AP Calculus BC FRQ Room

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 Method for Evaluating Limits

Consider the function $$h(x)=\frac{x^2-9}{x-3}.$$ Answer the following parts.

Algorithm Time Complexity

A recursive algorithm has an execution time that decreases with each iteration: the first iteration

Analysis of a Jump Discontinuity

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

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

Application of the Squeeze Theorem

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

Applying the Squeeze Theorem to a Trigonometric Function

Consider the function $$f(x)= x^2*\sin(\frac{1}{x})$$ for $$x \neq 0$$ with $$f(0)=0$$. Use the Sque

Calculating Tangent Line from Data

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

Evaluating a Rational Function Limit Using Algebraic Manipulation

Consider the function $$f(x)=\frac{x^2-9}{x-3}$$. Analyze the limit as $$x \to 3$$.

Exploring Removable and Nonremovable Discontinuities

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

Exponential Function Limit and Continuity

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

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

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 Water Tank Volume

The water volume in a tank over time is recorded and displayed in the graph provided. Due to a senso

Internet Data Packet Transmission and Error Rates

In a data transmission system, an error correction protocol improves the reliability of transmitted

Interplay of Polynomial Growth and Exponential Decay

Consider the function $$s(x)= x\cdot e^{-x}$$.

L'Hôpital's Rule for Indeterminate Forms

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

Limit at an Infinite Discontinuity

Consider the function $$g(x)= \frac{1}{(x-2)^2}$$. Analyze its behavior near the point where it is u

Limits and Absolute Value Functions

Examine the function $$f(x)= \frac{|x-3|}{x-3}$$ defined for $$x \neq 3$$.

Limits Involving Trigonometric Functions

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

Limits of Composite Trigonometric Functions

Let $$p(x)= \frac{\sin(3x)}{\sin(5x)}$$.

Limits with Composite Logarithmic Functions

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

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 Limits in a Piecewise Function

Consider the function $$f(x)=\begin{cases} \sqrt{x+4}, & x < 5, \\ 3*x-7, & x \ge 5. \end{cases}$$ A

Rational Functions and Limit at Infinity

Consider the rational function $$r(x)= \frac{2x^2+3x-1}{x^2-4}$$.

Removable Discontinuity in a Cubic Function

Consider the function $$f(x)=\frac{x^3-8}{x-2}$$ defined for $$x\neq2$$. Answer the following:

Removing a Removable Discontinuity in a Piecewise Function

Examine the function $$g(x)= \begin{cases} \frac{x^2-9}{x-3}, & x \neq 3 \\ m, & x=3 \end{cases}$$.

Using Power Series to Estimate a Trigonometric Function

The power series for $$\sin(x)$$ is $$Q(x)=\sum_{n=0}^{\infty} \frac{(-1)^n*x^{2*n+1}}{(2*n+1)!}.$$

Average and Instantaneous Growth Rates in a Bacterial Culture

A bacterial population is modeled by the function $$P(t)= e^{0.3*t} + 10$$, where $$t$$ is measured

Bacteria Culturing in a Bioreactor

In a bioreactor, the bacterial inflow (growth) rate is given by $$B_{in}(t)=\frac{15}{1+e^{-0.3*(t-5

Circular Motion Analysis

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

Derivative from First Principles: Reciprocal Function

Let $$f(x)= \frac{1}{x}$$.

Derivative Using Limit Definition

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

Differentiation in Biological Growth Models

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

Differentiation of Implicitly Defined Functions

An ellipse is defined by the equation $$\frac{x^2}{4} + \frac{y^2}{9} = 1$$. Use implicit differenti

Economic Model Rate Analysis

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

Error Bound Analysis for Cos(x) Approximations in Physical Experiments

In a controlled physics experiment, small angle approximations for $$\cos(x)$$ are critical. Analyze

Implicit Differentiation in Logarithmic Equations

Consider the relation given by $$x*\ln(y)+y*\ln(x)=5$$, where $$x>0$$ and $$y>0$$.

Instantaneous Velocity from a Displacement Function

A particle moves along a straight line with its position at time $$t$$ (in seconds) given by $$s(t)

Instantaneous vs. Average Rate of Change

Consider the trigonometric function $$f(x)= \sin(x)$$.

Limit Definition of Derivative for a Rational Function

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

Limit Definition of the Derivative for a Quadratic Function

Let $$f(x)= 5*x^2 - 4$$. Use the limit definition of the derivative to compute $$f'(x)$$.

Maclaurin Series for arctan(x) and Error Estimate

An engineer in signal processing needs the Maclaurin series for $$g(x)=\arctan(x)$$ and an understan

Motion Along a Line

An object moves along a line with its position given by $$s(t)=4*t^3 - 12*t^2 + 9*t$$, where $$s$$ i

Pharmacokinetics: Drug Concentration Analysis

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

Population Dynamics: Derivative and Series Analysis

A town's population is modeled by the continuous function $$P(t)= 1000e^{0.04t}$$, where t is in yea

Product and Quotient Rule Application

Consider the function $$f(x)=\frac{x*\ln(x)}{e^{x}+2}$$, defined for $$x>0$$. Analyze its behavior u

Product Rule in Differentiation

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

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

Revenue Change Analysis via the Product Rule

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

Secant and Tangent Approximations from a Graph

A function f(t) has been graphed from t = 0 to 10 seconds. Use the graph to estimate rates of change

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

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

Water Reservoir Depth Analysis

The depth of water (in meters) in a reservoir is modeled by $$d(t)=10+3*t-0.5*t^2$$, where $$t$$ is

Analysis of a Piecewise Function with Discontinuities

Consider the piecewise function $$ f(x) = \begin{cases} 2*x+1, & x < 1, \\ 3, & 1 \le x \le 2, \\ \s

Analyzing the Rate of Change in an Economic Model

Suppose the profit function is given by $$P(x)=e^{x}-4*\ln(x+2)$$, where x represents the number of

Biological Growth Model Differentiation

In a biological model, the concentration of a chemical is modeled by $$C(t)=e^{-0.5*t}+\ln(2*t+3)$$.

Chain, Product, and Implicit: A Motion Problem

A particle moves along a curve defined by the parametric equations $$x(t)=e^{-t}\cos(t)$$ and $$y(t)

Composite and Inverse Differentiation in an Electrical Circuit

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

Composite Function: Polynomial Exponent

Consider the function $$ f(x)= (2*x^2+3*x-5)^3 $$. Analyze the function's derivative and behavior.

Differentiation Involving an Inverse Function and Logarithms

Let $$f(x)= x^2+ 3*x+ 2$$, and let y be defined by $$y= \ln(f^{-1}(e^{x}))$$, where $$f^{-1}$$ is th

Engine Air-Fuel Mixture

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

Enzyme Kinetics in a Biochemical Reaction

In an enzymatic reaction, the substrate concentration $$S(t)$$ and the product concentration $$P(t)$

Implicit Differentiation in a Conic Section

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

Implicit Differentiation in a Cost-Production Model

In an economic model, the relationship between the production level $$x$$ (in units) and the average

Implicit Differentiation in a Hyperbola-like Equation

Consider the equation $$ x*y = 3*x - 4*y + 12 $$.

Implicit Differentiation in a Pressure-Temperature Experiment

In a chemistry experiment, the pressure $$P$$ (in atm) and temperature $$T$$ (in °C) of a system sat

Implicit Differentiation in an Elliptical Orbit

An orbit of a satellite is modeled by the ellipse $$4*x^2 + 9*y^2 = 36$$. At the point $$\left(1, \f

Implicit Differentiation in Circular Motion

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

Implicit Differentiation in Geometric Optics

A parabolic mirror used in a geometric optics experiment is described by the implicit equation $$x^2

Implicit Differentiation with Exponential and Trigonometric Mix

Consider the equation $$e^{x*y} + \sin(x) - y = 0$$. Differentiate implicitly with respect to $$x$$

Inverse Analysis of a Log-Polynomial Function

Consider the function $$f(x)=\ln(x^2+1)$$. Analyze its one-to-one property on the interval $$[0,\inf

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 Basics

Let $$f$$ be a one-to-one differentiable function with $$f(3)=5$$ and $$f'(3)=2$$, and let $$g$$ be

Inverse Function Differentiation for Cubic Functions

Let $$f(x)= x^3 + 2*x$$, and let $$g(x)$$ be its inverse function. Answer the following:

Inverse Function Differentiation in Navigation

A vehicle’s distance traveled is modeled by $$f(t)= t^3 + t$$, where $$t$$ represents time in hours.

Logarithmic and Composite Differentiation

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

Nested Composite Function Differentiation

Consider the function $$ h(x)= \sqrt{\cos(3*x^2+1)} $$.

Tangent Line for a Parametric Curve

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

Airplane Passing a Tower: A Related Rates Problem

An airplane is flying at a constant altitude of $$1000\; m$$ horizontally away from an observer on t

Analyzing Runner's Motion

A runner's displacement is modeled by the function $$s(t)=-t^3+9t^2+1$$, where s(t) is in meters and

Bacterial Growth and Linearization

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

Concavity and Acceleration in Motion

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

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

Cycloid Tangent Line

A cycloid is defined by the parametric equations $$x(t)=2*(t-\sin(t))$$ and $$y(t)=2*(1-\cos(t))$$ f

Economic Rates: Marginal Profit Analysis

A manufacturer’s profit (in dollars) from producing $$x$$ items is modeled by $$P(x)=500*x-2*x^2$$.

Estimating the Rate of Change from Reservoir Data

A reservoir's water level h (in meters) was recorded at different times, as shown in the table below

Exponential Cooling Rate Analysis

The temperature of a cup of coffee is modeled by $$T(t)=70+50e^{-0.05t}$$ (in °F), where t is in min

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 $$

Horizontal Tangents on Cubic Curve

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

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.

Inflating Balloon: Related Rates

A spherical balloon is being inflated such that its volume increases at a constant rate of 10 in³/s.

Inflating Spherical Balloon: A Related Rates Problem

A spherical balloon is being inflated so that its volume increases at a constant rate of $$12\; in^3

Linearization and Differentials: Approximating Function Values

Consider the function $$f(x)= x^4$$. Use linearization to estimate the value of the function for a s

Linearization in Engineering Load Estimation

In an engineering project, the load on a beam is modeled by $$L(x)=100*x^2+50*x+200$$, where $$L(x)$

Linearization of a Power Function

Let $$f(x)=x^4$$. Use linearization at $$x=4$$ with $$\Delta x=-0.02$$ to approximate $$(3.98)^4$$.

Minimum Time to Cross a River

A person must cross a 100-meter-wide river. They can swim at 2 m/s and run along the bank at 5 m/s.

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

Polar Coordinates: Arc Length of a Spiral

Consider the polar curve defined by $$r(\theta)=1+\frac{\theta}{2}$$ for $$0 \le \theta \le \pi$$.

Population Growth Rate

The population of a bacteria culture is given by $$P(t)= 500e^{0.03*t}$$, where $$t$$ is in hours. A

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

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

Water Tank Filling: Related Rates

A cylindrical tank has a fixed radius of 2 m. The volume of water in the tank is given by $$V=\pi*r^

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-Linear Function

Consider the function $$p(x)=e^x-4*x$$. Answer the following parts:

Analysis of Total Distance Traveled

A particle moves along a line with a velocity function given by $$v(t)=t^2-4*t+3$$ for $$t\in[0,5]$$

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.

Application of the Mean Value Theorem

Consider the function $$f(t)=t^3-3*t^2+2*t+5$$ representing the position (in meters) of a car along

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

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

Asymptotic Behavior and Limits of a Logarithmic Model

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

Car Motion: Velocity and Total Distance

A car’s position along a straight road is modeled by $$s(t) = t^3 - 6*t^2 + 9*t + 15$$ (in meters),

Concavity and Inflection Points

Examine the function $$p(x)= x^3-3*x^2-9*x+30$$ to determine its concavity and any inflection points

Curve Sketching with Second Derivative

Consider the function $$f(x) = x^4 - 4*x^3 + 6*x^2 - 4*x + 1$$.

Determining Absolute Extrema for a Trigonometric-Polynomial Function

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

Drug Dosage Infusion

A patient receives an intravenous drug infusion at a rate given by $$D(t)=4*\exp(-0.2*t)$$ mg/min. A

Dynamic Analysis Under Time-Varying Acceleration in Two Dimensions

A particle moves in the plane with acceleration given by $$\vec{a}(t)=\langle3\cos(t),-2\sin(t)\rang

Epidemic Infection Model

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

Graph Analysis of a Logarithmic Function

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

Interpreting a Velocity-Time Graph

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

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

Mean Value Theorem Application

Let $$f(x)=\ln(x)$$ be defined on the interval $$[1, e^2]$$. Answer the following parts using the Me

Mean Value Theorem in Motion

A car travels along a straight highway with its position given by $$s(t)=4*t^3 - 12*t^2 + 9*t + 5$$

Mean Value Theorem in River Flow

A river cross‐section’s depth (in meters) is modeled by the function $$f(x) = x^3 - 4*x^2 + 3*x + 5$

Optimization in a Geometric Setting: Garden Design

A farmer is designing a rectangular garden adjacent to a river. No fence is needed along the river s

Pharmaceutical Dosage and Metabolism

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

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: Modeling a Car's Journey

An object moves along a straight line and its position is given by $$s(t)= t^3-6*t^2+9*t$$ for $$t$$

Series Convergence and Integration in a Physical Model

A physical process is modeled by the power series $$g(x)=\sum_{n=1}^\infty \frac{2^n}{n!} * (x-3)^n$

Sign Chart Construction from the Derivative

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

Square Root Function Inverse Analysis

Consider the function $$f(x)= \sqrt{3*x+4}$$ defined for $$x \ge -\frac{4}{3}$$. Answer the followin

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

Wireless Signal Attenuation

A wireless signal, originally at an intensity of 80 units, passes through a series of walls. Each wa

Antiderivative with Initial Condition

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

Applying the Fundamental Theorem of Calculus

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

Approximating Energy Consumption Using Riemann Sums

A household’s power consumption (in kW) is recorded over an 8‐hour period. The following table shows

Approximating Water Volume Using Riemann Sums

A storm causes a varying inflow rate f(t) (in m³/h) into a reservoir. The inflow rate was recorded a

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 Curves

Consider the curves $$f(x)=x^2$$ and $$g(x)=4*x - 3$$. These curves intersect and enclose a region.

Area Under the Curve for a Quadratic Function

Consider the quadratic function $$h(x)= x^2 + 2*x$$. Find the area between the curve and the $$x$$-a

Bacterial Growth with Logarithmic Integration

A bacterial culture grows at a rate given by $$P'(t)=100/(t+2)$$ (in bacteria per hour). Given that

Definite Integration of a Polynomial Function

For the function $$f(x)=5*x^{3}$$ defined on the interval $$[1,2]$$, determine the antiderivative an

Drug Concentration in a Bloodstream

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

Estimating Integral from Tabular Data

Given the following table of values for $$F(t)$$ over time, estimate the integral $$\int F(t)\,dt$$

Evaluating a Complex Integral

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

Integration of a Trigonometric Product via U-Substitution

Evaluate the indefinite integral $$\int \sin(2*x)\cos(2*x)\,dx$$.

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}{

Logarithmic Functions in Ecosystem Models

Let \(f(t)= \ln(t+2)\) for \(t \ge 0\) model an ecosystem measurement. Answer the following question

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 $$t\in[0,8]$$

Particle Motion with Changing Velocity Signs

A particle is moving along a line with its velocity given by $$v(t)= 6 - 4*t$$ (in m/s) for t betwee

Particle Motion with Variable Acceleration and Displacement Analysis

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

Probability Density Function and Expected Value

Let the probability density function (pdf) be defined by $$f(x)=k*x*e^{-x}$$ for $$x\ge0$$.

Rainfall Accumulation and Runoff

During a storm, rainfall intensity is modeled by $$R(t)=3*t$$ inches per hour for $$0 \le t \le 4$$

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:

Sandpile Accumulation

At an industrial site, sand is continuously added to and removed from a pile. The addition rate is g

Taylor/Maclaurin Series Approximation and Error Analysis

Consider the function $$f(x)=\ln(1+x)$$. This function is infinitely differentiable at x = 0 and has

Trapezoidal Approximation for a Curved Function

Consider the function $$f(x)=x^2+2$$ on the interval [1, 5]. Answer the following:

Vehicle Distance Estimation from Velocity Data

A vehicle's velocity over time is recorded in the table provided. Use Riemann sums to estimate the v

Volume of a Solid with Known Cross-sectional Area

A solid extends from $$x=0$$ to $$x=5$$, and its cross-sectional area perpendicular to the x-axis is

Volume of a Solid with Square Cross-Sections

Consider the region bounded by the curve $$y=x^{2}$$ and the line $$y=4$$. Cross-sections taken perp

Analysis of a Nonlinear Differential Equation

Consider the nonlinear differential equation $$\frac{dy}{dx} = y^3-3*y$$.

Analysis of an Inverse Function from a Differential Equation Solution

Suppose a differential equation is solved to give an increasing function $$f(x)=\ln(2*x+3)$$ defined

Bacteria Growth with Antibiotic Treatment

A bacterial culture has a population $$N(t)$$ that grows at a rate proportional to its size, given b

Bacterial Growth with Time-Dependent Growth Rate

A bacterial culture grows according to the differential equation $$\frac{dP}{dt}=(r_0+r_1*t)P$$, whe

Capacitor Discharge in an RC Circuit

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

Car Engine Temperature Dynamics

The temperature $$T(t)$$ (in °C) of a car engine is modeled by the differential equation $$\frac{dT}

Chemical Reactor Mixing

In a chemical reactor, the concentration $$C(t)$$ (in M) of a chemical is governed by the equation $

Constructing and Interpreting a Slope Field

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

Cooling of a Smartphone Battery

A smartphone battery cools according to Newton’s law: $$\frac{dT}{dt} = -k*(T-T_{room})$$. Initially

Differential Equation Involving Logarithms

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

Economic Growth Model

An economy's output \(Y(t)\) is modeled by the differential equation $$\frac{dY}{dt}= a\,Y - b\,Y^2$

Exact Differential Equation

Consider the differential equation $$ (2*x*y + y^2)\,dx + (x^2+2*x*y)\,dy = 0 $$. Answer the followi

Exponential Growth via Slope Field Analysis

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

Exponential Growth with Variable Rate

A bacterial culture grows according to the differential equation $$\frac{dP}{dt}=k(t)P$$, where the

Free-Fall with Air Resistance

An object falling under gravity experiences air resistance proportional to the square of its velocit

Gas Pressure Dynamics

A container is being filled with gas such that the pressure $$P(t)$$ (in psi) increases at a constan

Implicit Differential Equations and Slope Fields

Consider the implicit differential equation $$x\frac{dy}{dx}+ y = e^x$$. Answer the following parts.

Implicit Differentiation from an Implicitly Defined Relation

Consider the implicit equation $$x^3 + y^3 - 3*x*y = 0$$ which defines $$y$$ as a function of $$x$$

Inverse Function Analysis Derived from a Differential Equation Solution

Consider the function $$f(x)=x^3+2$$. Although this function is provided outside of a differential e

Logistic Differential Equation Analysis

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

Logistic Growth Population Model

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

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

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

Newton's Law of Cooling

Newton's Law of Cooling is given by the differential equation $$\frac{dT}{dt} = -k*(T-T_a)$$, where

Picard Iteration for Approximate Solutions

Consider the initial value problem $$\frac{dy}{dt}=y+t, \quad y(0)=1$$. Use one iteration of the Pic

Population Growth with Harvesting

A fish population in a lake is modeled by the differential equation $$\frac{dP}{dt}= rP - H$$, where

Predator-Prey Model with Harvesting

Consider a simplified model for the prey population in a predator-prey system that includes constant

Radio Signal Strength Decay

A radio signal's strength $$S$$ decays with distance r according to the differential equation $$\fra

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 Parameter Identification

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

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

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

Slope Field and Sketching a Solution Curve

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

Solution and Analysis of a Linear Differential Equation with Equilibrium

Consider the differential equation $$\frac{dy}{dx} = 3*y - 2$$, with the initial condition $$y(0)=1$

Solution Curve from Slope Field

A differential equation is given by $$\frac{dy}{dx} = -y + \cos(x)$$. A slope field for this equatio

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 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

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 from Experimental Data

In an experiment, researchers recorded measurements for two functions, $$f(t)$$ and $$g(t)$$, repres

Area Between Curves in a Physical Context

The heights of two particles moving along parallel tracks are given by $$h_1(t)=t^2$$ and $$h_2(t)=4

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 and Linear Functions

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

Area Between Exponential Curves

Consider the functions $$f(x)=e^{-x}$$ and $$g(x)=e^{-2*x}$$ for $$x\ge0$$. Answer the following:

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 and Instantaneous Analysis in Periodic Motion

A particle moves along a line with its displacement given by $$s(t)= 4*\cos(2*t)$$ (in meters) for $

Average Chemical Concentration Analysis

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

Average Reaction Concentration in a Chemical Reaction

In a chemical reaction, the concentration of a reactant is modeled by $$C(t)=20*\exp(-0.5*t)$$ (in m

Average Value and Critical Points of a Trigonometric Function

Consider the function $$f(x)=\sin(2*x)+\cos(2*x)$$ on the interval $$\left[0,\frac{\pi}{2}\right]$$.

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

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 Rod

A thin rod of length 10 m has a linear density given by $$\rho(x)=3+0.4*x$$ (in kg/m) where $$x$$ is

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}{

Electric Current and Charge

An electric current in a circuit is defined by $$I(t)=4*\cos\left(\frac{\pi}{10}*t\right)$$ amperes,

Electrical Charge Distribution

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

Net Change and Direction of Motion

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

Particle Motion Analysis with Variable Acceleration

A particle moving along a straight line has an acceleration given by $$a(t)=4*e^{-t}-\sin(t)$$ (in m

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}

Solid of Revolution using Washer Method

The region bounded by the curves $$y = x^2$$ and $$y = 2 * x$$ is rotated about the x-axis. Answer t

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 a Solid of Revolution Between Curves

Consider the region bounded by the curves $$y=\sqrt{x}$$ and $$y=\frac{x}{2}$$ for $$x \in [0,4]$$.

Work Done in Pumping Water from a Parabolic Tank

A water tank has a parabolic cross-section described by $$y=x^2$$ (with y in meters, x in meters). T

Work Done in Pumping Water from a Tank

A cylindrical tank has a radius of $$3$$ meters and a height of $$10$$ meters. The tank is completel

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

Work Done with a Discontinuous Force Function

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

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 Quarter-Circle

Consider the circle defined parametrically by $$x(t)=\cos(t)$$ and $$y(t)=\sin(t)$$ for $$0 \le t \l

Arc Length of a Vector-Valued Function

Let $$ r(t)=\langle 3*t,\; 4*\sin(t) \rangle $$ for $$ 0 \le t \le \pi $$. Determine properties rela

Concavity and Inflection in Parametric Curves

A curve is defined by the parametric functions $$x(t)=t^3-3*t$$ and $$y(t)=t^2$$ for \(-2\le t\le2\)

Conversion and Differentiation of a Polar Curve

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

Curvature and Oscillation in Vector-Valued Functions

Given the vector function $$\vec{r}(t)=\langle \ln(t+1), \cos(t) \rangle$$ for $$t \ge 0$$, answer t

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

Differentiation and Integration of a Vector-Valued Function

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

Dynamics in Polar Coordinates

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

Inflow and Outflow in a Water Tank

A water tank has water entering at a rate given by $$I(t)=5+\sin(t)$$ (liters per minute) and water

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 +

Intersections in Polar Coordinates

Two polar curves are given by $$r = 3 - 2*\sin(\theta)$$ and $$r = 1 + \cos(\theta)$$.

Motion Along a Helix

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

Multi-Step Problem Involving Polar Integration and Conversion

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

Parametric Curve with a Loop and Tangent Analysis

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

Parametric Equations from Real-World Data

A moving vehicle’s position is modeled by the parametric equations $$ x(t)=3*t+1 $$ and $$ y(t)=t^2-

Particle Motion in the Plane

A particle moves in the plane with parametric equations $$x(t)= 3\cos(t)$$ and $$y(t)= 3\sin(t)$$ fo

Particle Motion in the Plane

Consider a particle whose motion in the plane is defined by the parametric equations $$x(t) = t^2 -

Periodic Motion: A Vector-Valued Function

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

Polar Boundary Conversion and Area

A region in the polar coordinate plane is defined by $$1 \le r \le 3$$ and $$0 \le \theta \le \frac{

Polar Differentiation and Tangent Lines

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

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