AP Calculus BC FRQ Room

Algebraic Manipulation with Radical Functions

Let $$f(x)= \frac{\sqrt{x+5}-3}{x-4}$$, defined for $$x\neq4$$. Answer the following:

Analysis of a Jump Discontinuity

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

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

Composite Function Involving Logarithm and Rational Expression

Consider the piecewise function $$ f(x)=\begin{cases} \frac{1}{x-1} & \text{if } x<2, \\ \ln(x-1) &

Compound Function Limits and Continuity Involving a Logarithm

Consider the function $$f(x)= \ln(|x-5|)$$, defined for $$x \neq 5$$. Analyze its behavior near x =

Continuity Analysis Using a Piecewise Defined Function

Let $$f(x)=\begin{cases} x^2, & x \leq 1 \\ 2x-1, & x> 1 \end{cases}$$.

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 Involving a Radical Expression

Examine the function $$f(x)= \begin{cases} \frac{\sqrt{x+4}-2}{x} & x \neq 0 \\ k & x=0 \end{cases}$

Continuity of Log‐Exponential Function

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

Defining a Function with a Unique Limit Behavior

Construct a function $$f(x)$$ that meets the following conditions: - It is defined and continuous fo

Determining Limits for a Function with Absolute Values and Parameters

Consider the function $$ f(x)= \begin{cases} \frac{|x-2|}{x-2}, & x \neq 2 \\ c, & x = 2 \end{cases

Evaluating Limits Involving Radical Expressions

Consider the function $$h(x)= \frac{\sqrt{4x+1}-3}{x-2}$$.

Higher‐Order Continuity in a Log‐Exponential Function

Define $$ f(x)= \begin{cases} \frac{e^x - 1 - \ln(1+x)}{x^3}, & x \neq 0 \\ D, & x = 0, \end{cases}

Implicitly Defined Curve and Its Tangent Line

Consider the circle defined by the equation $$x^2+y^2=16$$. Answer the following:

Indeterminate Forms in Log‐Exponential Context

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

Intermediate Value Theorem Application with a Cubic Function

A function f(x) is continuous on the interval [-2, 2] and its values at certain points are given in

Intermediate Value Theorem in a Continuous Function

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

Internet Data Packet Transmission and Error Rates

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

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

Limit Involving Log and Exponential Functions

Evaluate the limit $$\lim_{x \to 0^+} \frac{\ln(1+\sin(x))}{e^x-1},$$ and extend your investigation

Limits and Asymptotic Behavior of Rational Functions

Let $$k(x)=\frac{5*x^2-2*x+7}{x^2+4}.$$ Answer the following:

Limits and Continuity in Particle Motion

A particle moves along a straight line with velocity given by $$v(t)=\frac{t^2-4}{t-2}$$ for t ≠ 2 s

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-

Piecewise Inflow Function and Continuity Check

A water tank's inflow is measured by a piecewise function due to a change in sensor calibration at \

Radical Function Limit via Conjugate Multiplication

Consider the function $$f(x)=\frac{\sqrt{2*x+9}-3}{x}$$ defined for $$x \neq 0$$. Answer the followi

Removable Discontinuity in a Cubic Function

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

Removable Discontinuity in a Trigonometric Function

Consider the function $$f(x)=\begin{cases} \frac{\sin(x)}{x} & \text{if } x \neq 0, \\ L & \text{if

Series Representation and Convergence Analysis

Consider the power series $$S(x)=\sum_{n=1}^{\infty} \frac{(-1)^{n+1}*(x-2)^n}{n}.$$ (Calculator per

Temperature Change Analysis

The function $$T(t)$$ represents the temperature (in $$^\circ C$$) in a chemical reactor as a functi

Water Flow Measurement Analysis

A water flow sensor measures the flow rate $$Q(t)$$ (in cubic meters per second) of a stream at vari

Water Tank Inflow with Oscillatory Variation

A water tank is equipped with a sensor that records the inflow rate with a slight oscillatory error.

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

Application of Derivative to Relative Rates in Related Variables

Water is being pumped into a conical tank, and the volume of water is given by $$V=\frac{1}{3}\pi*r^

Average vs Instantaneous Rate of Change in Temperature Data

The table below shows the temperature (in °C) recorded at several times during an experiment. Use t

Average vs Instantaneous Rates

Consider the function $$f(x)=\frac{\sin(x)}{x}$$ for \(x\neq0\), with $$f(0)=1$$. Answer the followi

Car Motion: Velocity and Acceleration

A car’s position along a straight road is given by $$s(t)=t^3-9*t$$, where $$t$$ is in seconds and $

Chemical Reaction Rate Control

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

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 Using Limit Definition

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

Differentiation from First Principles

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

Engineering Analysis of Log-Exponential Function

In an engineering system, the output voltage is given by $$V(x)=\ln(4*x+1)*e^{-0.5*x}$$, where $$x$$

Estimating Temperature Change

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

Exploration of Derivative Notation and Higher Order Derivatives

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

Finding the Derivative of a Logarithmic Function

Consider the function $$g(x)=\ln(3*x+1)$$. Answer the following:

Graph Behavior of a Log-Exponential Function

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

Implicit Differentiation in a Geometric Context

Consider the circle defined by the equation $$x^2+y^2=25.$$ (a) Use implicit differentiation to f

Implicit Differentiation of a Circle

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

Implicit Differentiation: Conic with Mixed Terms

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

Inflection Points and Concavity Analysis

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

Instantaneous Rate of Change of a Trigonometric Function

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

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

Marginal Cost Analysis Using Composite Functions and the Chain Rule

A company's cost function is given by $$C(x)= e^{2*x} + \sqrt{x+5}$$, where x (in hundreds) represen

Parametric Analysis of a Curve

A particle moves along a path given by the parametric equations $$x(t)=t^2+1$$ and $$y(t)=t^3-3*t$$,

Polar Coordinates and Tangent Lines

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

Pollutant Levels in a Lake

A lake receives pollutants at a rate of $$P_{in}(t)=30-0.5*t$$ concentration units per day and a tre

Rainwater Harvesting System

A rainwater harvesting system collects water with an inflow rate of $$R_{in}(t)=100+25\sin((\pi*t)/1

Rate of Change in a Logarithmic Function

Consider the function $$f(x)=\frac{\ln(x)}{x}$$ defined for \(x>0\). Answer the following:

Related Rates in a Conical Tank

Water is draining from a conical tank. The tank has a total height of 10 m and its radius is always

Secant and Tangent Lines: Analysis of Rate of Change

Consider the function $$f(x)=x^3-6*x^2+9*x+1$$. This function represents a model of a certain proces

Sine Function Analysis

Let $$g(x)=3*\sin(x)+2$$, where $$x$$ is in radians. Analyze its rate of change.

Taylor Expansion of a Polynomial Function Centered at x = 1

Given the polynomial function $$f(x)=3+2*x- x^2+4*x^3$$, analyze its Taylor series expansion centere

Temperature Change Rate

The temperature in a chemical reactor is modeled by $$T(t)=\frac{\sin(2*t)}{t}$$ for \(t>0\), where

Temperature Change with Provided Data

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

Vector Function and Motion Analysis

A particle moves according to the vector function $$\vec{r}(t)=\langle 2*\cos(t), 2*\sin(t)\rangle$$

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

Chemical Mixing: Implicit Relationships and Composite Rates

In a chemical mix tank, the solute amount (in grams) and the concentration (in mg/L) are related by

Composite Differentiation in Biological Growth

A biologist models the temperature $$T$$ (in °C) of a culture over time $$t$$ (in hours) by the func

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 $

Implicit Differentiation in a Conic Section

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

Implicit Differentiation Involving Exponential Functions

Consider the relation defined implicitly by $$e^{x*y} + x^2 - y^2 = 7$$.

Implicit Differentiation with Logarithmic Equation

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

Implicit Differentiation with Trigonometric Equation

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

Implicit Differentiation: Second Derivatives of a Circle

Given the circle $$x^2+y^2=10$$, answer the following parts:

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 for the Natural Logarithm

Consider the function $$f(x) = \ln(x+1)$$ for $$x > -1$$ and let $$g$$ be its inverse function. Anal

Inverse Function Differentiation in a Trigonometric Context

Let $$f(x)= \sin(x) + x$$, defined on the interval $$[0, \frac{\pi}{2}]$$, and let $$g$$ be its inve

Inverse Function Differentiation in Thermodynamics

In a thermodynamics experiment, a differentiable one-to-one function $$f$$ describes the temperature

Inverse Function Differentiation with Composite Trigonometric Functions

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

Inverse Functions in Economic Modeling

Let the cost function be given by $$f(x)= 4*x + \sqrt{x}$$, where x represents the number of items p

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

Second Derivative via Implicit Differentiation

Given the relation $$x^2 + x*y + y^2 = 7$$, answer the following:

Temperature Modeling and Composite Functions

A weather balloon ascends and the temperature at altitude x (in kilometers) is modeled by $$T(x) = \

Air Conditioning Refrigerant Balance

An air conditioning system is charged with refrigerant at a rate given by $$I(t)=12-0.5t$$ (kg/min)

Bacterial Growth and Linearization

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

Biology: Logistic Population Growth Analysis

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

Continuity in a Piecewise-Defined Function

Let $$g(x)= \begin{cases} x^2 - 1 & \text{if } x < 1 \\ 2*x + k & \text{if } x \ge 1 \end{cases}$$.

Cooling Coffee Temperature Change

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

Derivative of Concentration in a Chemical Reaction

The concentration of a reactant in a chemical reaction is modeled by $$C(t)=\frac{8e^{-0.5t}}{1+e^{-

Economic Marginal Cost Analysis

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

Economic Model: Revenue and Cost Rates

A company's revenue (in thousands of dollars) is modeled by $$R(x)=120-4*x^2+0.5*x^3$$, where $$x$$

Engineering Linearization for Error Approximation

An engineer is working with the function $$f(x)= \sqrt{x}$$ where \(x\) is a measured quantity. To s

Graphical Interpretation of Slope and Instantaneous Rate

A graph (provided below) displays a linear function representing a physical quantity over time. Use

Instantaneous vs. Average Rate of Change in Temperature

A rod's temperature along its length is modeled by $$T(x)=20\ln(x+1)+e^{-x}$$, where x (in meters) i

Interpreting Position Graphs: From Position to Velocity

A graph of position (in meters) versus time (in seconds) is provided in the stimulus. The graph show

Linearization to Estimate Change in Electrical Resistance

The resistance of a wire is modeled by $$R(T)=R_0(1+\alpha*T)$$, where $$R_0=100$$ ohms and $$\alpha

Marginal Cost Analysis

A company’s cost function is given by $$C(x)=100+25*x+4*x^2$$, where $$x$$ represents the number of

Maximizing Revenue in a Business Model

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

Optimization of a Rectangular Enclosure

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

Optimizing a Cylindrical Can Design

A manufacturer wants to design a cylindrical can with a fixed surface area of $$600\pi$$ cm² in orde

Ozone Layer Recovery Simulation

In a simulation of ozone layer dynamics, ozone is produced at a rate of $$I(t)=\frac{25}{t+1}$$ (Dob

Particle Motion Analysis

A particle moves along a straight line and its position at time $$t$$ seconds is given by $$s(t)= t^

Projectile Motion Analysis

A projectile is launched such that its horizontal and vertical positions are modeled by the parametr

Rational Function Inversion

Consider the rational function $$f(x)=\frac{2*x+3}{x-1}$$. Analyze its inverse.

Security Camera Angle Change

A security camera is mounted on a 15 m tall tower. Let $$x$$ denote the horizontal distance from the

Series Analysis in Acoustics

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

Series Identification and Approximation

Consider the series $$F(x)= \sum_{n=0}^{\infty} \frac{(-3)^n (x-1)^n}{n!}$$. Answer the following:

Series Representation of a CDF

A cumulative distribution function (CDF) is given by $$F(x)= \sum_{n=0}^{\infty} \frac{(-1)^n (x-2)^

Varying Acceleration and Particle Motion

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

Vehicle Motion on a Curved Path

A vehicle moving along a straight road has its position given by $$s(t)= 4*t^3 - 24*t^2 + 36*t + 5$$

Air Pollution Control in an Enclosed Space

In an enclosed environment, contaminated air enters at a rate of $$I(t)=15-\frac{t}{2}$$ m³/min and

Analysis of a Rational Function and Its Inverse

Consider the function $$f(x)= \frac{2*x+3}{x-1}$$ defined for $$x \neq 1$$. Answer the following par

Analysis of Relative Extrema and Increasing/Decreasing Intervals

A particle moves along a line with position given by $$s(x)=x^3-6*x^2+9*x+4$$, where $$x$$ represent

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.

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

Application of Rolle's Theorem

Consider the function $$g(x)=x^3-3x$$ on the interval $$[-\sqrt{3},\sqrt{3}]$$. Answer the following

Application of the Extreme Value Theorem in Economics

A company's revenue is modeled by $$R(x)= -2*x^2+40*x+100$$, where $$x$$ is the number of units sold

Concavity and Inflection Points Analysis

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

Convergence and Differentiation of a Series with Polynomial Coefficients

The function $$P(x)=\sum_{n=0}^\infty \frac{n^2 * (x-1)^n}{3^n}$$ is used to model stress in a mater

Curve Sketching Using Derivatives

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

Differentiability of a Piecewise Function

Consider the piecewise function $$r(x)=\begin{cases} x^2, & x \le 2 \\ 4*x-4, & x > 2 \end{cases}$$.

Extreme Value Theorem in Temperature Variation

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

Inverse Derivative Analysis of a Quartic Polynomial

Consider the function $$f(x)= x^4 - 4*x^2 + 2$$ defined for $$x \ge 0$$. Answer the following.

Inverse Function in a Physical Context

Suppose $$f(t)= t^3 + 2*t + 1$$ represents the displacement (in meters) of an object over time t (in

Investigating a Composite Function Involving Logarithms and Exponentials

Let $$f(x)= \ln(e^x + x^2)$$. Analyze the function by addressing the following parts:

Loan Amortization with Increasing Payments

A loan of $$20000$$ dollars is to be repaid in equal installments over 10 years. However, the repaym

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$

Parameter-Dependent Concavity Conditions

Consider the function $$ f(x)=x^3+a*x^2+2x,$$ where $$a$$ is a real parameter. Answer the following

Radius of Convergence and Series Manipulation in Substitution

Let $$f(x)=\sum_{n=0}^\infty c_n * (x-2)^n$$ be a power series with radius of convergence $$R = 4$$.

Rate of Change in a Chemical Reaction

The concentration of a reactant in a chemical reaction is modeled by $$C(t)=10*e^{-0.5*t} + 2$$ (in

Rational Function Discontinuities

Consider the rational function $$ R(x)=\frac{(x-3)(x+2)}{(x-3)(x-1)}.$$ Answer the following parts:

Related Rates: Expanding Balloon

A spherical balloon is being inflated so that its volume $$V$$ increases at a constant rate of $$\fr

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 $$t$$

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 $$\ln\left(\frac{1+x}{1-x}\right)$$

Let $$f(x)=\ln\left(\frac{1+x}{1-x}\right)$$. Derive its Taylor series expansion about $$x=0$$, dete

Accumulated Change Prediction

A population grows continuously at a rate proportional to its size. Specifically, the growth rate is

Accumulated Population Change from a Growth Rate Function

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

Antiderivative with an Initial Condition

Given the function $$f(x)=6*x$$, find a function $$F(x)$$ such that $$F'(x)=f(x)$$ and $$F(2)=5$$.

Antiderivative with Initial Condition

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

Approximating an Exponential Integral via Riemann Sums

Consider the function $$h(x)=e^{-x}$$ on the interval $$[0,2]$$. A table of values is provided below

Chemical Reaction: Rate of Concentration Change

A chemical reaction features a rate of change of concentration given by $$R(t)= 5*e^{-0.5*t}$$ (in m

Composite Functions and Inverses

Consider \(f(x)= x^2+1\) for \(x \ge 0\). Answer the following questions regarding \(f\) and its inv

Convergence of an Improper Integral

Consider the function $$f(x)=\frac{1}{x*(\ln(x))^2}$$ for $$x > 1$$.

Cooling of a Hot Object

An object cools in a room with ambient temperature 20°C according to Newton's Law of Cooling, modele

Cost Function Accumulation

A manufacturer’s marginal cost function is given by $$C'(x)= 0.1*x + 5$$ dollars per unit, where x

Definite Integral Involving an Inverse Function

Evaluate the definite integral $$\int_{1}^{4} \frac{1}{\sqrt{x}}\,dx$$ and explain the significance

Definite Integral via the Fundamental Theorem of Calculus

Consider the linear function $$f(x)=2*x+3$$ defined on the interval $$[1,4]$$. A graph of the functi

Differentiation and Integration of a Power Series

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

Distance Traveled by a Particle

A particle has a velocity function given by $$v(t)=t^2-4*t+3$$ (in m/s) for $$t\in [0,5]$$ seconds.

Economic Surplus: Area between Supply and Demand Curves

In an economic model, the demand function is given by $$D(x)=10 - x^2$$ and the supply function by $

Estimating Chemical Production via Riemann Sums

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

Estimating Integral from Tabular Data

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

Estimating Rainfall Accumulation

Rainfall intensity measurements (in mm/hr) at various times are recorded in the table. Use Riemann s

Evaluating a Complex Integral

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

Evaluating an Integral via U-Substitution

Evaluate the integral $$\int_{1}^{5} (x-4)^{10}\,dx$$ using u-substitution.

Graphical Analysis of Riemann Sums

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

Integration of a Rational Function via Partial Fractions

Evaluate the indefinite integral $$\int \frac{2*x+3}{x^2+x-2}\,dx$$ by using partial fractions.

Integration via Substitution and Numerical Methods

Evaluate the integral $$\int_0^2 \frac{2*x}{\sqrt{1+x^2}}\,dx$$.

Riemann Sum Approximation of Area

Given the following table of values for the function $$f(x)$$ on the interval $$[0,4]$$, use Riemann

Riemann Sum from a Table: Plant Growth Data

A function $$f(t)$$ describes the height (in cm) of a plant over time (in days). The measurements ar

U-Substitution in Accumulation Functions

In a chemical reactor, the accumulation rate of a substance is given by $$R(x)= 3*(x-2)^4$$ units pe

Volume of Water Flow in a River

The water flow rate through a river, given in cubic meters per second, is measured at different time

Autonomous Differential Equations and Stability Analysis

An autonomous differential equation has the form $$\frac{dy}{dt} = f(y)$$ with critical points at $$

Bacteria Culture with Regular Removal

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

Chemical Reaction Kinetics

A first-order chemical reaction has its concentration $$C(t)$$ (in mol/L) governed by the differenti

Cooling Cup of Coffee

A cup of coffee at an initial temperature of $$95^\circ C$$ is placed in a room. For the first 5 min

Cooling with Time-Varying Ambient Temperature

An object cools according to the modified Newton's Law of Cooling given by $$\frac{dT}{dt}= -k*(T-T_

Differential Equation Involving Logarithms

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

Environmental Modeling Using Differential Equations

The concentration $$C(t)$$ of a pollutant in a lake is modeled by the differential equation $$\frac{

Exact Differential Equation

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

Exact Differential Equation

Examine the differential equation $$ (2*x*y + y^2)\,dx + (x^2 + 2*x*y)\,dy = 0 $$. Determine if the

FRQ 7: Projectile Motion with Air Resistance

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

FRQ 9: Epidemiological Model Differential Equation

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

FRQ 11: Linear Differential Equation via Integrating Factor

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

Implicit Differentiation in a Differential Equation Context

Suppose the function $$y(x)$$ satisfies the implicit equation $$x\,e^{y}+y^2=7$$. Differentiate both

Infectious Disease Spread Model

In a closed population of N individuals, the number of infected individuals $$I(t)$$ is modeled by t

Interpreting Slope Fields for a Differential Equation

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

Logistic Model in Product Adoption

A company models the adoption rate of a new product using the logistic equation $$\frac{dP}{dt} = 0.

Mixing Problem in a Tank

A tank initially contains 50 liters of pure water. A brine solution with a salt concentration of $$3

Mixing Problem in a Water Tank

A tank initially contains $$100$$ liters of saltwater with $$5$$ kg of salt dissolved in it. Pure wa

Particle Motion with Damping

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

Phase-Plane Analysis of a Nonlinear Differential Equation

Consider the logistic differential equation $$\frac{dy}{dt} = y(1-y)$$, which models a normalized po

Population Saturation Model

Consider the differential equation $$\frac{dy}{dt}= \frac{k}{1+y^2}$$ with the initial condition $$y

Predator-Prey Model with Harvesting

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

Simplified Predator-Prey Model

A simplified model for a predator population is given by the differential equation $$\frac{dP}{dt} =

Slope Field Analysis and Solution Curve Sketching for $$\frac{dy}{dx}= x - y$$

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

Solution Curve Sketching Using Slope Fields

Given the differential equation $$\frac{dy}{dx} = x - y$$, a slope field is provided. Use the field

Temperature Change with Variable Ambient Temperature

A heated object is cooling in an environment where the ambient temperature changes over time. For $$

Accumulation of Rainwater in a Reservoir

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

Advanced Parameter-Dependent Integration Problem

Consider the function $$g(x)=e^{-a*x}$$, where $$a>0$$ and $$x$$ lies within $$[0,b]$$. The average

Arc Length of a Logarithmic Curve

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

Area Between a Rational Function and Its Asymptote

Consider the function $$f(x)=\frac{2*x+3}{x+1}$$ and its horizontal asymptote $$y=2$$ over the inter

Area Between Curves: Parabolic & Linear Regions

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

Average Force on a Beam

A beam experiences a varying force along its length given by $$F(x)=20 - 0.5*x$$ (in kN) where $$x$$

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

Average Temperature Calculation

A city's temperature during a day is modeled by $$T(t)=10+5*\sin\left(\frac{\pi*t}{12}\right)$$ for

Average Temperature of a Day

In a certain city, the temperature during the day is modeled by a continuous function $$T(t)$$ for $

Average Value and Monotonicity of an Oscillatory Function

Consider the function $$f(x)=\sin(2*x)+1$$ defined on the interval $$[0,\pi]$$.

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

Charging a Battery: Inflow and Outflow Currents

A battery is charged and discharged simultaneously. The charging current is given by $$I_{in}(t)=10+

Error Analysis in Taylor Polynomial Approximations

Let $$h(x)= \cos(3*x)$$. Analyze the error involved when approximating $$h(x)$$ by its third-degree

Implicit Differentiation with Exponential Terms

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

Implicit Function Differentiation

Consider the implicitly defined function $$\sin(x * y) + x^2 = \ln(y)$$. Answer the following:

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: Position, Velocity, and Acceleration

A particle moves along a straight line with acceleration $$a(t)=4-2*t$$ (in m/s²), initial velocity

Position and Velocity from Tabulated Data

A particle’s velocity (in m/s) is measured at discrete time intervals as shown in the table. Use the

River Cross Section Area

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

Salt Concentration in a Mixing Tank

A tank initially contains 50 L of water with 5 g of salt. A salt solution with a concentration of 0.

Savings Account with Decreasing Deposits

An individual opens a savings account with an initial deposit of $1000 in the first month. Every sub

Volume by Cross‐Sectional Area in a Variable Tank

A tank has a variable cross‐section. For a water level at height $$y$$ (in cm), the width of the tan

Volume of a Solid by the Washer Method

The region bounded by $$y=x^2$$ and $$y=4$$ is rotated about the x-axis, forming a solid with a hole

Volume of a Solid with Square Cross Sections

Consider the region bounded by the curve $$f(x)= 4 - x^2$$ and the x-axis for $$x \in [-2,2]$$. A so

Volume Using the Shell Method

Consider the region bounded by $$y=x^2$$ and $$y=4$$ with $$x\ge0$$. This region is rotated about th

Work Done by a Variable Force

A force acting on an object along a displacement is given by $$F(x)=3*x^2 -2*x+1$$ (in Newtons), whe

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

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

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

Area Between Polar Curves

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

Area Between Polar Curves

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

Area Enclosed by a Polar Curve

Let the polar curve be defined by $$r=3\sin(\theta)$$ with $$0\le \theta \le \pi$$. Answer the follo

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 Between Polar and Cartesian Coordinates

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

Conversion of Parametric to Polar: Motion Analysis

An object moves along a path given by the parametric equations $$x(t)=t^2$$ and $$y(t)=2*t$$ for $$t

Discontinuities in a Piecewise-Defined Function

Consider the function $$f(x)= \begin{cases} \frac{x^2-4}{x-2} & x < 2 \\ 3 & x = 2 \\ x+1 & x > 2 \e

Distance Traveled in a Turning Curve

A curve is defined by the parametric equations $$x(t)=4*\sin(t)$$ and $$y(t)=4*\cos(t)$$ for $$0\le

Exponential and Logarithmic Dynamics in a Polar Equation

Consider the polar curve defined by $$r=e^{\theta}$$. Answer the following:

Exponential Decay in Vector-Valued Functions

A particle moves in the plane with its position given by the vector-valued function $$\vec{r}(t)=\la

Helical Motion with Damping

A particle moves along a helical path with damping, described by the vector function $$\vec{r}(t)= \

Integrating a Vector-Valued Function

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

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

Maclaurin Series for Trigonometric Functions

Let $$f(x)=\sin(x)$$.

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 in a Damped Force Field

A particle moves in the plane according to the vector function $$\mathbf{r}(t)=\langle e^{-t}\cos(t)

Parametric Motion Analysis

A particle moves with its position given by the parametric equations $$x(t) = t^2 - 4*t$$ and $$y(t)

Parametric Plotting and Cusps

Let the parametric equations be $$ x(t)=t-\sin(t) $$ and $$ y(t)=1-\cos(t) $$ for $$ 0 \le t \le 2\p

Particle Motion in the Plane

A particle's position is given by the parametric equations $$x(t)=3*t^2-2*t$$ and $$y(t)=4*t^2+t-1$$

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)=

Polar and Parametric Form Conversion

A curve is given in polar form by $$r(\theta)=\frac{2}{1+\cos(\theta)}$$. This curve represents a co

Polar Coordinate Area Calculation

Consider the polar curve $$r=4*\sin(θ)$$ for $$0 \le θ \le \pi$$. This equation represents a circle.

Polar Curve Sketching and Area Estimation

A polar curve is described by sample data given in the table below.

Polar Plots and Intersection Points in Design

A designer creates a pattern using the polar equations $$r=5\cos(θ)$$ and $$r=5\sin(θ)$$. Analyze th

Satellite Orbit: Vector-Valued Functions

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

Tangent Line to a Polar Curve

Consider the polar curve $$r=5-2\cos(\theta)$$. Answer the following parts.

Vector Functions and Work Done Along a Path

A force field is given by $$\mathbf{F}(x,y)=\langle x*y, x^2 \rangle$$. A particle moves along the p

Vector-Valued Function and Particle Motion

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

Vector-Valued 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 and Kinematics

A particle moves in space with its position given by the vector-valued function $$\vec{r}(t)= \langl

Vector-Valued Functions: Position, Velocity, and Acceleration

Let $$\textbf{r}(t)= \langle e^t, \ln(t+1) \rangle$$ represent the position of a particle in the pla

Work Done Along a Path in a Force Field

A force field is given by \(\mathbf{F}(x,y)=\langle x,\,y \rangle\), and a particle moves along a pa

Work Done by a Force along a Path

A force acting on an object is given by the vector function $$\vec{F}(t)= \langle 3t,\; 2,\; t^2 \ra