AP Calculus BC FRQ Room

Approximating Limits Using Tabulated Values

The function g(x) is sampled near x = 2 and the following values are recorded: | x | g(x) | |--

Complex Rational Function and Continuity Analysis

Let $$R(x)= \frac{x^3-8}{x-2}$$ for $$x \neq 2$$.

Composite Function and Continuity

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

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 Involving Logarithmic and Polynomial Expressions

Consider the function $$f(x)= \begin{cases} \frac{\ln(x+2)}{x} & \text{if } x<0 \\ (x+1)^2 & \text{i

Continuity and the Intermediate Value Theorem in Temperature Modeling

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

Continuity in a Parametric Function Context

A particle moves such that its coordinates are given by the parametric equations: $$x(t)= t^2-4$$ an

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:

Evaluating a Rational Function Limit Using Algebraic Manipulation

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

Evaluating Limits Involving Exponential and Rational Functions

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

Graphical Analysis of Removable Discontinuity

A graph of a function f is provided (see stimulus). The graph shows that f has a hole at (2, 4) whil

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

Limit Behavior in a Container Optimization Problem

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

Limit Definition of the Derivative for a Polynomial Function

Let $$f(x)=3*x^2-2*x+1$$. Use the limit definition of the derivative to find $$f'(2)$$.

Limit Evaluation Involving Trigonometric Functions

Let $$f(x)=\frac{\sin(4*x)}{\tan(2*x)}$$ for $$x\neq0$$, with f(0) defined separately. Answer the

Limits Involving Radical Functions

Examine the function $$m(x)=\frac{\sqrt{x}-2}{x-4}$$.

Limits with a Parameter in a Trigonometric Function

Consider the function $$f(x)= \begin{cases} \frac{\sin(a*x)}{x} & x \neq 0 \\ b & x=0 \end{cases}$$,

Logarithmic Function Limits

Consider the function $$f(x)=\frac{\ln(1+3*x)}{x}$$ for $$x \neq 0$$. Answer the following:

One-Sided Limits in a Real-World Profit Model

A company’s profit function is given by $$ P(x)=\begin{cases} 0.05*x+100, & x<1000 \\ 0.08*x+50,

Oscillatory Behavior and Limits

Consider the function $$f(x)=x\sin(1/x)$$ for x \neq 0, with f(0) defined to be 0. Use the following

Piecewise Function Critical Analysis

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

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

Removable Discontinuity in a Cubic Function

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

Removing a Removable Discontinuity in a Piecewise Function

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

Trigonometric Limits Analysis

Evaluate the following limits involving trigonometric functions.

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

Biochemical Reaction Rates: Derivative from Experimental Data

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

Chemical Reaction Rate Analysis

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

Chemical Reaction Rate Control

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

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

Cost Optimization in Production: Derivative Application

A company's cost function is given by $$K(x)= 0.1x^3 - 2x^2 + 15x + 100$$, where x represents the nu

Differentiability of a Piecewise Function

Consider the piecewise function $$f(x)=\begin{cases} x^2, & x < 1 \\ 2*x, & x \ge 1 \end{cases}$$. A

Differentiating a Series Representing a Function

Consider the function defined by the infinite series $$S(x)= \sum_{n=0}^\infty \frac{(-1)^n * x^{2*

Differentiating Composite Functions using the Chain Rule

Consider the function $$S(x)=\sin(3*x^2+2)$$ which might model the stress on a structure as a functi

Differentiation in Exponential Growth Models

A population is modeled by $$P(t)=P_0e^{r*t}$$ with the initial population $$P_0=500$$ and growth ra

Differentiation in Polar Coordinates

Consider the polar curve defined by $$r(\theta)= 1+\cos(\theta).$$ (a) Use the formula for polar

Heat Transfer in a Rod: Modeling and Differentiation

The temperature distribution along a rod is given by $$T(x)= 100 - 2x^2 + 0.5x^3$$, where x is in me

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 with Inverse Functions

Suppose a differentiable function $$f$$ satisfies the equation $$f(x) + f^(-1)(x) = 2*x$$ for all x

Implicit Differentiation: Cost Allocation Model

A company's cost allocation between two departments is modeled by the equation $$x^2 + x*y + y^2 = 1

Implicit Differentiation: Inverse Trigonometric Equation

Consider the function defined implicitly by $$\arctan(y) + y = x$$.

Instantaneous Rate of Change in Motion

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

Limit Definition of the Derivative for a Quadratic Function

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

Maclaurin Series for e^x Approximation

Consider the function $$f(x)=e^x$$, which models many growth processes in nature. Use its Maclaurin

Manufacturing Cost Function and Instantaneous Rate

The total cost (in dollars) to produce x units of a product is given by $$C(x)= 0.2x^3 - 3x^2 + 50x

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

Tangent Line Approximation vs. Taylor Series for ln(1+x)

An engineer studying the function $$f(x)=\ln(1+x)$$ is comparing the tangent line approximation with

Tangent Lines and Related Approximations

For the function $$f(x)= \sin(x) + x^2,$$ (a) Compute \(f'(0)\). (b) Write the equation of the t

Testing Differentiability at a Junction Point

Consider the function $$f(x)= \begin{cases} x+2 & x < 3 \\ 5 & x = 3 \\ 2*x-1 & x > 3 \end{cases}$$.

Traffic Flow and Instantaneous Rate

The number of cars passing through an intersection per minute is modeled by $$F(t)= 3t^2 + 2t + 10$$

Trigonometric Function Differentiation

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

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

Analysis of a Composite Chemical Concentration Model

The concentration of a chemical in a reaction is modeled by the composite function $$C(t)= \ln(0.5*t

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 Chain Rule with Exponential and Trigonometric Functions

Consider the function $$f(x) = e^{\cos(x)}$$. Analyze its derivative and explain the role of the cha

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 Function and Its Second Derivative

A temperature profile is modeled by a composite function: $$T(t) = h(m(t))$$, where $$m(t)= 3*t^2 +

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 Involving Inverse Trigonometric Functions

Consider the function $$f(x)= \arctan(\sqrt{x})$$.

Implicit Differentiation in a Conical Sand Pile Problem

A conical sand pile has a constant ratio between its radius and height given by $$r= \frac{1}{2}*h$$

Implicit Differentiation in an Elliptical Orbit

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

Implicit Differentiation with Logarithmic Equation

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

Implicit Differentiation with Logarithmic Functions

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

Inverse Function Derivative Calculation

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

Inverse Function Differentiation in a Logarithmic Scenario

Let $$f(x)= \ln(x+2) + x$$, which is a one-to-one differentiable function. It is known that $$f(0)=

Inverse Function Differentiation with a Cubic Function

Let $$f(x)= x^3+ x + 1$$ be a one-to-one function, and let $$g$$ be its inverse function. Answer the

Inverse of a Shifted Logarithmic Function

Analyze the function $$f(x)=\ln(x-1)+2$$ defined for $$x>1$$ and its inverse.

Inverse Trigonometric Function in a Navigation Problem

A navigator uses the function $$\theta(x)=\arcsin\left(\frac{x}{10}\right)$$ to determine the angle

Inverse Trigonometric Functions in Navigation

A ship navigates such that its angular position relative to a fixed reference is given by $$\theta =

Inverse Trigonometric Functions: Analysis and Application

Consider the function $$f(x) = \arctan(3*x)$$. Analyze its rate of change and the equation of the ta

Optimization in Manufacturing Material

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

Related Rates in an Inflating Balloon

The volume of a spherical balloon is given by $$V= \frac{4}{3}*\pi*r^3$$, where r is the radius. Sup

Reservoir Levels and Evaporation Rates

A reservoir is being filled with water from an inflow while losing water through controlled release

Revenue Model and Inverse Analysis

A company's revenue (in millions) is modeled by $$R(x)=\sqrt{x+4}+3$$, where x represents production

Tangent Line to a Circle via Implicit Differentiation

Consider the circle defined by $$x^2 + y^2 = 25$$. At the point $$(3, -4)$$, determine the slope of

Analysis of Particle Motion

A particle’s velocity is given by $$v(t)= 4t^3 - 3t^2 + 2$$. Analyze the particle’s motion by invest

Bacterial Population Growth Analysis

A laboratory culture of bacteria has an initial population of $$P_0=1000$$ and grows according to th

Circular Motion and Angular Rate

A point moves along a circle of radius 5 meters. Its angular position is given by $$\theta(t)=2*t^2-

City Population Migration

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

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

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

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),

Engineering Linearization for Error Approximation

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

Error Propagation in Circular Disk Area Measurement

A circular disk has a measured diameter of 10 cm with a possible error of ±0.05 cm. The area of the

Estimating the Rate of Change from Reservoir Data

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

Expanding Circular Ripple

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

Fuel Consumption Rate Analysis

The fuel consumption of a car (in gallons per 100 miles) is modeled by $$C(v)=0.05*v^2+1$$, where $$

Graphical Interpretation of Slope and Instantaneous Rate

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

Integration of Flow Rates Using the Trapezoidal Rule

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

Interpreting the Derivative in Straight Line Motion

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

Inverse Trigonometric Composition

Consider the function $$f(x)=2*\sin(x)-1$$ defined on $$\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$$.

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 in Chemical Reaction Rates

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

Ladder Sliding Down a Wall

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

Linearization and Differentials: Approximating Function Values

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

Linearization in Engineering Load Estimation

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

Maclaurin Series for ln(1+x)

Consider the function $$f(x)= \ln(1+x)$$. Its Maclaurin series may be used to approximate values of

Motion along a Curved Path

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

Motion with Non-Uniform Acceleration

A particle moves along a straight line and its position is given by $$s(t)= 2*t^3 - 9*t^2 + 12*t + 3

Optimal Dimensions of a Cylinder with Fixed Volume

A closed right circular cylinder must have a volume of $$200\pi$$ cubic centimeters. The surface are

Particle Motion with Measured Positions

A particle moves along a straight line with its velocity given by $$v(t)=4*t-1$$ for $$t \ge 0$$ (in

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)

Security Camera Angle Change

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

Series Integration in Fluid Flow Modeling

The flow rate of a fluid is modeled by $$Q(t)= \sum_{n=0}^{\infty} (-1)^n \frac{(0.1t)^{n+1}}{n+1}$$

Vector Function: Particle Motion in the Plane

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

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 an Exponential-Linear Function

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

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 the Mean Value Theorem

Let $$f(x)=\frac{x}{x^2+1}$$ be defined on the interval $$[0,2]$$. Answer the following questions us

Arc Length Approximation

Let $$f(x) = \sqrt{x}$$ be defined on the interval [1,9].

Area Enclosed by a Polar Curve

Consider the polar curve defined by $$r(\theta) = 2 + 2*\sin(\theta)$$. This curve represents a lima

Bank Account Growth and Instantaneous Rate

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

Combining Series and Integration to Analyze a Population Model

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

Concavity in an Economic Model

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

Economic Production Optimization

A company’s cost function is given by $$C(x) = 0.5*x^3 - 3*x^2 + 4*x + 200$$, where x represents the

Elasticity Analysis of a Demand Function

The demand function for a product is given by $$Q(p) = 100 - 5*p + 0.2*p^2$$, where p (in dollars) i

Exploring Inverses of a Trigonometric Transformation

Consider the function $$f(x)= 2*\tan(x) + x$$ defined on the interval $$(-\pi/4, \pi/4)$$. Answer th

Extreme Value Theorem: Finding Global Extrema

Consider the function $$f(x)= x^3-6*x^2+9*x+2$$ on the closed interval $$[0,4]$$. Use the Extreme Va

Function Behavior Analysis

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

Ink Drop Diffusion and Intensity Loss

When a drop of ink is placed in water, it spreads out in concentric rings. The intensity of the ink

Inverse Function and Critical Points in a Business Context

A company models its profit (in thousands of dollars) by $$f(x)= \ln(4*x+7)$$ for $$x \ge 0$$, where

Maclaurin Approximation for $$\ln(1+2*x)$$

Consider the function $$f(x)=\ln(1+2*x)$$. In this problem, you will generate the Maclaurin series f

Maximizing the Area of an Inscribed Rectangle

A rectangle is inscribed in a circle of radius $$5$$. Determine the dimensions of the rectangle that

Mean Value Theorem in Motion

A car travels along a straight road and its position is modeled by $$s(x) = x^2$$ (in kilometers), w

Modeling Real World with the Mean Value Theorem

A car travels along a straight road with its position at time $$t$$ (in seconds) given by $$ s(t)=0.

Optimization Problem: Designing a Box

A company needs to design an open-top box with a square base that has a volume of $$32\,m^3$$. The c

Projectile Motion and Maximum Height

A projectile is launched with its height (in meters) given by the function $$h(t) = -5*t^2 + 20*t +

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

Relative Motion in Two Dimensions

A point moves in the plane along the path defined by $$x(t)=t^2-4*t+3$$ and $$y(t)=\ln(t+1)$$ for $$

Series Convergence and Differentiation in Thermodynamics

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

Series Representation in a Biological Growth Model

A biological process is approximated by the series $$B(t)=\sum_{n=0}^\infty (-1)^n * \frac{(0.3*t)^n

Taylor Series for $$\ln(1+3*x)$$

Let $$f(x)=\ln(1+3*x)$$. Develop its Maclaurin series up to the 3rd degree, determine the radius of

Taylor Series for $$\sqrt{x}$$ Centered at $$x=4$$

For the function $$f(x)=\sqrt{x}$$, find the Taylor series expansion centered at $$x=4$$ including t

Volume of a Solid of Revolution Using the Washer Method

Find the volume of the solid obtained by revolving the region bounded by $$y=\sqrt{x}$$, $$y=\frac{x

Advanced U-Substitution with a Quadratic Expression

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

Application of the Fundamental Theorem

Consider the function $$f(x)=x^2+2*x$$ defined on the interval $$[1,4]$$. Evaluate the definite inte

Arc Length of a Power Function

Find the arc length of the curve $$y=\frac{2}{3}*x^{3/2}$$ for $$x$$ in the interval $$[1,9]$$. Answ

Chemical Reactor Concentration

In a chemical reactor, a reactant enters at a rate of $$C_{in}(t)=5+t$$ grams per minute and is simu

Comparing Riemann Sums with Definite Integral in Estimating Distance

A vehicle's velocity (in m/s) is recorded at discrete times during a trip. Use these data to estimat

Convergence of an Improper Integral Representing Accumulation

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

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.

Drug Concentration in a Bloodstream

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

Error Analysis in Riemann Sum Approximations

Consider approximating the integral $$\int_{0}^{2} x^3\,dx$$ using a left-hand Riemann sum with $$n$

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

Evaluation of an Improper Integral

Consider the integral $$\int_{1}^{\infty} \frac{1}{x^{2}} dx$$. Answer the following:

Implicit Differentiation Involving an Integral

Consider the relationship $$y^2 + \int_{1}^{x} \cos(t)\, dt = 4$$. Answer the following parts.

Improper Integral Evaluation

Evaluate the integral $$\int_{1}^{\infty} \frac{1}{t^2}\, dt$$ by answering the following parts.

Integration by Substitution and Inverse Functions

Consider the function $$f(x)= (x-4)^2 + 3$$ for $$x \ge 4$$. Answer the following questions about $$

Logistic Growth and Population Integration

A population grows according to the logistic differential equation $$\frac{dP}{dt}=k*P\left(1-\frac

Minimizing Material for a Container

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

Optimizing the Inflow Rate Strategy

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

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 [

Riemann Sum Approximations: Midpoint vs. Trapezoidal

Consider the function $$f(x)=e^(-x)$$ on the interval [0, 3]. Compare the approximations for the def

Temperature Change Analysis

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

Temperature Change in a Material

A laser heats a material such that its temperature changes at a rate given by $$\frac{dT}{dt} = 2*\s

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 a Solid by the Shell Method

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

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

Work Done by a Variable Force

A variable force given by $$F(x)=4*x^2$$ (in Newtons) is applied along a straight line over the disp

Work Done by a Variable Force

A force acting along a displacement is given by $$F(x)=5*x^2-2*x$$ (in Newtons), where x is measured

Work Done by an Exponential Force

A variable force acting along the x-axis is given by $$F(x)=5 * e^(0.5 * x)$$ (in Newtons) for 0 \(\

Analysis of an Inverse Function from a Differential Equation Solution

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

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 and Separable Differential Equation

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

Chemical Reaction in a Closed System

The concentration $$C(t)$$ of a reactant in a closed system decreases according to the differential

Differential Equation in a Gravitational Context

Consider the differential equation $$\frac{dv}{dt}= -G\,\frac{M}{(R+t)^2}$$, which models a simplifi

Differential Equation Involving Logarithms

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

Differential Equations in Compound Interest

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

Drug Concentration in the Bloodstream

A drug is administered intravenously, and its concentration in the bloodstream is modeled by the dif

Epidemic Spread Modeling

An epidemic in a closed population of $$N=10000$$ individuals is modeled by the logistic equation $$

Exact Differential Equation

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

Exact Differential Equations

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

Exact Differential Equations

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

FRQ 2: Separable Differential Equation with Initial Condition

Consider the differential equation $$\frac{dy}{dx} = \frac{4*x}{y}$$ with the initial condition $$y(

FRQ 3: Population Growth and Logistic Model

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

FRQ 20: Epidemic Decay with Intervention

After strict intervention measures, the number of active cases in an epidemic decays according to th

Gas Pressure Dynamics

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

Integration Factor Method

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

Logistic Differential Equation Analysis

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

Mixing Tank with Variable Inflow

A tank initially contains 50 L of water with 5 kg of salt dissolved in it. A brine solution with a s

Newton's Law of Cooling

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

Piecewise Differential Equation with Discontinuities

Consider the following piecewise differential equation defined for a function $$y(x)$$: For $$x < 2

Population Dynamics with Harvesting

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

Radioactive Decay Data Analysis

A radioactive substance is decaying over time. The following table shows the measured mass (in grams

Radioactive Decay with Constant Source

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

Related Rates: Conical Tank Overflow

A conical tank has a height of $$10\,m$$ and a base radius of $$4\,m$$. Water is draining from the t

Separable Differential Equation and Slope Field Analysis

Consider the differential equation $$\frac{dy}{dx} = \frac{4*x}{y}$$ with the initial condition $$y(

Separable Differential Equation with Absolute Values

Consider the differential equation $$\frac{dy}{dx} = \frac{|x|}{y}$$ with the condition that $$y>0$$

Separable Differential Equation with Initial Condition

Consider the differential equation $$\frac{dy}{dx} = \frac{4*x}{y}$$ with the initial condition $$y(

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,

Analyzing a Motion Graph from Data

The following table represents the instantaneous velocity (in m/s) of a vehicle over a 6-second inte

Analyzing Acceleration Data from Discrete Measurements

A vehicle’s acceleration (in m/s²) is recorded at discrete time intervals as given in the table. Use

Arc Length and Average Speed for a Parametric Curve

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

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: Supply and Demand Analysis

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

Area Between Nonlinear Curves

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

Area between Parabola and Tangent

Consider the parabola defined by $$y^2 = 4 * x$$. Let $$P = (1, 2)$$ be a point on the parabola. Ans

Area Under an Exponential Decay Curve

Consider the function $$f(x)=e^{-x}$$ on the interval $$[0,1]$$. Answer the following:

Average Cost Function in Production

A factory’s cost function for producing $$x$$ units is modeled by $$C(x)=0.5*x^2+3*x+100$$, where $$

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 Speed from a Variable Acceleration Scenario

A particle moves along the x-axis under an acceleration given by $$a(t)= 3*t - 2$$ (in m/s²) and has

Average Temperature in a City

An urban planner recorded the temperature variation over a 24‐hour period modeled by $$T(t)=10+5*\si

Average Value and Monotonicity of an Oscillatory Function

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

Center of Mass of a Plate

A lamina occupies the rectangular region defined by $$0 \le x \le 2$$ and $$0 \le y \le 3$$, with a

Center of Mass of a Thin Rod

A thin rod extends from $$x=0$$ to $$x=4$$ m and has a density function $$\lambda(x)=1+\frac{\ln(x+2

Chemical Mixing in a Tank

A tank initially contains 100 liters of water. A chemical solution with a concentration of 0.5 g/l f

Determining the Arc Length of a Curve

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

Drone Motion Analysis

A drone’s vertical acceleration is modeled by $$a(t) = 6 - 2*t$$ (in m/s²) for time $$t$$ in seconds

Electrical Charge Distribution

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

Implicit Differentiation with Exponential Terms

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

Kinematics: Motion with Variable Acceleration

A particle moves along a straight line with acceleration $$a(t)=4-2*t$$ (in m/s²). The particle has

Optimizing the Thickness of a Cooling Plate

The local heat conduction efficiency at a point on a cooling plate is modeled by the function $$A(x)

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

River Cross Section Area

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

Series Convergence and Approximation

Consider the function defined by the infinite series $$f(x)=\sum_{n=0}^{\infty} \frac{(-1)^n*x^{2*n}

Temperature Modeling: Applying the Average Value Theorem

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

Total Charge in an Electrical Circuit

In an electrical circuit, the current is given by $$I(t)=5*\cos(0.5*t)$$ (in amperes), where \(t\) i

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 by Shell Method: Rotating a Region

Consider the region bounded by the curves $$y=x$$ and $$y=x^2$$. This region is rotated about the y-

Volume of a Solid of Revolution Between Curves

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

Volume of a Water Tank with Varying Cross-Sectional Area

A water tank has a cross-sectional area given by $$A(x)=3*x^2+2$$ in square meters, where $$x$$ (in

Work Done by a Variable Force

A force acting on an object is given by $$F(x)=5+3*x$$ (in newtons), where $$x$$ is the displacement

Analysis of Vector Trajectories

A particle in the plane follows the path given by $$\mathbf{r}(t)=\langle \ln(t+1), \sqrt{t} \rangle

Arc Length of a Polar Curve

Consider the polar curve given by $$r=2+\cos(\theta)$$ for $$0\le \theta \le \pi$$. Answer the follo

Comparative Particle Motion

Two particles follow the paths given by: Particle A: $$x_A(t)=t^2,\, y_A(t)=2*t$$ and Particle B: $$

Computing the Area Between Two Polar Curves

Consider the polar curves given by $$R(\theta)=3+2*\cos(\theta)$$ (outer curve) and $$r(\theta)=1+\c

Curvature of a Parametric Curve

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

Integrating a Vector-Valued Function

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

Lissajous Figures and Their Properties

A Lissajous curve is defined by the parametric equations $$x(t)=5*\sin(3*t)$$ and $$y(t)=5*\cos(2*t)

Motion Along an Elliptical Path

Consider a particle moving along the curve defined by $$ x(t)=2*\cos(t) $$ and $$ y(t)=3*\sin(t) $$

Motion of a Particle in the Plane

A particle moves in the plane with parametric equations $$x(t)=t^2-4*t$$ and $$y(t)=2*t^3-6*t^2$$ fo

Parametric and Polar Conversion Challenge

Consider the parametric equations $$x(t)= \frac{1-t^2}{1+t^2}$$ and $$y(t)= \frac{2*t}{1+t^2}$$ for

Parametric Equations and Tangent Lines

A curve is defined parametrically by $$x(t)=t^3-3t$$ and $$y(t)=t^2+2$$, where $$t$$ is a real numbe

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 $$

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.

Projectile Motion Modeled by Vector-Valued Functions

A projectile is launched with an initial velocity vector $$\vec{v}_0=\langle 10, 20 \rangle$$ (in m/

Projectile Motion via Parametric Equations

A projectile is launched with initial speed $$v_0 = 20\,m/s$$ at an angle of $$45^\circ$$. Its motio

Spiral Intersection on the X-Axis

Consider the spiral defined by the parametric equations $$x(t) = e^{-t}\cos(2*t)$$ and $$y(t)= e^{-t

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 Fields and Particle Trajectories

A particle moves in the plane with velocity given by $$\vec{v}(t)=\langle \frac{e^{t}}{t+1}, \ln(t+2

Vector-Valued Functions and Curvature

Let the vector-valued function be $$\vec{r}(t)= \langle t, t^2, t^3 \rangle$$.

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*

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