AP Calculus BC FRQ Room

Absolute Value Limit Analysis II

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

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

Analyzing Discontinuities in a Piecewise Function

Consider the function $$f(x)= \begin{cases}\frac{x^2-1}{x-1}, & x \neq 1 \\ 3, & x=1\end{cases}$$.

Application of the Intermediate Value Theorem in Temperature Change

A laboratory experiment records the temperature $$T(t)$$ in a reaction over time $$t$$ (in minutes).

Application of the Squeeze Theorem with Trigonometric Functions

Let $$f(x)= x^2 \sin\left(\frac{1}{x}\right)$$ for $$x\neq0$$, and $$f(0)=0$$. Analyze the behavior

Applying Algebraic Techniques to Evaluate Limits

Examine the limit $$\lim_{x\to4} \frac{\sqrt{x+5}-3}{x-4}$$. Answer the following: (a) Evaluate the

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

Approaching Vertical Asymptotes

Consider the function $$g(x)=\frac{3}{(x-2)^2}-\frac{1}{x-2}.$$ Answer the following:

Comparing Methods for Limit Evaluation

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

Composite Function and Continuity

Consider the piecewise function $$ g(x)=\begin{cases} x^2 & \text{if } x<2, \\ 3x-2 & \text{if } x\

Computing a Limit Using Algebraic Manipulation

Evaluate the limit $$\lim_{x\to2} \frac{x^2-4}{x-2}$$ using algebraic manipulation.

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 of an Integral-Defined Function

Consider the function defined by the integral $$F(x)= \int_{0}^{x} \frac{t}{t^2+1} \; dt$$.

Drainage Rate with a Removable Discontinuity

A drainage system is modeled by the function $$R_{out}(t)=\frac{t^2-2\,t-15}{t-5}$$ liters per minut

End Behavior Analysis of a Rational Function

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

End Behavior and Horizontal Asymptote Analysis

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

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

Estimating Limits from Tabulated Data

A function $$g(x)$$ is experimentally measured near $$x=2$$. Use the following data to estimate $$\l

Evaluating a Limit with Algebraic Manipulation

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

Evaluating Limits Involving Absolute Value Functions

Consider the function $$f(x)= \frac{|x-4|}{x-4}$$.

Fuel Efficiency and Speed Graph Analysis

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

Graphical Analysis of Limits and Asymptotic Behavior

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

Graphical Analysis of Water Tank Volume

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

Inflow Function with a Vertical Asymptote

A water reservoir is fed by an inflow given by $$R_{in}(t)=\frac{50\,t}{t-5}$$ liters per minute, de

Jump Discontinuity Analysis using Table Data

A function f is defined by experimental measurements near $$x=2$$. Use the table provided to answer

Limit at an Infinite Discontinuity

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

Limit Evaluation Involving Radicals and Rationalization

Evaluate the limit $$\lim_{x \to 4} \frac{\sqrt{x}-2}{x-4}$$.

Limits at Infinity and Horizontal Asymptotes

Examine the function $$h(x)=\frac{2*x^3-5*x+1}{4*x^3+3*x^2-2}$$.

Limits Involving Absolute Value

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

Limits with Infinite Discontinuities

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

Maclaurin Polynomial Approximation and Error Analysis for $$\ln(1+x)$$

Consider the function $$f(x)=\ln(1+x)$$. You are asked to approximate $$f(0.5)$$ using its Maclaurin

Mixed Function Inflow Limit Analysis

Consider the water inflow function defined by $$R(t)=10+\frac{\sqrt{t+4}-2}{t}$$ for \(t\neq0\). Det

Piecewise Function Continuity

Consider the function defined by $$ f(x)=\begin{cases} \frac{x^2-4}{x-2}, & x\neq2\\ k, & x=2 \en

Piecewise Function Critical Analysis

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

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

Trigonometric Limits

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

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:

Advanced Analysis of a Composite Piecewise Function

Consider the function $$g(x)= \begin{cases} \frac{2*x^2-8}{x-2} & x \neq 2 \\ 5 & x=2 \end{cases}$$

Analysis of a Quadratic Function

Consider the function $$f(x)=3*x^2 - 2*x + 5$$. Using the limit definition of the derivative, answer

Chemical Mixing Tank

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

Chemical Reaction Rate Analysis

The concentration of a reactant in a chemical reaction (in M) is recorded over time (in seconds) as

Complex Rational Differentiation

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

Composite Function Differentiation and Taylor Series for $$e^{\sin(x)}$$

Consider the composite function $$f(x)=e^{\sin(x)}$$. A physicist needs to approximate this function

Differentiation of Functions with Variable Exponents

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

Fuel Storage Tank

A fuel storage tank receives oil at a rate of $$F_{in}(t)=40\sqrt{t+1}$$ liters per hour and loses o

Graphical Derivative Analysis

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

Implicit Differentiation on an Ellipse

Consider the ellipse defined by $$4*x^2 + 9*y^2 = 36$$.

Implicit Differentiation: Exponential-Polynomial Equation

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

Investment Return Rates: Continuous vs. Discrete Comparison

An investment's value grows continuously according to $$V(t)= 5000e^{0.07t}$$, where t is in years.

Linearization and Tangent Approximations

Let $$f(x)=e^{-x}$$ represent a cost decay function over time. Use linear approximation near $$x=0$$

Logarithmic Differentiation

Let $$T(x)= (x^2+1)^{3*x}$$ model a quantity with variable growth characteristics. Use logarithmic d

Maclaurin Polynomial for √(1+x)

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

Manufacturing Production Rates

A factory produces items at a rate given by $$P_{in}(t)=\frac{200}{1+e^{-0.3*(t-4)}}$$ items per hou

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

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

Population Model Rate Analysis

A city's population is modeled by $$P(x)=2000+500\ln(x)$$, where $$x$$ represents years since a base

Rate Function Involving Logarithms

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

Savings Account Growth: From Discrete Deposits to Continuous Derivatives

An individual deposits $$P$$ dollars at the beginning of each month into an account that earns a con

Secant and Tangent Approximations from a Graph

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

Traffic Flow Analysis

A highway on-ramp has vehicles entering at a rate of $$V_{in}(t)=30+2*t$$ vehicles per minute and ve

Biological Growth Model Differentiation

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

Chain 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 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

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 Functions in Biological Growth

Let a model for bacteria growth be represented by $$f(t)=e^{2*t}$$, and let the effect of nutrient c

Composite Implicit Differentiation Involving Trigonometric and Polynomial Terms

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

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 in an Economic Cost Function

The cost of producing $$q$$ units is modeled by $$C(q)= (5*q)^{3/2} + 200*\ln(1+q)$$. Differentiate

Differentiation Involving Absolute Values and Composite Functions

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

Drug Concentration in the Bloodstream

A drug is infused into a patient's bloodstream at a rate given by the composite function $$R(t)=k(m(

Implicit Differentiation and Inverse Functions in a Trigonometric Equation

Consider the equation $$x*y + \sin(x+y)= 1$$ which defines y implicitly as a function of x.

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 in Mixed Function Equation

Consider the relation $$x^2*y+\sin(y)=5*x$$. Analyze this relation using implicit differentiation.

Implicit Differentiation Involving a Mixed Function

Consider the equation $$x*e^{y}+y*\ln(x)=10$$, where x > 0 and y is defined implicitly as a function

Implicit Differentiation with Logarithmic Functions

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

Implicit Differentiation with Trigonometric Components

Consider the equation $$x*\sqrt{y} + \cos(y) = x^2$$, where $$y$$ is a function of $$x$$. Differenti

Implicit Differentiation with Trigonometric Equation

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

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 Function Analysis for Exponential Functions

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

Inverse Function Differentiation with Composite Trigonometric Functions

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

Inverse Trigonometric Differentiation

Differentiate the function $$ y= \arctan\left(\frac{2*x}{1-x}\right) $$.

Navigation on a Curved Path: Boat's Eastward Velocity

A boat's location in polar coordinates is described by $$r(t)= \sqrt{4*t+1}$$ and its direction by $

Physics Lab: Logarithmic Chain Rule in a Kinetics Experiment

In a kinetics experiment, the reactant concentration is modeled by $$C(t)=\ln(3*e^{2*t}+4)$$, where

Rocket Fuel Consumption Analysis

A rocket’s fuel consumption rate is modeled by the composite function $$C(t)=n(m(t))$$, where $$m(t)

Water Tank Composite Rate Analysis

A water tank receives water from an inflow pipe where the inflow rate is given by the composite func

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]$

Chemical Concentration Rate Analysis

The concentration of a chemical in a reactor is given by $$C(t)=\frac{5*t}{t+2}$$ M (moles per liter

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

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

Cooling Temperature Model

The temperature of a heated object cooling in a room is modeled by $$T(t)= 80 + 120*e^{-0.25*t}$$, w

Cubic Curve Linearization

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

Deceleration of a Vehicle on a Straight Road

A vehicle travels along a straight road with velocity function $$v(t)=30-4*t$$ (m/s) for $$0 \le t \

Draining Conical Tank

Water is draining from a conical tank at a rate of $$5$$ m³/min. The tank has a height of $$10$$ m a

Draining Hemispherical Tank

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

Drug Concentration Dynamics

The concentration of a drug in the bloodstream is modeled by $$C(t)= 50*e^{-0.2*t} + 10$$ (in mg/L),

Expanding Circular Ripple

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

Forensic Gas Leakage Analysis

A gas tank under investigation shows leakage at a rate of $$O(t)=3t$$ (liters per minute) while it i

Graphical Analysis of an Inverse Function

Consider a continuously differentiable and strictly increasing function $$f(x)$$ as depicted in the

Graphical Analysis of Derivatives

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

Inflating Spherical Balloon

A spherical balloon is being inflated such that its volume is given by $$V= \frac{4}{3}*\pi*r^3$$, w

Inverse of a Trigonometric Function

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

Ladder Sliding Down a Wall

A 10-meter ladder leans against a vertical wall and begins to slide. The bottom slides away from the

Limit Evaluation via L'Hopital's Rule

Evaluate the limit: $$L=\lim_{x\to 0}\frac{e^{2x}-1}{\ln(1+3x)}$$. Answer the following:

Linearization of Implicit Equation

Consider the implicit equation $$x^2 + y^2 - 2*x*y = 1$$, which defines $$y$$ as a function of $$x$$

Marginal Cost and Revenue Analysis

A company’s cost function is given by $$C(x)=100+25*x+4*x^2$$, where $$C(x)$$ is measured in dollars

Particle Motion Analysis Using Cubic Position Function

Consider a particle moving along a straight line with its position given by $$x(t)=t^3 - 6*t^2 + 9*t

Piecewise Velocity and Acceleration Analysis

A particle moves along a straight line with its velocity given by $$ v(t)= \begin{cases} t^2-4*t+3,

Radical Function Inversion

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

Related Rates: Expanding Circular Ripple

A circular ripple in a pond expands such that its area, given by $$A=\pi r^2$$, is increasing at a c

Revenue Function and Marginal Revenue

A company’s revenue (in thousands of dollars) is modeled as a function of units sold (in thousands)

Savings Account Dynamics

A bank account receives deposits at a rate of $$I(t)=50+10t$$ (dollars per month) and experiences wi

Series Approximation in an Exponential Population Model

A population is modeled by $$P(t)= 1000 \times \sum_{n=0}^{\infty} \frac{(0.05t)^n}{n!}$$, which is

Series-Based Analysis of Experimental Data

An experiment models a measurement function as $$g(x)= \sum_{n=0}^{\infty} \frac{(-1)^n (x/4)^n}{n+1

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)

Absolute Extrema and the Candidate’s Test

Let $$f(x)=x^3-3x^2-9x+5$$ be defined on the closed interval $$[-2,5]$$. Answer the following parts:

Aircraft Climb Analysis

An aircraft's vertical motion is modeled by a vertical velocity function given by $$v(t)=20-2*t$$ (i

Analysis of Total Distance Traveled

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

Application of Rolle's Theorem to a Trigonometric Function

Consider the function $$f(x)=\cos(x)$$ on the interval [0,π]. Answer the following parts:

Determining Absolute Extrema for a Trigonometric-Polynomial Function

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

Extreme Value Theorem for a Piecewise Function

Let $$h(x)$$ be defined on $$[-2,4]$$ as $$ h(x)= \begin{cases} -x^2+4 & \text{if } x \le 1, \\ 2x-

Implicit Differentiation in Economic Context

Consider the curve defined implicitly by $$x*y + y^2 = 12$$, representing an economic relationship b

Instantaneous vs. Average Rates in a Real-World Model

A company’s monthly revenue is modeled by $$ R(t)=0.5t^3-4t^2+12t+100, \quad 0 \le t \le 6,$$ where

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

Linear Particle Motion Analysis

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

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 Application

Let \( f(x) = x^3 - 3*x^2 + 2 \) be defined on the closed interval \([0,3]\). Answer the following p

Mean Value Theorem in a Temperature Model

The temperature over a day (in °C) is modeled by $$T(t)=10+8*\sin\left(\frac{\pi*t}{12}\right)$$ for

Modeling Exponential Population Growth

A population is modeled by the function $$P(t)=500*e^{0.2*t}$$, where \(t\) is measured in years.

Projectile Motion Analysis

A projectile is launched at a 45° angle with an initial speed of 20 m/s. Its motion is modeled by th

Real-World Modeling: Radioactive Decay with Logarithmic Adjustment

A radioactive substance decays according to $$N(t)= N_0\,e^{-0.03t}$$. In an experiment, the recorde

Second Derivative Test for Critical Points

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

Series Approximation in Engineering: Oscillation Amplitude

An engineer models the oscillation amplitude by $$A(t)=\sin(0.2*t)\,e^{-0.05*t}$$. Derive the Maclau

Skier's Speed Analysis

A skier's speed (in m/s) on a slope was recorded at various time intervals. Use the data provided to

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

Vector Analysis of Particle Motion

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

Accumulated Change Prediction

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

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.

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

Area Estimation with Riemann Sums

A water flow rate function f(x) (in m³/s) is measured at various times. The table below shows the me

Area Under a Piecewise-Defined Curve with a Jump Discontinuity

Consider the function $$ g(x)= \begin{cases} 2x+1 & \text{if } 0 \le x < 2, \\ 3x-2 & \text{if } 2 \

Bacteria Population Accumulation

A bacteria culture grows at a rate given by $$r(t)=0.5*t^2-t+3$$ (in thousand bacteria per hour) for

Charging a Battery

An electric battery is charged with a variable current given by $$I(t)=4+2*\sin\left(\frac{\pi*t}{6}

Comprehensive Integration of a Polynomial Function

Consider the function $$f(x)=(x-3)(x+2)^2$$ on the interval $$[1,5]$$. This problem involves multipl

Consumer Surplus via Integration

In an economic model, the demand function is given by $$p(x)= 20 - 0.5*x$$, where p is the price in

Convergence of an Improper Integral

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

Determining the Average Value via Integration

Find the average value of the function $$f(x)=3*x^2-2*x+1$$ on the interval $$[1,4]$$.

Economic Applications: Consumer and Producer Surplus

In a market, the demand function is given by $$D(p)=100-2p$$ and the supply function by $$S(p)=20+3p

Estimating Area Under a Curve Using Riemann Sums

A function $$f(x)$$ is defined on the interval $$[0,6]$$. The following table provides the values of

Integration by Parts: Logarithmic Function

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

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

Investment Growth Analysis with Exponentials

An investment grows according to the function $$f(t)= 100*e^{0.05*t}$$ for $$t \ge 0$$ (in years). A

Midpoint Riemann Sum Approximation

Estimate the area under the curve $$f(x)=x^{3} - 2*x + 1$$ on the interval $$[0,3]$$ using a midpoin

Minimizing Material for a Container

A company wants to design an open-top rectangular container with a square base that must have a volu

Modeling Water Inflow Using Integration

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

Recovering Position from Velocity

A particle moves along a straight line with a velocity given by $$v(t)=6*t-2$$ (in m/s) for $$t\in [

Revenue Estimation Using the Trapezoidal Rule

A company recorded its daily revenue (in thousands of dollars) over four days. Use the data in the t

Riemann Sum Approximation of Area

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

Taylor/Maclaurin Series Approximation and Error Analysis

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

Temperature Change Analysis

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

Trigonometric Integral via U-Substitution

Evaluate the integral $$\int_{0}^{\pi/2} \sin(2*x)\,dx$$ using an appropriate substitution.

Volume of a Solid by the Shell Method

Consider the region bounded by the curve $$y=\sqrt{x}$$, the x-axis, and the vertical line $$x=4$$.

Bank Account Growth with Continuous Compounding

A bank account balance $$A(t)$$ grows according to the differential equation $$\frac{dA}{dt}= r*A$$,

Chemical Reaction Kinetics

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

Chemical Reaction Rate

In a chemical reaction, the concentration $$C(t)$$ of a reactant decreases according to the first-or

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 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_

Disease Spread Model

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

Economic Growth Model

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

Exact Differential Equations

Consider the differential equation $$ (2*x + y) + (x + 3*y)\,\frac{dy}{dx} = 0$$.

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 7: Projectile Motion with Air Resistance

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

Infectious Disease Spread Model

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

Integrating Factor for a Non-Exact Differential Equation

Consider the differential equation $$ (y - x)\,dx + (y + 2*x)\,dy = 0 $$. This equation is not exact

Inverse Function Analysis Derived from a Differential Equation Solution

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

Investment Growth with Nonlinear Dynamics

The rate of change of an investment amount $$I$$ is modeled by the nonlinear differential equation $

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

Modeling Cooling and Heating: Temperature Differential Equation

Suppose the temperature of an object changes according to the differential equation $$\frac{dT}{dt}

Modeling Temperature in a Biological Specimen

A biological specimen initially at $$37^\circ C$$ is cooling in an environment where the ideal ambie

Newton's Law of Cooling

An object cools according to Newton's Law of Cooling: $$\frac{dT}{dt} = -k*(T-20)$$, where the ambie

Newton's Law of Cooling

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

Newton’s Law of Cooling Application

An object is cooling in a room with ambient temperature $$T_a=20^\circ C$$. Its temperature $$T(t)$$

Non-linear Differential Equation Modeling Spill Rate

Water leaks from a reservoir at a rate proportional to the square root of the volume. This is modele

Separable Differential Equation with a Logarithmic Integral

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

Separable Differential Equation with Initial Condition

Solve the differential equation $$\frac{dy}{dx} = \frac{x}{y}$$ subject to the initial condition $$y

Slope Field Analysis and DE Solutions

Consider the differential equation $$\frac{dy}{dx} = x$$. The equation has a slope field as represen

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

Slope Field and Solution Curve Sketching

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

Solving a Linear Differential Equation using an Integrating Factor

Consider the linear differential equation $$\frac{dy}{dx} + \frac{2}{x} * y = \frac{\sin(x)}{x}$$ wi

Tank Draining Problem

A tank with a variable cross-sectional area is being emptied. The height \(h(t)\) of the water satis

Accumulated Change in a Population Model

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

Analyzing Convergence of a Taylor Series

Consider the function $$g(x)= e^{-x^2}$$. Analyze the Maclaurin series for this function.

Arc Length in Polar Coordinates

Find the length of the curve defined in polar coordinates by $$r(θ)= 1+ \cos(θ)$$ for $$θ \in [0, 2\

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 of One Petal of a Polar Curve

The polar curve defined by $$r = \cos(2\theta)$$ forms a rose with four petals. Find the area of one

Average Temperature Over a Day

A research team studies the variation in water temperature in a lake over a 24‐hour period. The temp

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

Cyclist's Journey: Displacement versus Total Distance

A cyclist's velocity is given by $$v(t)=\sin(t)$$ (in m/s) for $$t\in[0,2\pi]$$. Answer the followin

Designing a Bridge Arch

A bridge arch is modeled by the curve $$y = 10 - 0.25*x^2$$, where $$x$$ is measured in meters and $

Electric Charge Distribution Along a Rod

A rod of length 10 m has a linear charge density given by $$\lambda(x) = 3e^{-0.5*x}$$ coulombs per

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 $$0 \le t \le 6$$.

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 Motion with Constant Acceleration

A ball is thrown upward and moves under the constant acceleration due to gravity $$a(t)=-9.8$$ (in m

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.

Sand Pile Dynamics

Sand is being added to a pile at a rate given by $$A(t)=8-0.5*t$$ (kg/min) for $$0\le t\le12$$ minut

Surface Area of a Rotated Curve

Consider the curve $$y=x^3$$ on the interval $$[0,2]$$. This curve is rotated about the x-axis, form

Volume by Cross-Section: Rotated Region

Consider the region bounded by the curves $$y=x+2$$ and $$y=x^2$$, with the intersection points form

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

Work Done by a Variable Force

A variable force acting along a straight line is given by $$F(x) = \frac{10}{x+2}$$ (in Newtons). Fi

Analyzing a Walker's Path: A Vector-Valued Function

A pedestrian's path is modeled by the vector function $$\vec{r}(t)= \langle t^2 - 4, \sqrt{t+5} \ran

Arc Length and Curvature Comparison

Consider two curves given by: $$C_1: x(t)=\ln(t),\, y(t)=\sqrt{t}$$ for $$1\leq t\leq e$$, and $$C_2

Arc Length of a Decaying Spiral

Consider the parametric equations $$x(t)= e^{-t}\cos(t)$$ and $$y(t)= e^{-t}\sin(t)$$ for $$t \ge 0$

Arc Length of a Parametric Curve

Consider the curve defined by the parametric equations $$x(t)=3*\cos(t)$$ and $$y(t)=3*\sin(t)$$ for

Arc Length of a Parametric Curve

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

Arc Length of a Parametric Curve

Consider the curve defined by $$x(t)= 3*\sin(t)$$ and $$y(t)= 3*\cos(t)$$ for $$0 \le t \le \frac{\p

Arc Length of a Parametric Curve

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

Arc Length of a Polar Curve

Consider the polar curve $$r(\theta)= 1+\cos(\theta)$$ for \(0 \le \theta \le \pi\).

Area Between Polar Curves: Annulus with a Hole

Two polar curves are given by \(R(\theta)=3\) and \(r(\theta)=2+\cos(\theta)\) for \(0\le\theta\le2\

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

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

Concavity and Inflection in Parametric Curves

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

Conversion and Analysis of Polar and Rectangular Forms

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

Conversion from Polar to Cartesian Coordinates

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

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

Curvature of a Vector-Valued Function

Let $$\vec{r}(t)=\langle t, t^2, \ln(t) \rangle$$ for \(t>0\). The curvature \(\kappa(t)\) is given

Dynamics in Polar Coordinates

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

Inner Loop of a Limaçon in Polar Coordinates

The polar curve given by \(r=1+2\cos(\theta)\) forms a limaçon with an inner loop. Answer the follow

Oscillatory Behavior in Vector-Valued Functions

Examine the vector-valued function $$\mathbf{r}(t)=\langle \cos(2*t), \sin(3*t), \cos(t)\sin(2*t) \r

Parametric Equations and Intersection Points

Consider the curves defined parametrically by $$x_1(t)=t^2, \; y_1(t)=2t$$ and $$x_2(s)=s+1, \; y_2(

Parametric Representation of an Ellipse

An ellipse is represented by the parametric equations $$x(t)=4\cos(t)$$ and $$y(t)=3\sin(t)$$ for $$

Parametric Representation of Circular Motion

An object moves along a circle of radius $$5$$, with its position given by $$x(t)=5\cos(t)$$ and $$y

Particle Motion with Logarithmic Component

A particle moves along a path given by $$x(t)= \frac{t}{t+1}$$ and $$y(t)= \ln(t+1)$$, where $$t \ge

Polar Boundary Conversion and Area

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

Polar Coordinate Area Calculation

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

Polar Coordinates: Analysis of $$r = 2+\cos(\theta)$$

The polar curve $$r= 2+\cos(\theta)$$ is given. Analyze various aspects of this curve.

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

Polar to Parametric Conversion and Arc Length

A curve is defined in polar coordinates by $$r= 1+\sin(\theta)$$. Convert and analyze the curve.

Projectile Motion with Air Resistance: Parametric Analysis

A projectile is launched with air resistance, and its motion is modeled by the parametric equations:

Projectile Motion with Parametric Equations

A ball is launched from ground level with an initial speed of $$20 \text{ m/s}$$ at an angle of $$\f

Relative Motion of Two Objects

Two objects A and B move in the plane with positions given by the vector functions $$\vec{r}_A(t)= \

Roller Coaster Design: Parametric Path

A roller coaster is modeled by the parametric equations $$x(t)=t-\cos(t)$$ and $$y(t)=t-\sin(t)$$ fo

Spiral Path Analysis

A spiral is defined by the vector-valued function $$r(t) = \langle e^{-t}*\cos(t), e^{-t}*\sin(t) \r

Vector-Valued Function and Derivatives

Consider the vector-valued function given by $$ r(t)=\langle e^t*\cos(t),\; e^t*\sin(t) \rangle $$ f

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: Motion in the Plane

The position of a particle in space is given by $$\vec{r}(t)=\langle e^t, \ln(1+t), t^2 \rangle$$ fo

Vector-Valued Integrals in Motion

A particle's acceleration is given by the vector function $$\vec{a}(t)=<\ln(t),\, t^{-1},\, e^{t}>$$