antiderivative with Initial Condition

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

Antiderivatives and the Fundamental Theorem

Suppose a continuous function $$h(x)$$ is defined on [2, 8] and its graph (provided) shows that it i

Arc Length Calculation

Find the arc length of the curve $$y=\frac{1}{3}x^{3/2}$$ from $$x=0$$ to $$x=9$$.

Area Between Inverse Functions

Consider the functions $$f(x)=\sqrt{x}$$ and $$g(x)=x-2$$.

Area Estimation with Riemann Sums

Consider the function $$f(x)=x^2-4*x+3$$ on the interval $$[1,5]$$. Using a partition of 4 equal sub

Area Under a Parametric Curve

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

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

Chemical Reactor Concentration

In a chemical reactor, a reactant enters at a rate of $$C_{in}(t)=5+t$$ grams per minute and is simu

Comparing Integration Approximations: Simpson's Rule and Trapezoidal Rule

A student approximates the definite integral $$\int_{0}^{4} (x^2+1)\,dx$$ using both the trapezoidal

Determining Constant in a Height Function

A ball is thrown upward with a constant acceleration of $$a(t)= -9.8$$ m/s² and an initial velocity

Error Analysis in Riemann Sum Approximations

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

Estimating Chemical Production via Riemann Sums

In a laboratory experiment, the reaction rate of a chemical process is recorded at various times. Th

Finding Area Between Two Curves

Consider the curves $$f(x)=x^2$$ and $$g(x)=4*x$$.

Improper Integral Convergence

Examine the convergence of the improper integral $$\int_1^\infty \frac{1}{x^p}\,dx$$.

Integration of a Piecewise-Defined Function

Define the function $$f(x)$$ as follows: $$f(x)= \begin{cases} 2*x, & 0\le x < 3 \\ 12, & 3 \le x \

Integration of a Trigonometric Function by Two Methods

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

Integration Using U-Substitution

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

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}{

Mechanical Systems: Total Change and Inverse Analysis

Consider the function \(f(x)= x^3 + 3*x\) defined for all real \(x\), modeling a mechanical system.

Modeling Water Inflow Using Integration

Water flows into a tank at a rate given by $$R(t)=4-0.5*t$$ (in liters per minute) for $$t\in[0,8]$$

Population Increase from a Discontinuous Growth Rate

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

Rainfall Accumulation Over Time

A storm produces rainfall at a rate modeled by the function $$r(t)=6 * t^(1/2)$$, where $$0 \le t \l

Total Rainfall Accumulation from a Discontinuous Rate Function

Rain falls at a rate (in mm/hr) given by $$ R(t)= \begin{cases} 3t, & 0 \le t < 2, \\ 5, & t = 2, \\

Trapezoidal and Riemann Sums from Tabular Data

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

U-Substitution Integration Challenge

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

Work Done by a Variable Force

A force acting along a displacement is given by $$F(x)=5*x^2-2*x$$ (in Newtons), where x is measured

Bacteria Culture with Regular Removal

A bacterial culture has a population $$B(t)$$ that grows at a continuous rate of $$12\%$$ per hour,

Capacitor Discharge in an RC Circuit

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

Chemical Reaction in a Closed System

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

Chemical Reaction Rate

A chemical reaction causes the concentration $$A(t)$$ of a reactant to decrease according to the rat

Compound Interest with Continuous Payment

An investment account grows with a continuous compound interest rate $$r$$ and also receives continu

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

Direction Field Analysis for Differential Equation

Consider the differential equation $$\frac{dy}{dx} = y\,(1-y)$$. A direction field for this equation

Direction Fields and Isoclines

Examine the differential equation $$\frac{dy}{dx} = \frac{x+y}{x-y}$$. A direction field displaying

Drug Concentration in the Bloodstream

A drug is administered intravenously, and its concentration in the bloodstream is modeled by the dif

Epidemic Spread Modeling

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

Estimating Instantaneous Rate from a Table

A function $$f(x)$$ is defined by the following table of values:

Euler's Method Approximation

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

Euler's Method Approximation

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

Exponential Population Growth and Doubling Time

A certain population is modeled by the differential equation $$\frac{dP}{dt} = k*P$$. This equation

Flow Rate in River Pollution Modeling

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

FRQ 1: Slope Field Analysis for $$\frac{dy}{dx}=x$$

Consider the differential equation $$\frac{dy}{dx}=x$$. Answer the following parts.

FRQ 3: Population Growth and Logistic Model

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

FRQ 7: Projectile Motion with Air Resistance

A projectile is launched vertically upward with an initial velocity of 50 m/s. Its vertical motion i

FRQ 12: Bacterial Growth with Limiting Resources

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

Loan Balance with Continuous Interest and Payments

A loan has a balance $$L(t)$$ (in dollars) that accrues interest continuously at a rate of $$5\%$$ p

Logistic Growth in Population Dynamics

The population of a small town is modeled by the logistic differential equation $$\frac{dP}{dt}=rP\l

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 Tank with Variable Inflow

A tank initially contains 50 L of water with 5 kg of salt dissolved in it. A brine solution with a s

Motion Under Gravity with Air Resistance

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

Separable and Implicit Solution for $$\frac{dy}{dx}= \frac{x}{1+y^2}$$

Consider the differential equation $$\frac{dy}{dx}= \frac{x}{1+y^2}$$, which is defined for all real

Separable Differential Equation with Parameter Identification

A chemical reaction is modeled by the differential equation $$\frac{dC}{dt} = -a*C$$, where $$C(t)$$

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:

Slope Field and Solution Curve Sketching

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

Temperature Control in a Chemical Reaction Vessel

In a chemical reactor, the temperature $$T(t)$$ is regulated by both natural cooling and an external

Accumulated Change in a Population Model

A population of insects grows at a rate given by $$P'(t)=10e^{-0.2*t}$$, where $$t$$ is in days and

Analyzing a Reservoir's Volume Over Time

Water flows into a reservoir at a variable rate given by $$R(t)=50e^{-0.1*t}$$ m³/hour and simultane

Area Between Curves in Window Design

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

Area Between Curves: Enclosed Region

The curves $$f(x)=x^2$$ and $$g(x)=x+2$$ enclose a region. Answer the following:

Area Between Curves: Parabolic & Linear Regions

Consider the curves $$f(x)=5*x-x^2$$ and $$g(x)=x$$. Answer the following questions regarding the re

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

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 Temperature Function

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

Averaging Chemical Concentration in a Reactor

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

Bonus Payout: Geometric Series vs. Integral Approximation

A company issues monthly bonuses that decrease by 20% each month. The bonus in the first month is $5

Center of Mass of a Plate

A lamina occupies the rectangular region defined by $$0 \le x \le 2$$ and $$0 \le y \le 3$$, with a

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

Comparing Average and Instantaneous Rates of Change

For the quadratic function $$f(x)= 3*x^2 - 4*x + 1$$ on the interval $$[1,3]$$, investigate both its

Cost Analysis of a Water Channel

A water channel has a cross-sectional shape defined by the region bounded by $$y=\sqrt{x}$$ and $$y=

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[

Electric Charge Accumulation

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

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

Fluid Force on a Submerged Plate

A vertical plate submerged in water experiences a force due to fluid pressure given by $$F(y)=\rho*g

Implicit Differentiation with Trigonometric Function

Consider the equation $$\cos(x * y) + x = y$$. Answer the following:

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

Optimizing the Shape of a Parabolic Container

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

Particle Motion: Position, Velocity, and Acceleration

A particle moves along a straight line with an acceleration given by $$a(t)= 4 - 2*t$$ (in m/s²). Th

Radioactive Decay Accumulation

The rate of decay of a radioactive substance is given by $$R(t)=100*e^{-0.3*t}$$ decays per day. Ans

River Cross Section Area

The cross-sectional boundaries of a river are modeled by the curves $$y = 5 * x - x^2$$ and $$y = x$

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-

Volume of a Hollow Cylinder Using the Shell Method

A hollow cylindrical tube of height 5 m is formed by rotating the rectangular region bounded by $$x

Volume of Revolution between sin(x) and cos(x)

Consider the region bounded by $$y = \sin(x)$$ and $$y = \cos(x)$$ over the interval where they inte

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 Pumping Water

A water tank is shaped like an inverted circular cone with a height of $$10$$ m and a top radius of

Arc Length of a Parabolic Curve

The parametric curve defined by $$x(t)=t$$ and $$y(t)=t^2$$ models a portion of a parabolic path for

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 an Elliptical Curve

The parametric equations $$x(t)= 4\cos(t)$$ and $$y(t)= 3\sin(t)$$, for $$0 \le t \le \frac{\pi}{2}$

Area Between Polar Curves

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

Area Enclosed by a Polar Curve: Lemniscate

The lemniscate is defined by the polar equation $$r^2=8\cos(2\theta)$$.

Conversion Between Polar and Cartesian Coordinates

Given the polar equation $$r=4\cos(\theta)$$, explore its conversion and properties.

Conversion from Polar to Cartesian Coordinates

The polar equation $$r(\theta)=4*\cos(\theta)$$ represents a curve.

Curve Analysis and Optimization in a Bus Route

A bus follows a route described by the parametric equations $$x(t)=t^3-3*t$$ and $$y(t)=2*t^2-t$$, w

Enclosed Area of a Parametric Curve

A closed curve is given by the parametric equations $$x(t)=3*\cos(t)-\cos(3*t)$$ and $$y(t)=3*\sin(t

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 and Area Between Polar Curves

Two polar curves are given by $$r_1(\theta)=2\sin(\theta)$$ and $$r_2(\theta)=1+\cos(\theta)$$.

Intersection of Parametric Curves

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

Parametric and Polar Conversion Challenge

Consider the parametric equations $$x(t)= \frac{1-t^2}{1+t^2}$$ and $$y(t)= \frac{2*t}{1+t^2}$$ for

Particle Motion with Uniform Angular Change

A particle moves in the polar coordinate plane with its distance given by $$r(t)= 3*t$$ and its angl

Polar Differentiation and Tangent Lines

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

Reparameterization between Parametric and Polar Forms

A curve is described by the parametric equations $$x(t)=a*t$$ and $$y(t)=b*t^2$$, where $$a$$ and $$

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),

Symmetry and Area in Polar Coordinates

Consider the polar curve given by $$r=2\cos(\theta)$$. Answer the following:

Symmetry and Self-Intersection of a Parametric Curve

Consider the parametric curve defined by $$x(t)= \sin(t)$$ and $$y(t)= \sin(2*t)$$ for $$t \in [0, \

Vector Fields and Particle Trajectories

A particle moves in the plane with velocity given by $$\vec{v}(t)=\langle \frac{e^{t}}{t+1}, \ln(t+2

Vector-Valued Fourier Series Representation

The vector function $$\mathbf{r}(t)=\langle \cos(t), \sin(t), 0 \rangle$$ for $$t \in [-\pi,\pi]$$ c

Vector-Valued Function and Particle Motion

Consider the vector-valued function $$\vec{r}(t)= \langle e^t, \sin(t), \cos(t) \rangle$$ representi

Vector-Valued Function with Constant Acceleration

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

Vector-Valued Functions in Motion

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