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

Analyzing Limits of a Combined Exponential‐Log Function

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

Asymptotic Behavior in Rational Functions

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

Caffeine Metabolism in the Human Body

A person consumes a cup of coffee containing 100 mg of caffeine at the start, and then drinks one cu

Continuity and the Intermediate Value Theorem in Temperature Modeling

A temperature function $$T(t)$$, continuous for $$t\in[0,24]$$ (in hours), satisfies $$T(6)=10^\circ

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

Electricity Consumption Rate Analysis

A table provides the instantaneous electricity consumption, $$E(t)$$ (in kW), at various times durin

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

Exponential Function Limits at Infinity

Consider the function $$f(x)=\frac{e^{2*x} - e^{x}}{e^{2*x}+e^{x}}$$. 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 Application

Let $$f(x)=x^3-4*x+1$$, which is continuous on the real numbers. Answer the following:

Intermediate Value Theorem in Temperature Analysis

A city's temperature during a day is modeled by a continuous function $$T(t)$$, where t (in hours) l

Investigating Limits Involving Nested Rational Expressions

Evaluate the limit $$\lim_{x\to3} \frac{\frac{x^2-9}{x-3}}{x-2}$$. (a) Simplify the expression and e

Limits at Infinity and Horizontal Asymptotes

Consider the rational function $$g(x)= \frac{4*x^3-x+2}{2*x^3+3*x^2-5}$$.

Limits Involving Absolute Value

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

Limits Involving Absolute Value Functions

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

Limits Involving Radicals

Consider the function $$f(x)=\frac{\sqrt{x+4}-2}{x}$$ defined for $$x \neq 0$$. Answer the following

Limits Involving Trigonometric Functions and the Squeeze Theorem

Examine the following trigonometric limits: (a) Evaluate $$\lim_{x\to0} \frac{\sin(4*x)}{x}$$. (b) E

Modeling Temperature Change with Continuity

A city’s temperature throughout the day is modeled by the continuous function $$T(t)=\frac{1}{2}t^2-

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

Non-Existence of a Limit due to Oscillation

Consider the function $$h(x)= \sin(\frac{1}{x})$$. Answer the following regarding its limit as x app

One-Sided Limits and Jump Discontinuity Analysis

Consider the piecewise function $$ f(x)= \begin{cases} x+2, & x < 1 \\ 3-x, & x \ge 1 \end{cases} $

Physical Applications: Temperature Continuity

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

Piecewise Functions and Continuity

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

Pond Ecosystem Nutrient Levels

In a pond ecosystem, nutrient input occurs from periodic runoff events. Each runoff adds 20 kg of nu

Reciprocal Function Behavior and Asymptotes

Examine the function $$f(x)= \frac{1}{x-1}$$.

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\

Squeeze Theorem in Oscillatory Functions

Consider the function $$f(x)= x\sin(1/x)$$ for $$x \neq 0$$ and define $$f(0)=0$$.

Trigonometric Rate Function Analysis

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

Analyzing Motion Through Derivatives

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

Applying Product and Quotient Rules

For the function $$h(x)=\frac{(3*x^2+2)*(x-4)}{x+1}$$, determine its derivative by appropriately app

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

Biochemical Reaction Rates: Derivative from Experimental Data

The concentration of a reactant in a chemical reaction is modeled by $$C(t)= 8 - 5t + t^2$$ (in M) w

Calculating Velocity and Acceleration from a Position Function

A car’s position along a straight road is given by the function $$s(t)= 0.5*t^3 - 3*t^2 + 4*t + 2$$

Car Acceleration: Secant and Tangent Slope

A car's position along a straight road is given by $$s(t)= 2t^3 - 9t^2 + 12t$$, where s is in meters

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 from First Principles: Quadratic Function

Consider the function $$f(x)= 3*x^2 + 2*x - 5$$. Use the limit definition of the derivative to compu

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

Differentiation from First Principles

Let $$h(x)=3*x^2+2*x-1$$. Use the limit definition of the derivative to analyze this function.

Differentiation of Functions with Variable Exponents

Consider the function $$Z(x)=x^{\sin(x)}$$ which represents a complex growth model. Differentiate th

Differentiation of Inverse Functions

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

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 Estimation of Tangent Slopes

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

Implicit Differentiation and Tangent Line Slope

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

Implicit Differentiation of a Circle

Given the equation of a circle $$x^2 + y^2 = 25$$,

Implicit Differentiation: Exponential-Polynomial Equation

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

Implicit Differentiation: Square Root Equation

Consider the curve defined by $$\sqrt{x} + \sqrt{y} = \sqrt{10}$$, where $$x, y \ge 0$$.

Inflection Points and Concavity Analysis

Consider the function $$f(x)=x^3 - 6*x^2+9*x+1$$. Answer the following:

Limit Definition of the Derivative for a Trigonometric Function

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

Maclaurin Polynomial for √(1+x)

A scientist approximates the function $$f(x)=\sqrt{1+x}$$ for small values of x using its Maclaurin

Optimization in Engineering Design

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

Population Growth Approximation using Taylor Series

A biologist models population growth with the exponential function $$P(t)=e^{0.05*t}$$. To estimate

Population Growth Rates

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

Rate Function Involving Logarithms

Consider the function $$h(x)=\ln(x+3)$$.

Rate of Change and Change in Market Trends

A company’s profit (in thousands of dollars) is modeled by the quadratic function $$P(x)=-2*x^2+40*x

Secant and Tangent Slope Analysis

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

Tangent Line Approximation

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

Tangent Line Estimation from Experimental Graph Data

A function $$f(x)$$ is represented by the following graph of experimental data approximating $$f(x)=

Tangent Line to a Curve

Consider the function $$f(x)=\sqrt{x+4}$$ modeling a physical quantity. Analyze the behavior at $$x=

Tangent Line to a Logarithmic Function

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

Temperature Function Analysis

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

Velocity Function from a Cubic Position Function

An object’s position along a line is modeled by $$s(t) = t^3 - 6*t^2 + 9*t$$, where s is in meters a

Vibration Analysis: Rate of Change in Oscillatory Motion

The displacement of a vibrating string is given by $$f(t)= 3\sin(4t) - 2\cos(4t)$$, where t is in se

Analyzing a Composite Function with Nested Radicals

Consider the function $$h(x)=\sqrt{1+\sqrt{2+3x}}$$. Answer the following parts:

Chain Rule and Higher-Order Derivatives

Given the function $$f(x)= \ln(\sqrt{1 + e^{3*x}})$$, answer the following parts:

Chain Rule for Inverse Trigonometric Functions in Optics

In an optics experiment, the angle of incidence $$\theta(t)$$ (in radians) is modeled by $$\theta(t)

Coffee Cooling Dynamics using Inverse Function Differentiation

A cup of coffee cools according to the model $$T=100-a\,\ln(t+1)$$, where $$T$$ is the temperature i

Complex Composite and Implicit Function Analysis

Consider the equation $$e^{x*y}+\ln(x+y)=2$$, where y is defined implicitly as a function of x. Answ

Composite Exponential Logarithmic Function Analysis

Consider the function $$f(x)=\ln(2*e^{3*x}+5)$$ which models a logarithmic transformation of an expo

Composite Function Modeling with Chain Rule

A chemical reaction rate is modeled by the composite function $$R(x)=f(g(x))$$ where $$f(u)=\sin(u)$

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 Population Growth Function

A population model is given by $$P(t)= e^{3\sqrt{t+1}}$$, where $$t$$ is measured in years. Analyze

Differentiation of an Inverse Trigonometric Composite Function

Let $$f(x)= \arctan(e^{2*x})$$. Answer the following parts:

Implicit Differentiation in Exponential Equation

Consider the equation $$e^{x*y}+x^2-y^3=0$$ that relates x and y. Answer the following parts:

Implicit Differentiation with Exponential and Trigonometric Components

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

Implicit Differentiation with Product and Chain Rule in a Thermal Expansion Model

A material's length $$L$$ (in meters) under thermal expansion satisfies the equation $$L - \sin(L *

Inverse Analysis of a Composite Exponential-Trigonometric Function

Let $$f(x)=e^x+\cos(x)$$. Analyze the behavior of its inverse function under appropriate domain rest

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 Derivative Calculation

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

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 Trigonometric Differentiation

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

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$

Related Rates: Ladder Sliding Down a Wall

A ladder of length $$10\, m$$ leans against a wall such that its position is governed by $$x^2 + y^2

Second Derivative of an Implicit Function

The curve is defined implicitly by the equation $$x^2*y+y^3=4$$. Answer the following parts:

Trigonometric Composite Inverse Function Analysis

Consider the function $$f(x)=\sin(x)+x$$ defined on the interval $$\left[-\frac{\pi}{2}, \frac{\pi}{

Analyzing Motion on a Curved Path

A particle moves along a path defined by $$x(t)=\cos(t)$$ and $$y(t)=\sin(t)$$ for $$t \in [0,2\pi]$

Arc Length Calculation

Consider the curve $$y = \sqrt{x}$$ for $$x \in [1, 4]$$. Determine the arc length of the curve.

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

Boat Crossing a River: Relative Motion

A boat must cross a 100 m wide river. The boat's speed relative to the water is 5 m/s (directly acro

Business Profit Optimization

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

City Population Migration

A city's population is influenced by immigration at a rate of $$I(t)=100e^{-0.2t}$$ (people per year

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

Cooling Coffee Temperature

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

Draining Hemispherical Tank

A hemispherical tank of radius $$5$$ m is draining. The volume of water in the tank is given by $$V

Graphical Analysis of Derivatives

A function $$f(x)$$ is plotted on the graph provided below. Using this graph, answer the following:

Horizontal Tangents on Cubic Curve

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

Integration of Flow Rates Using the Trapezoidal Rule

A tank is being filled with water, and the flow rate Q (in L/min) is recorded at several time interv

Inverse of a Trigonometric Function

Consider the function $$f(x)=\sin(x)$$ defined on the restricted domain $$\left[-\frac{\pi}{2},\frac

Inversion in a Light Intensity Decay Model

A laboratory experiment records the decay of light intensity over time, modeled by $$f(t)=80*e^{-0.2

L'Hôpital’s Rule in Chemical Reaction Rates

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

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 Approximation

Let $$f(x)=x^4$$. Linearization can be used to approximate small changes in a function's values. Ans

Minimizing Travel Time in Mixed Terrain

A hiker travels from point A to point B. On a flat plain the hiker walks at 5 km/h, but on an uphill

Motion along a Curved Path

A particle moves along the curve defined by $$y=\sqrt{x}$$. At the moment when $$x=9$$ and the x-coo

Motion on a Straight Line with a Logarithmic Position Function

A particle moves along a straight line with its position given by $$s(t)=\ln(t+2)+t^2$$ (in meters),

Polar Curve: Slope of the Tangent Line

Consider the polar curve defined by $$r(\theta)=10e^{-0.1*\theta}$$.

Pollution Accumulation in a Lake

A lake is subject to pollution with pollutants entering at a rate of $$I(t)=3e^{0.1t}$$ (kg per day)

Production Cost Analysis

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

Series Approximation for a Displacement Function

A displacement function is modeled by $$s(t)= \sum_{n=1}^{\infty} \frac{(-1)^{n+1} t^n}{n}$$, which

Sliding Ladder

A 10 m long ladder rests against a vertical wall. Let $$x$$ be the distance from the foot of the lad

Solids of Revolution: Washer vs Shell Methods

Consider the region enclosed by $$y = \sin(x)$$ and $$y = \cos(x)$$ for $$0 \le x \le \frac{\pi}{4}$

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

Water Tank Flow Analysis

A water tank receives water from an inlet at a rate given by $$I(t)=4+\cos(t)$$ (liters per minute)

Analyzing Inverses in a Rate of Change Scenario

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

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

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

Area Between Curves and Rates of Change

An irrigation canal has a cross-sectional shape described by \( y=4-x^2 \) for \( |x| \le 2 \). The

Bacterial Culture with Periodic Removal

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

Bank Account Growth and Instantaneous Rate

A bank account balance is modeled by the function $$B(t) = 1000*e^{0.05*t}$$, where t (in years) rep

Chemical Reaction Rate

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

Combining Series and Integration to Analyze a Population Model

A population's growth rate is approximated by the series $$P'(t)=\sum_{n=0}^\infty \frac{t^n}{(n+1)!

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

Concavity in an Economic Model

Consider the function $$f(x)= x^3 - 3*x^2 + 2$$, which represents a simplified economic trend over t

Discounted Cash Flow Analysis

A project is expected to return cash flows that decrease by 10% each year from an initial cash flow

Epidemic Infection Model

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

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.

Mean Value Theorem Application for Mixed Log-Exponential Function

Let $$h(x)= \ln(x) + e^{-x}$$ be defined on the interval [1,3]. Analyze the function using the Mean

Parameter Estimation in a Log-Linear Model

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

Projectile Motion Analysis

A projectile is launched vertically with its height given by $$s(t) = -16*t^2 + 64*t + 80$$ (in feet

Rate of Change and Inverse Functions

Let $$f(x)=x^3 + 3*x + 1$$, which is one-to-one. Investigate the rate of change of \(f(x)\) and its

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 $$t$$

Roller Coaster Height Analysis

A roller coaster's height (in meters) as a function of time (in seconds) is given by $$h(t) = -0.5*t

Salt Tank Mixing Problem

In a mixing tank, salt is added at a constant rate of $$A(t)=10$$ grams/min while the salt solution

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:

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

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}

Travel Distance from Speed Data

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

Wastewater Treatment Reservoir

At a wastewater treatment reservoir, wastewater enters at a rate of $$W_{in}(t)=12+2*t$$ m³/min and

Water Tank Rate of Change

The volume of water in a tank is modeled by $$V(t)= t^3 - 6*t^2 + 9*t$$ (in cubic meters), where $$t

Accumulated Displacement from a Velocity Function

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

Applying the Fundamental Theorem of Calculus

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

Area and Volume for an Exponential Function Region

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

Average Value of a Function on an Interval

Let $$f(x)=\ln(x)$$ be defined on the interval $$[1,e]$$. Determine the average value of $$f(x)$$ on

Bacteria Growth with Nutrient Supply

A bacterial culture in a laboratory is provided with nutrients at a rate of $$N(t)=6*\ln(t+1)$$ mg/m

Bacterial Population Growth from Accumulated Rate

A bacteria population grows according to the rate function $$r(t)=k*t*e^{-t}$$ (in cells/hour) for \

Continuous Antiderivative for a Piecewise Function

A function $$f(x)$$ is defined piecewise as follows: for $$x<2$$, $$f(x)=3*x^2$$, and for $$x\ge2$$,

Convergence of an Improper Integral Representing Accumulation

Consider the improper integral $$\int_{1}^{\infty} \frac{1}{t^2}dt$$, which can be interpreted as th

Cost Accumulation via Integration

A manufacturing process has a marginal cost function given by $$MC(x)= 4 + 3*x$$ dollars per item, w

Definite Integration of a Polynomial Function

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

Differentiation and Integration of a Power Series

Consider the function given by the power series $$f(x)=\sum_{n=0}^\infty \frac{x^n}{2^n}$$.

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 \

Economics: Accumulated Earnings

A company’s instantaneous revenue rate (in dollars per day) is modeled by the function $$R(t)=1000\s

Filling a Tank: Antiderivative with Initial Value

Water is entering a tank at a rate given by $$r(t)= \frac{2}{t+1}$$ liters per minute. The initial

Integrated Growth in Economic Modeling

A company experiences revenue growth at an instantaneous rate given by $$r(t)=0.5*t+2$$ (in millions

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

Particle Motion and the Fundamental Theorem of Calculus

A particle moves along a straight line with its velocity given by $$v(t)=3*t^2-12*t+9$$ (in m/s) for

Particle Motion with Variable Acceleration

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

Population Growth: Rate to Accumulation

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

Scooter Motion with Variable Acceleration

A scooter's acceleration is given by $$a(t)= 2*t - 1$$ (m/s²) for $$t \in [0,5]$$, with an initial v

Temperature Change Analysis

A series of temperature readings (in °C) are recorded over the day as shown in the table. Analyze th

Work Done by a Variable Force

A variable force acting along a track is given by $$F(x)=6*\sqrt{x}$$ (in Newtons). Compute the work

Autocatalytic Reaction Dynamics

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

Complex Related Rates Problem Involving a Moving Ladder

A 10-meter ladder leans against a vertical wall. The bottom of the ladder slides away from the wall

Cooling of a Metal Rod

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

Differential Equations in Compound Interest

An investment account grows with continuously compounded interest following $$\frac{dA}{dt}=rA$$, wh

Direction Fields and Isoclines

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

Disease Spread Model

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

Economic Investment Growth Model with Regular Deposits

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

Forced Oscillation in a Damped System

Consider the differential equation $$\frac{dx}{dt}=-0.2*x+\sin(t)$$ with initial condition $$x(0)=1$

FRQ 4: Newton's Law of Cooling

A cup of coffee cools according to Newton's Law of Cooling, where the temperature $$T(t)$$ satisfies

FRQ 6: Radioactive Decay

A radioactive substance decays according to the differential equation $$\frac{dN}{dt}=-\lambda * N$$

FRQ 9: Epidemiological Model Differential Equation

An epidemic evolves according to the differential equation $$\frac{dI}{dt}=r*I*(M-I)$$, where $$I(t)

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

FRQ 17: Slope Field Analysis and Particular Solution

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

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 Populations

A population of organisms is modeled by the logistic differential equation $$\frac{dP}{dt} = rP \lef

Logistic Model with Harvesting

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

Logistic Population Model

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

Mixing Problem in a Tank

A tank initially contains a certain amount of salt dissolved in 100 L of water. Brine with a known s

Mixing Problem in a Tank

A tank initially contains 100 L of water with 5 kg of dissolved salt. Brine containing 0.1 kg of sal

Mixing Problem with Constant Rates

A tank contains $$200\,L$$ of a well-mixed saline solution with $$5\,kg$$ of salt initially. Brine w

Modeling Medication Concentration in the Bloodstream

A patient receives an intravenous drug at a constant rate $$R$$ (mg/min) and the drug is eliminated

Particle Motion in the Plane with Non-constant Acceleration

A particle moves in the $$xy$$-plane with an acceleration vector given by $$a(t)=\langle 2, e^t \ran

Particle Motion with Damping

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

Particle Motion with Variable Acceleration

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

Particle Motion with Variable Acceleration

A particle moving along a straight line has an acceleration given by $$a(t) = 6 - 4*t$$ (in m/s²). A

Reservoir Contaminant Dilution

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

Separable DE: Basic SIPPY Problem

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

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 and Solution Curve Sketching

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

Solving a Nonlinear Differential Equation by Separation

Given the differential equation $$\frac{dy}{dx} = \frac{1+y^2}{1+x^2}$$ with the initial condition $

Analysis of Particle Motion in the Plane

A particle’s position is given by the vector function $$\mathbf{r}(t)=\langle e^{-t},\,\sin(t)\rangl

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: 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 Nonlinear Curves

Consider the functions $$f(x)=e^{-x}$$ and $$g(x)=\frac{x}{3}$$. Determine the area between these tw

Area Between Two Curves in a Water Channel

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

Area Between Two Curves: Parabola and Line

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

Average Temperature Over a Day

The temperature in a city over a 24-hour period is modeled by $$T(t)=10+5\sin\left(\frac{\pi}{12}*t\

Average Value of a Piecewise Function

Consider the function $$g(x)$$ defined piecewise on the interval $$[0,6]$$ by $$g(x)=\begin{cases} x

Average Value of a Velocity Function

The velocity of a car is modeled by $$v(t)=3*t^2-12*t+9$$ (m/s) for $$t\in[0,5]$$ seconds. Answer th

Average Value of a Velocity Function

A particle moves along a line with its velocity given by $$v(t)= 2*\cos(t) + \sin(t)$$ for $$t \in [

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 Function from Marginal Cost

A manufacturing process has a marginal cost function given by $$MC(q)=3*\sqrt{q}$$, where $$q$$ (in

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^

Draining a Conical Tank Related Rates

Water is draining from a conical tank that has a height of $$8$$ meters and a top radius of $$3$$ me

Integral Approximation Using Taylor Series

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

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

Position from Velocity Function

A particle moves along a horizontal line with a velocity function given by $$v(t)=4*\cos(t) - 1$$ fo

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

Temperature Modeling: Applying the Average Value Theorem

The temperature of a chemical solution in a tank is modeled by $$T(t)=20+5\cos(0.5*t)$$ (°C) for $$t

Volume by Cross-Sectional Area (Square Cross-Sections)

A solid has a base in the xy-plane bounded by the curve $$y=\sqrt{x}$$ and the x-axis for $$x\in[0,4

Volume by Shell Method: Rotated Parabolic Region

Consider the region in the first quadrant bounded by the curve $$y=x^2$$ and the horizontal line $$y

Volume of a Solid with Equilateral Triangle Cross Sections

Consider the region bounded by $$y= \sqrt{x}$$ and $$y=0$$ for $$x \in [0,1]$$. A solid is formed by

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

Work to Pump Water from a Tank

A cylindrical tank of radius 2 ft and height 10 ft is partially filled with water to a depth of 8 ft

Analyzing Oscillatory Motion in Parametric Form

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

Arc Length of a Cycloid

A cycloid is generated by a circle of radius \(r=1\) rolling along a straight line. The cycloid is g

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 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 \

Arc Length of a Vector-Valued Curve

A vector-valued function is given by $$\mathbf{r}(t)=\langle e^t,\, \sin(t),\, \cos(t) \rangle$$ for

Area Between Polar Curves

Consider the polar curves defined by $$r_1= 4$$ and $$r_2= 2+2\cos(\theta)$$. Find the area of the r

Area between Two Polar Curves

Given two polar curves: $$r_1 = 1+\cos(\theta)$$ and $$r_2 = 2\cos(\theta)$$, consider the region wh

Area Between Two Polar Curves

Consider the polar curves $$ r_1=2*\sin(\theta) $$ and $$ r_2=\sin(\theta) $$. Determine the area of

Comparing Representations: Parametric and Polar

A curve is represented by the parametric equations $$x(t)=3\cos(t)-\sin(t)$$ and $$y(t)=3\sin(t)+\co

Conversion and Analysis of Polar and Rectangular Forms

Consider the polar equation $$r=3e^{\theta}$$. Answer the following:

Conversion to Cartesian and Analysis of a Parametric Curve

Consider the parametric equations $$x(t)= 2*t + 1$$ and $$y(t)= (t - 1)^2$$ for $$-2 \le t \le 3$$.

Converting Polar to Cartesian and Computing Slope

The polar curve is given by the equation $$r=4\cos(\theta)$$. Answer the following:

Finding the Slope of a Tangent to a Parametric Curve

Consider the parametric equations $$x(t)=e^t$$ and $$y(t)=e^{-t}$$, where $$t \in \mathbb{R}$$.

Integrating a Vector-Valued Function

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

Kinematics in Polar Coordinates

An object moves so that its position in polar coordinates is given by $$r(t)= 4 - t$$ and $$\theta(t

Motion Along a Parametric Curve

Consider the parametric equations $$x(t)= 2*t - 3$$ and $$y(t)= t^2$$, where $$t$$ represents time i

Motion Analysis via a Vector-Valued Function

An object's position is described by the vector function $$\mathbf{r}(t)= \langle e^{-t}, \; \ln(1+t

Multi-Step Problem Involving Polar Integration and Conversion

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

Parametric Equations and Tangent Slopes

Consider the parametric equations $$x(t)= t^3 - 3*t$$ and $$y(t)= t^2$$, for $$t \in [-2, 2]$$. Anal

Parametric Intersection and Enclosed Area

Consider the curves C₁ given by $$x=\cos(t)$$, $$y=\sin(t)$$ for $$0 \le t \le 2\pi$$, and C₂ given

Parametric Motion and Change of Direction

A particle moves along a path defined by the parametric equations $$x(t)=t^3-3t$$ and $$y(t)=2t^2$$

Particle Motion in Circular Motion

A particle moves along a circular path given by the parametric equations $$x(t)= 5\cos(t)$$ and $$y(

Particle Motion in the Plane

Consider a particle whose motion in the plane is defined by the parametric equations $$x(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

Polar Equations and Slope Analysis

Given the polar curve $$r = 4\sin(\theta)$$, analyze its Cartesian form and tangent properties.

Polar Spiral: Area and Arc Length

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

Sensitivity Analysis and Linear Approximation using Implicit Differentiation

The variables $$x$$ and $$y$$ satisfy the equation $$xy+\ln(y)=5$$.

Tangent Line to a Parametric Curve

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

Vector-Valued Function Integration

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

Vector-Valued Functions in Motion

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