AP Calculus BC FRQ Room

Algorithm Time Complexity

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

Analyzing a Function with a Removable Discontinuity

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

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

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 in a Piecewise Function with Polynomial and Trigonometric Components

Consider the function $$f(x)= \begin{cases} x^2-1 & \text{if } x < \pi \\ \sin(x) & \text{if } x \ge

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

Continuity in Piecewise Functions with Parameters

A function is defined piecewise by $$f(x)=\begin{cases}a*x^2+3,& x<1,\\ b*x+1,& x\ge 1.\end{cases}$$

Continuity of a Trigonometric Function Near Zero

Consider the function defined by $$ f(x)= \begin{cases} \frac{\sin(5*x)}{x}, & x \neq 0 \\ L, & x =

Environmental Pollution Modeling

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

Epsilon-Delta Style (Conceptual) Analysis

Consider the function $$f(x)=\begin{cases} 3*x+2, & x\neq1, \\ 6, & x=1. \end{cases}$$ Answer the

Evaluating a Complex Limit for Continuous Extension

Consider the function $$ f(x)= \begin{cases} \frac{\ln(1+x+e^x) - (x+e^x-1)}{x^2}, & x \neq 0 \\ C,

Evaluating a Limit Involving a Radical and Trigonometric Component

Consider the function $$f(x)= \frac{\sqrt{1+x}-\sqrt{1-x}}{x}$$. Answer the following:

Evaluating Limits Involving Exponential and Rational Functions

Consider the limits involving exponential and polynomial functions. (a) Evaluate $$\lim_{x\to\infty}

Intermediate Value Theorem in Water Tank Levels

The water volume \(V(t)\) in a tank is a continuous function on the interval \([0,10]\) minutes. It

Investigating Limits and Areas Under Curves

Consider the region bounded by the curve $$y=\frac{1}{x}$$, the vertical line $$x=1$$, and the verti

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

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

Limit Behavior in a Container Optimization Problem

A manufacturer designs a closed cylindrical container with a fixed volume $$V$$ (in cubic units). Th

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

Parameter Determination for Continuity

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

Physical Applications: Temperature Continuity

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

Population Growth and Limits

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

Rational Function and Removable Discontinuity

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

Real-World Temperature Sensor Analysis

A temperature sensor is modeled by the function $$T(t)=\frac{t^2-9}{t-3}$$ for t ≠ 3 (with t in minu

Squeeze Theorem Application

Consider the function $$f(x)=\frac{\sin(3*x)}{x}$$ defined for x ≠ 0.

Squeeze Theorem with Oscillatory Behavior

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

Using the Squeeze Theorem for Trigonometric Limits

Let the function $$f(x)=x^2*\sin(1/x)$$ for x \neq 0 and define f(0)=0. Use the Squeeze Theorem to a

Vertical Asymptote Analysis in a Rational Function

Consider the function $$g(x)=\frac{x+1}{x-3}$$, which is undefined at $$x=3$$. Answer the following:

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

Analysis of a Piecewise Function

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

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

Chemical Reaction Rate Control

During a chemical reaction in a reactor, reactants enter at a rate of $$R_{in}(t)=\frac{10*t}{t+2}$$

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

Composite Function Behavior

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

Differentiation of an Exponential Function

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

Graph Interpretation: Average vs Instantaneous Rates

A function is represented in the table below. Analyze the difference between average and instantaneo

Implicit Differentiation: Conic with Mixed Terms

Consider the curve defined by $$x*y + y^2 = 6$$.

Implicit Differentiation: Square Root Equation

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

Interpreting Graphical Slope Data

A laboratory experiment measures the velocity (in m/s) of a moving object over time. A graph of the

Logarithmic Differentiation: Equating Powers

Consider the equation $$y^x = x^y$$ that relates $$x$$ and $$y$$ implicitly.

Particle Motion on a Straight Line: Average and Instantaneous Rates

A particle moving along a straight line has its position given by $$s(t)=t^3 - 6*t^2 + 9*t + 4$$ for

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

Position Recovery from a Velocity Function

A particle moving along a straight line has a velocity function given by $$v(t)=6-3*t$$ (in m/s) for

Profit Rate Analysis in Economics

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

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 and Normal Lines to a Curve

Given the function $$p(x)=\ln(x)$$ defined for $$x > 0$$, analyze its rate of change at a specific p

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

Tangent Line to a Curve

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

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

Temperature Function Analysis

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

Analyzing an Implicit Function with Mixed Variables

Consider the curve defined by $$x^3 + x*y + y^3 = 3$$. Analyze the derivative of the curve at a give

Chain Rule and Inverse Trigonometric Differentiation

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

Chain Rule Application: Differentiating a Nested Trigonometric Function

Consider the function $$f(x) = \sin(\cos(2*x))$$. Analyze its derivative using the chain rule.

Chain Rule for a Multi-layered Composite Function

Let $$f(x)= \sqrt{\ln((3*x+2)^5)}$$. Answer the following:

Chain Rule in Economic Utility Functions

A consumer's utility function is given by $$U(x,y)=\sqrt{x+y^2}$$, where x and y represent quantitie

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 Temperature Change in a Chemical Reaction

A chemical reaction in a laboratory is modeled by the composite temperature function $$R(t)= f(g(t))

Differentiation Involving Absolute Values and Composite Functions

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

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

Differentiation of an Inverse Trigonometric Composite Function

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

Differentiation of an Inverse Trigonometric Form

Consider the function $$f(x)=\arcsin\left(\frac{2*x}{1+x^2}\right)$$. Answer the following parts.

Differentiation of Composite Inverse Trigonometric Function involving a Rational Function

Differentiate the function $$f(x)= \arccos\left(\frac{3*x}{1+x^2}\right)$$ with respect to $$x$$ and

Differentiation of the Inverse Function in a Mechanics Experiment

An object's displacement is described by a one-to-one differentiable function \(s(t)\). It is given

Implicit Differentiation in a Nonlinear Trigonometric Equation

Consider the equation $$ x^2+\sin(x*y)= y^2 $$.

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 Involving Product and Logarithm

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

Implicit Differentiation of a Circle

Consider the circle described by $$x^2 + y^2 = 25$$. A table of select points on the circle is given

Implicit Differentiation on a Trigonometric Curve

Consider the curve defined implicitly by $$\sin(x+y) = x^2$$.

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 *

Implicit Differentiation: Second Derivative of Exponential-Trigonometric Equation

Consider the equation $$e^{x*y} + \sin(y) - x^2 = 0$$ where $$y$$ is defined implicitly as a functio

Inverse Analysis of an Exponential-Linear Function

Consider the function $$f(x)=e^{x}+x$$ defined for all real numbers. Analyze its inverse function.

Inverse Analysis via Implicit Differentiation for a Transcendental Equation

Consider the equation $$e^{x*y}+x-y=0$$ defining y implicitly as a function of x near a point where

Inverse Derivative via Chain Rule for a Logarithmic-Exponential Function

Let $$f(x)=\ln(1+e^x)$$. Analyze its inverse derivative.

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 Derivatives in a Sensor Model

An instrument outputs a reading defined by $$f(x)= x^3 + 2$$, where $$x$$ represents the voltage inp

Inverse Function Differentiation in a Logarithmic Context

Let $$f(x)= \ln(x+2) - x$$, and let $$g$$ be its inverse function. Answer the following:

Inverse of a Composite Function

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

Logarithmic and Composite Differentiation

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

Logarithmic and Exponential Composite Function with Transformation

Let $$g(x)=\ln((3*x+1)^2)-e^{x}$$. Answer the following questions.

Multilayer Composite Differentiation in a Climate Model

A climate model gives the temperature $$T(t)$$ (in °C) as a function of time $$t$$ (in years) by $$T

Nested Composite Function Differentiation

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

Optimization in Manufacturing Material

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

Parametric Curve Analysis with Composite Functions

A curve is defined by the parametric equations $$x(t)=\ln(1+t^2)$$ and $$y(t)=\sqrt{t+4}$$, where t

Parametric Equations and the Chain Rule

A particle moves in the plane according to the parametric equations $$x(t)= e^{2*t}$$ and $$y(t)= \l

Polar and Composite Differentiation: Arc Slope for a Polar Curve

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

Vector Function Trajectory Analysis

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

Analysis of a Piecewise Function with Discontinuities

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

Analyzing Experimental Temperature Data

A laboratory experiment records the temperature of a chemical reaction over time. The temperature (i

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

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

Cost Function Analysis in Production

A company's cost for producing $$x$$ items is given by $$C(x)=0.5*x^3-4*x^2+10*x+500$$ dollars.

Cubic Curve Linearization

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

Differentiation of a Product Involving Exponentials and Logarithms

Consider the function $$f(t)=e^{-t}\ln(t+2)$$, defined for t > -2. Answer the following:

Draining Hemispherical Tank

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

Expanding Circular Ripple

A stone is thrown in a pond, creating circular ripples. The area of the circle defined by the ripple

Expanding Pool Rates

The area $$A$$ of a circular swimming pool is given by $$A=\pi*r^2$$. The pool is being filled so th

Horizontal Tangents on Cubic Curve

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

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 Application

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

L’Hôpital’s Rule for an Exponential Ratio

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

Ladder Sliding Problem

A 10-meter ladder is leaning against a vertical wall. The bottom of the ladder is pulled away from t

Limit Evaluation Using L'Hôpital's Rule

Evaluate the limit $$\lim_{x \to \infty} \frac{5*x^3 - 4*x^2 + 1}{7*x^3 + 2*x - 6}$$. Answer the fol

Linear Account Growth in Finance

The amount in a savings account during a promotional period is given by the linear function $$A(t)=1

Marginal Analysis in Economics

The cost function for producing $$x$$ items is given by $$C(x)= 0.1*x^3 - 2*x^2 + 20*x + 100$$ dolla

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),

Parametric Motion in the Plane

A particle moves in the plane according to the parametric equations $$x(t)=t^2-2*t$$ and $$y(t)=3*t-

Pool Water Volume Change

The volume of water in a pool is described by the function $$V(t)=8*t^2-32*t+4$$, where $$V$$ is mea

Population Growth Rate

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

Population Growth Rate Analysis

A population grows exponentially according to $$P(t)=1200e^{0.15t}$$, where t is measured in months.

Radical Function Inversion

Let $$f(x)=\sqrt{2*x+5}$$ represent a measurement function. Analyze its inverse.

Seasonal Reservoir Dynamics

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

Series Solution of a Drug Concentration Model

The drug concentration in the bloodstream is modeled by $$C(t)= \sum_{n=0}^{\infty} \frac{(-t)^n}{n!

Spherical Balloon Inflation

A spherical balloon is being inflated such that its volume increases at a constant rate of $$\frac{d

Tangent Lines in Motion Analysis

A particle's position is given by $$s(t)=t^3 - 6t^2 + 9t + 5$$. Analyze the tangent lines to the gra

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)

Analysis of a Quartic Function as a Perfect Power

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

Bacterial Culture with Periodic Removal

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

Bouncing Ball with Energy Loss

A ball is dropped from a height of 100 meters. Each time it bounces, it reaches 60% of the height fr

Chemical Reaction Rate

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

Chemical Reactor Rate Analysis

In a chemical reactor, a reactant is added at a rate given by $$A(t)=8*\sqrt{t}$$ grams/min and is s

Concavity and Inflection Points Analysis

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

Concavity in an Economic Model

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

Derivatives and Inverses

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

Echoes in an Auditorium

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

Exponential Decay in Velocity

A particle’s velocity is modeled by the function $$v(t)=10e^{-0.5*t}-3$$ (in m/s) for $$t\ge0$$.

Inverse Function Derivative for a Piecewise Function

Suppose f is defined piecewise by $$f(x)= x^2$$ for $$x \ge 0$$ and $$f(x)= -x$$ for $$x < 0$$. Cons

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

Logarithmic Function Derivative Analysis

Consider the function $$f(x)= \ln(x^2+1)$$. Answer the following questions about its behavior.

Logistic Growth Model Analysis

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

Mean Value Theorem with a Trigonometric Function

Let $$f(x)=\sin(x)$$ be defined on the interval $$[0,\pi]$$. Answer the following parts:

Optimization in Particle Motion

A particle moves along a line with position given by $$ s(t)=t^3-6t^2+9t+4, \quad t\ge0.$$ Answer t

Piecewise Function with Absolute Value

Consider the function defined by $$ g(x)=\begin{cases} |x-1| & \text{if } x<2, \\ 3x-5 & \text{if }

Region Area and Volume: Polynomial and Linear Function

A region in the x-y plane is bounded by the curves $$f(x)=x^2$$ and $$g(x)=2 - x$$. Answer the follo

Related Rates: Draining Conical Tank

Water is draining from a conical tank with a height of \(10\,m\) and a top diameter of \(8\,m\). Wat

Revenue Optimization in Business

A company’s price-demand function is given by $$P(x)= 50 - 0.5*x$$, where $$x$$ is the number of uni

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 Differentiation in Thermodynamics

In a thermodynamic process, the temperature $$T(x)=\sum_{n=0}^\infty \frac{(-2)^n}{n+1} * (x-5)^n$$

Stress Analysis in Engineering Structures

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

Taylor Series for an Integral Function: $$F(x)=\int_0^x \sin(t^2)\,dt$$

Because the antiderivative of $$\sin(t^2)$$ cannot be expressed in closed form, use its power series

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

Vector Analysis of Particle Motion

A particle moves in the plane with its position given by the vector function $$\mathbf{r}(t) = \lang

Wireless Signal Attenuation

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

Advanced Inflow/Outflow Dynamics

A reservoir receives water from a river at a rate given by $$f(t)=50*(1+0.1*t)$$ cubic meters per ho

Advanced U-Substitution with a Quadratic Expression

Evaluate the indefinite integral $$\int \frac{2*x}{\sqrt{x^{2}+1}}\,dx$$ using u-substitution.

Antiderivatives and the Fundamental Theorem

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

Area Estimation Using Trapezoidal Sums from Tabulated Data

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

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 \

Displacement vs. Total Distance Traveled

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

Flow of Traffic on a Bridge

Cars cross a bridge at a rate modeled by $$R(t)=300+50*\cos\left(\frac{\pi*t}{6}\right)$$ vehicles p

Integration by Parts: Logarithmic Function

Evaluate the definite integral $$\int_{1}^{3} x*\ln(x) dx$$ using integration by parts. Answer the f

Integration via Partial Fractions

Evaluate the integral $$\int_{0}^{1} \frac{2*x+3}{(x+1)(x+2)} dx$$. Answer the following:

Interpreting the Constant of Integration in Cooling

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

Logarithmic Functions in Ecosystem Models

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

Midpoint Riemann Sum for $$f(x)=\frac{1}{1+x^2}$$

Consider the function $$f(x)=\frac{1}{1+x^2}$$ on the interval $$[-1,1]$$. Use the midpoint Riemann

Non-Uniform Subinterval Riemann Sum

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

Optimizing the Inflow Rate Strategy

A municipality is redesigning its water distribution system. The water inflow rate is modeled by $$F

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 in the Plane

A particle moves in the plane with its acceleration components given by $$a_x(t)=4-2*t$$ and $$a_y(t

Population Growth from Birth Rate

In a small town, the birth rate is modeled by $$B(t)= \frac{100}{1+t^2}$$ people per year, where $$t

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

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

Trapezoidal Approximation of a Definite Integral from Tabular Data

The table below shows the height H(t) (in meters) of a liquid in a tank at specific times. Use a tra

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

Water Pollution Accumulation

In a river, a pollutant is introduced at a rate $$P_{in}(t)=8-0.5*t$$ (kg/min) and is simultaneously

Water Tank Inflow and Outflow

A water tank begins operation at t = 0 with an initial volume of 0 liters. Water flows in through an

Bacteria Growth with Antibiotic Treatment

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

Capacitor Discharge in an RC Circuit

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

Cooling Model Using Newton's Law

Newton's law of cooling states that the temperature T of an object changes at a rate proportional to

Estimating Instantaneous Rate from a Table

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

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

Free-Fall with Air Resistance

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

FRQ 9: Epidemiological Model Differential Equation

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

Growth and Decay with External Forcing Term

Consider the non-homogeneous differential equation $$\frac{dy}{dt} = k*y + f(t)$$ where $$f(t)$$ rep

Implicit Differentiation and Homogeneous Equation

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

Implicit Solution of a Separable Differential Equation

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

Investment Account Growth with Fees

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

Logistic Growth Model in Population Dynamics

A population is modeled by the logistic differential equation $$\frac{dy}{dt} = 0.5*y\left(1-\frac{y

Logistic Population Growth Model

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

Mixing Problem with Time-Dependent Inflow

A tank initially contains $$100$$ L of salt water with a salt concentration of $$0.5$$ kg/L. Pure wa

Modeling Ambient Temperature Change

The ambient temperature $$T(t)$$ of a city changes according to the differential equation $$\frac{dT

Modeling the Spread of a Disease Using Differential Equations

Suppose the spread of a disease in a population is modeled by the differential equation $$\frac{dI}{

Newton's Law of Cooling

A hot liquid cools in a room maintained at a constant temperature $$T_{room}$$. The temperature $$T(

Non-linear Differential Equation using Separation of Variables

Consider the differential equation $$\frac{dy}{dx}= \frac{x*y}{x^2+1}$$. Answer the following questi

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,

Pollutant Concentration in a Lake

A lake receives a pollutant at a constant rate of $$5$$ kg/day and the pollutant is removed at a rat

Population Dynamics with Harvesting

Consider a population model that includes constant harvesting, given by the differential equation $$

Radioactive Decay with Constant Source

A radioactive material is produced at a constant rate S while simultaneously decaying. This process

Separable Differential Equation with Parameter Identification

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

Spring-Mass System with Damping

A spring-mass system with damping is modeled by the differential equation $$m\frac{d^2y}{dt^2}+ c\fr

Water Pollution with Seasonal Variation

A river receives a pollutant with a time-varying influx modeled by $$I(t)=20+5\cos(0.5*t)$$ kg/day,

Accumulated Rainfall

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

Arc Length of a Logarithmic Curve

Determine the arc length of the curve $$f(x)= \ln(x)$$ on the interval $$[1,e]$$.

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 a Parabola and a Line

Consider the curves given by $$y=5*x-x^2$$ and $$y=x$$. These curves intersect at certain $$x$$-valu

Area Between Curves from Experimental Data

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

Area Between Curves: Park Design

A park layout is bounded by two curves: $$f(x)=10-x^2$$ and $$g(x)=2*x+2$$. Answer the following par

Area Between Curves: Supply and Demand Analysis

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

Area Between Economic Curves

In an economic model, two cost functions are given by $$C_1(x)=100-2*x$$ and $$C_2(x)=60-x$$, where

Area Between Exponential Curves

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

Area Under a Curve with a Discontinuity

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

Average Population in a Logistic Model

A population is modeled by a logistic function $$P(t)=\frac{500}{1+2e^{-0.3*t}}$$, where $$t$$ is me

Center of Mass of a Nonuniform Rod

A thin rod extends from $$x=0$$ to $$x=3$$ and has a linear density given by $$\delta(x)=1+x$$ (in k

Center of Mass of a Rod with Variable Density

A thin rod of length 10 meters lies along the x-axis from $$x=0$$ to $$x=10$$. Its density is given

Complex Integral Evaluation with Exponential Function

Evaluate the integral $$I=\int_1^e \frac{2*\ln(t)}{t}dt$$ by applying a suitable substitution.

Cost Function from Marginal Cost

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

Determining the Arc Length of a Curve

Consider the curve defined by $$y=\frac{1}{2}*e^{x/2}$$ over the interval $$[0,2]$$.

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[

Fluid Pressure on a Submerged Plate

A vertical rectangular plate with a width of 3 ft and a height of 10 ft is submerged in water so tha

Implicit Differentiation with Exponential Terms

Consider the equation $$e^{x * y} + x^2 * y = y^3$$. Answer the following:

Population Change via Rate Integration

A small town’s population changes at a rate given by $$P'(t)=100*e^{-0.3*t}$$ (persons per year) wit

Projectile Maximum Height

A ball is thrown upward with an acceleration of $$a(t)=-9.8$$ m/s², an initial velocity of $$v(0)=20

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

Series and Integration Combined: Error Bound in Integration

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

Surface Area of a Solid of Revolution

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

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 of a Rotated Region via Washer Method

Consider the region bounded by the curves $$y=x$$ and $$y=\sqrt{x}$$ along with the vertical line $$

Volume of a Solid: ln(x) Region Rotated

Consider the region in the $$xy$$-plane bounded by $$y=\ln(x)$$, $$y=0$$, $$x=1$$, and $$x=e$$. This

Water Pumping System: Instantaneous Versus Average Rates

A water pumping system operates with an inflow function $$I(t)=12+8*t$$ (liters/min) and an outflow

Work Done on a Non-linear Spring

A non-linear spring exerts a force given by $$F(x) = 3 * x^2 + 2 * x$$ (in Newtons), where $$x$$ (in

Analyzing a Cycloid

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

Arc Length of a Cycloid

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

Arc Length of a Parametric Curve

Consider the parametric equations $$x(t) = t^2$$ and $$y(t) = t^3$$ for $$0 \le t \le 2$$.

Arc Length of a Parametric Curve

Consider the curve defined by $$x(t)=t^3-3*t$$ and $$y(t)=t^2+2$$ for $$t \in [0,2]$$.

Area between Two Polar Curves

Given the polar curves $$R(\theta)=3$$ and $$r(\theta)=2$$ for $$0 \le \theta \le 2\pi$$, find the a

Area Between Two Polar Curves

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

Catching a Thief: A Parametric Pursuit Problem

A police car and a thief are moving along a straight road. Initially, both are on the same road with

Differentiability of a Piecewise-Defined Vector Function

Consider the vector-valued function $$\textbf{r}(t)= \begin{cases} \langle t, t^2 \rangle & \text{i

Double Integration in Polar Coordinates for Mass Distribution

A thin lamina occupies the region in the first quadrant defined in polar coordinates by $$0\le r\le2

Equivalence of Parametric and Polar Circle Representations

A circle is represented by the parametric equations $$x(t)=3\cos(t)$$ and $$y(t)=3\sin(t)$$ for $$0\

Exponential Growth in Parametric Representation

A model for population growth is given by the parametric equations $$x(t)=t$$ and $$y(t)=e^{0.3t}$$,

Integration of Vector-Valued Acceleration

A particle's acceleration is given by the vector function $$\mathbf{a}(t)=\langle 2*t,\; 6-3*t \rang

Intersection of Parametric Curves

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

Logarithmic Exponential Transformations in Polar Graphs

Consider the polar equation $$r=2\ln(3+\cos(\theta))$$. Answer the following:

Oscillatory Motion in a Vector-Valued Function

Consider the vector-valued function $$\vec{r}(t)= \langle \sin(2*t), \cos(3*t) \rangle$$ for $$t \in

Parametric Curves and Concavity

Consider the parametric equations $$x(t)= \sin(t)$$ and $$y(t)= \cos(2*t)$$ for $$t \in [0, 2\pi]$$.

Parametric Egg Curve Analysis

An egg-shaped curve is modeled by the parametric equations $$x(t)= \cos(t)+0.5\cos(2t)$$ and $$y(t)=

Parametric Spiral Curve Analysis

The curve defined by $$x(t)=t\cos(t)$$ and $$y(t)=t\sin(t)$$ for $$t \in [0,4\pi]$$ represents a spi

Particle Motion on an Elliptical Arc

A particle moves along a curve described by the parametric equations $$x(t)= 2*cos(t)$$ and $$y(t)=

Particle Trajectory in Parametric Motion

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

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{

Projectile Motion in Parametric Form

A projectile is launched with an initial speed of $$20\,m/s$$ at an angle of $$30^\circ$$ above the

Taylor/Maclaurin Series: Approximation and Error Analysis

Let $$f(x)=\ln(1+x)$$. Without using a calculator, generate the third-degree Maclaurin polynomial fo

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 Analysis

Let the vector-valued function be given by $$\vec{r}(t)=<e^{t},\, \sin(t),\, \cos(t)>$$ for $$0\leq

Vector-Valued Functions and 3D Projectile Motion

An object's position in three dimensions is given by $$\mathbf{r}(t)=\langle 3t, 4t, 10t-5t^2 \rangl

Vector-Valued Functions: Tangent and Normal Components

A car’s motion on a plane is described by the vector-valued function $$\mathbf{r}(t)=\langle t^2, 4*