AP Calculus BC FRQ Room

Analyzing a Piecewise Defined Function Near a Boundary

Consider the function $$g(x)=\begin{cases} \frac{x^2-4}{x-2} & \text{if } x<2, \\ 2*x+1 & \text{if

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

Asymptotic Behavior and Horizontal Limits

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

Asymptotic Behavior of a Water Flow Function

In a reservoir, the net water flow rate is modeled by the rational function $$R(t)=\frac{6\,t^2+5\,t

Calculating Tangent Line from Data

The table below gives a function $$f(x)$$ representing the distance (in meters) of a moving object f

Continuity Across Piecewise‐Defined Functions with Mixed Components

Let $$ f(x)= \begin{cases} e^{\sin(x)} - \cos(x), & x < 0, \\ \ln(1+x) + x^2, & 0 \le x < 2, \\

Continuity Analysis of an Integral Function

Let $$F(x)=\int_{0}^{x} f(t)\,dt,$$ where $$ f(t)= \begin{cases} t+1, & t < 2 \\ 3, & t \ge 2 \end{

Continuity Analysis Using a Piecewise Defined Function

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

Continuity for a Logarithmic Transform Function

Consider the function $$f(x)= \ln\Bigl(\frac{e^{3x}-1}{x}\Bigr)$$ for $$x \neq 0$$ and define $$f(0)

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

Electricity Consumption Rate Analysis

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

Epsilon-Delta Proof for a Polynomial Function

Let $$f(x)=x^2+3*x+2$$. Answer the following:

Experimental Data Limit Estimation from a Table

Using the table below, estimate the behavior of a function f(x) as x approaches 1.

Exploring the Squeeze Theorem

Define the function $$ f(x)= \begin{cases} x^2*\cos\left(\frac{1}{x}\right), & x\neq0 \\ 0, & x=0

Exponential Function Limits at Infinity

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

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

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

Let $$g(x)=x^3+2*x-1$$ be defined on the interval [0, 1].

Limits Involving Infinity and Vertical Asymptotes

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

Limits of Composite Trigonometric Functions

Let $$p(x)= \frac{\sin(3x)}{\sin(5x)}$$.

Modeling with a Removable Discontinuity

A water reservoir’s inflow rate is modeled by the function $$R(t)=\frac{t^2-16}{t-4}$$ liters per mi

One-Sided Limits and Discontinuities

Consider the function $$p(x)=\begin{cases} x^2+1, & x<2, \\ 4*x-3, & x\ge2. \end{cases}$$ Answer t

One-Sided Limits and Jump Discontinuities

Consider the piecewise function $$j(x)=\begin{cases}x+2 & \text{if } x< 3,\\ 5-x & \text{if } x\ge 3

Piecewise Function Critical Analysis

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

Squeeze Theorem with a Log Function

Let $$f(x)= x\,\ln\Bigl(1+\frac{1}{x}\Bigr)$$ for $$x > 0$$. Use the Squeeze Theorem to determine $$

Squeeze Theorem with an Oscillatory Factor

Consider the function $$f(x)= x*\cos(\frac{1}{x})$$ for $$x \neq 0$$, with f(0) defined as 0. Use th

Calculating Velocity and Acceleration from a Position Function

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

Derivative Using Limit Definition

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

Graphical Estimation of Tangent Slopes

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

Growth Rate of a Bacterial Colony

The radius of a bacterial colony is modeled by $$r(t)= \sqrt{4*t+1}$$, where t (in hours) represents

Implicit Differentiation in Logarithmic Equations

Consider the relation given by $$x*\ln(y)+y*\ln(x)=5$$, where $$x>0$$ and $$y>0$$.

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 the Derivative for a Quadratic Function

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

Linearization and Tangent Approximations

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

Optimization in a Chemical Reaction

The rate of a chemical reaction is modeled by the function $$R(x)=x*e^{-x}+\ln(x+2)$$, where $$x$$ r

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: Constant Area Rectangle

A rectangle maintains a constant area of $$A = l*w = 50$$ m², where the length l and width w vary wi

Related Rates: Sweeping Spotlight

A spotlight located at the origin rotates at a constant rate of $$2 \text{ rad/s}$$. A wall is posit

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

Taylor Series of ln(x) Centered at x = 1

A researcher studies the natural logarithm function $$f(x)=\ln(x)$$ by constructing its Taylor serie

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

Traffic Flow and Instantaneous Rate

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

Water Tank: Inflow-Outflow Dynamics

A water tank initially contains $$1000$$ liters of water. Water enters the tank at a rate of $$R_{in

Bacterial Culture: Nutrient Inflow vs Waste Outflow

In a bioreactor, the nutrient inflow rate is given by $$N(t)=\ln(2*t+1)$$ (in units/min). The waste

Chain Rule and Higher-Order Derivatives

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

Coffee Cooling Dynamics using Inverse Function Differentiation

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

Combined Differentiation: Composite, Implicit, and Inverse Analysis

A complex system is modeled by the equation $$\sqrt{1+xy}+\ln(x+y)=3,$$ which relates the variables

Composite, Implicit, and Inverse: A Multi-Method Analysis

Let $$F(x)=\sqrt{\ln(5*x+9)}$$ for all x such that $$5*x+9>0$$, and let y = F(x) with g as the inver

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 of a Product Involving Inverse Trigonometric Components

Let $$m(x)=\arctan(x)+x*\arcsin\left(\frac{x}{2}\right)$$, where the domain of arcsin requires $$-2\

Differentiation of an Inverse Trigonometric Composite Function

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

Differentiation of an Inverse Trigonometric Composite Function

Let $$y = \arcsin(\sqrt{x})$$. Answer the following:

Differentiation of Inverse Trigonometric Functions

Consider the function $$f(x)= \sin(x)$$ for $$x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$$ and

Differentiation of the Inverse Function in a Mechanics Experiment

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

Engine Air-Fuel Mixture

In an engine, the fuel injection rate is modeled by the composite function $$F(t)=w(z(t))$$, where $

Fuel Tank Dynamics

A fuel storage tank is being filled by a pump at a rate given by the composite function $$P(t)=(4*t+

Implicit Differentiation and Concavity of a Logarithmic Curve

The curve defined implicitly by $$y^3 + x*y - \ln(x+y) = 5$$ is given. Use implicit differentiation

Implicit Differentiation for an Elliptical Path

An ellipse is given by the equation $$4*x^2 + y^2 = 16$$. Answer the following parts.

Implicit Differentiation of a Composite Equation

Given the implicit relation $$x^2*y + \sin(y) = x$$, answer the following:

Implicit Differentiation with Logarithmic Equation

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

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: Second Derivatives of a Circle

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

Implicit Equation with Logarithmic and Exponential Terms

The relation $$\ln(x+y)+e^{x-y}=3$$ defines y implicitly as a function of x. Answer the following pa

Inverse Function Differentiation in Economics

In an economic model, the price function $$f(x)$$ is differentiable and one-to-one, mapping the quan

Inverse Trigonometric Differentiation

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

Population Dynamics in a Fishery

A lake is being stocked with fish as part of a conservation program. The number of fish added per da

Reservoir Levels and Evaporation Rates

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

Second Derivative of an Implicit Function

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

Air Pressure Change in a Sealed Container

The air pressure in a sealed container is modeled by $$P(t)=\frac{100}{1+e^{-0.5*(t-10)}}$$, where $

Analyzing Concavity through the Second Derivative

A particle’s position is given by $$x(t)=\ln(t^2+1)$$, where $$t$$ is in seconds.

Area and Volume of Bounded Polynomials

Consider the region in the first quadrant bounded by the curves $$y = x^2$$ and $$y = 4 - x$$. Use t

Balloon Inflation and Related Rates

A spherical balloon is being inflated such that its volume increases at a constant rate of $$12\;in^

Biological Growth Rate

A bacterial culture grows according to the model $$P(t)= 500*e^{0.8*t}$$, where \(P(t)\) is the popu

Boat Crossing a River: Relative Motion

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

Comparison of Series Convergence and Error Analysis

Consider the series $$S(x)= \sum_{n=0}^{\infty} \frac{(-1)^n x^n}{2^n}$$ and $$T(x)= \sum_{n=0}^{\in

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 Pool Rates

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

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 $$

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

Inversion of an Absolute Value Function

Consider the function $$f(x)=|x-3|+2$$ with the domain restricted to $$x\ge3$$. Analyze its inverse.

L'Hôpital's Analysis

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

Linearization Approximation Problem

Given the function $$f(x)=\sqrt{x+4}$$, use linearization to approximate the value of $$f(5.1)$$.

Linearization for Approximating Function Values

Let $$f(x)= \sqrt{x}$$. Use linearization at $$x=10$$ to approximate $$\sqrt{10.1}$$. Answer the fol

Logarithmic Differentiation and Asymptotic Behavior

Let $$f(x)=\frac{(x^2+1)^3}{e^{2x}(x-1)^2}$$ for x > 1. Answer the following:

Maximizing a Rectangular Enclosure Area

A farmer has 100 m of fencing to enclose a rectangular area. Answer the following:

Maximizing Efficiency: Derivative Analysis in a Production Process

The efficiency of a production process is modeled by $$E(x)=50+10*\ln(x)-0.5*x$$, where $$x$$ repres

Maximizing the Area of an Inscribed Rectangle

A rectangle is inscribed in a semicircle of radius $$R$$, with its base along the diameter. The rect

Minimum Time to Cross a River

A person must cross a 100-meter-wide river. They can swim at 2 m/s and run along the bank at 5 m/s.

Mixing a Saline Solution: Related Rates

A tank contains a saline solution with a constant volume of 50 liters. Salt is added at a rate of 2

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

Particle Motion Along a Line with Polynomial Velocity

A particle moves along the x-axis with velocity $$v(t)=4*t^3-9*t^2+6*t-1$$ (m/s). Given that $$s(0)=

Pool Water Volume Change

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

Projectile Motion Analysis

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

Rational Function Particle Motion Analysis

A particle moves along a straight line with its position given by $$s(t)=\frac{t^2+1}{t-1}$$, where

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

Related Rates: Inflating Spherical Balloon

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

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

Solids of Revolution: Washer vs Shell Methods

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

Temperature Change of Coffee: Exponential Cooling

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

Vehicle Motion on a Curved Path

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

Vertical Projectile Motion

An object is thrown vertically upward with an initial velocity of 20 m/s and experiences a constant

Amusement Park Ride Braking Distance

An amusement park ride uses a sequence of friction pads to stop a roller coaster. The first pad diss

Analysis of a Cubic Function

Consider the function $$f(x)=x^3-6*x^2+9*x+2$$. Using this function, answer the following parts.

Analysis of an Absolute Value Function

Consider the function $$f(x)=|x^2-4|$$. Answer the following parts:

Application of Rolle's Theorem

Consider the function $$f(x) = x^2 - 4*x + 4$$ on the interval $$[0,4]$$.

Bacterial Culture with Periodic Removal

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

Concavity and Inflection Points in a Trigonometric Function

Consider the function $$f(x)=\sin(x)-\frac{1}{2}*x$$ on the interval [0, 2π]. Answer the following p

Cumulative Angular Displacement Analysis

A rotating wheel has an angular acceleration given by $$\alpha(t)=4-0.6*t$$ (in rad/s²), with an ini

Derivative Sign Chart and Function Behavior

Given the function $$ f(x)=\frac{x}{x^2+1},$$ answer the following parts:

Differentiability of a Piecewise Function

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

Echoes in an Auditorium

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

Epidemic Infection Model

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

Graph Analysis of Experimental Data

A set of experimental measurements was recorded over time. Analyze the following data regarding the

Graph Interpretation of a Function's Second Derivative

Using the provided graph of the second derivative, analyze the concavity of the original function $$

Increasing/Decreasing Intervals for a Rational Function

Consider the function $$f(x) = \frac{x^2}{x+2}$$, defined for $$x > -2$$ (with $$x \neq -2$$).

Integration of a Series Representing an Economic Model

An economist models the marginal cost by the power series $$MC(q)=\sum_{n=0}^\infty (-1)^n * \frac{q

Inverse Analysis for a Function with Multiple Transformations

Consider the function $$f(x)= 2*(x-1)^3 + 3$$ for all real numbers. Answer the following parts.

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

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

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 Motion

A car travels along a straight highway with its position given by $$s(t)=4*t^3 - 12*t^2 + 9*t + 5$$

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.

Profit Maximization in Business

A company’s profit function is given by $$P(x) = -2*x^3 + 30*x^2 - 100*x + 150$$, where x represents

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

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:

Relative Extrema Using the First Derivative Test

Consider the function $$ f(x)=e^{-x^2}.$$ Answer the following parts:

Taylor Polynomial for $$\cos(x)$$ Centered at $$x=\pi/4$$

Consider the function $$f(x)=\cos(x)$$. You will generate the second degree Taylor polynomial for f(

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

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

Water Tank Volume Analysis

Water is being added to a tank at a varying rate given by $$r(t) = 3*t^2 - 12*t + 15$$ (in liters/mi

Accumulated Change Prediction

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

Advanced U-Substitution with a Quadratic Expression

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

Antiderivative Application in Crop Growth

A crop field grows at a rate modeled by the function $$G'(t)=4*t-3$$ (in square meters per week). Th

Antiderivatives and the Fundamental Theorem

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

Arc Length of $$y=x^{3/2}$$ on $$[0,4]$$

The curve defined by $$y=x^{3/2}$$ is given for $$x\in[0,4]$$. The arc length of a curve is determin

Arc Length of a Power Function

Find the arc length of the curve $$y=\frac{1}{3}*x^{3/2}$$ on the interval $$[0,9]$$.

Area and Volume of a Region Bounded by Trigonometric Functions

Consider the curves $$y=\sin(x)$$ and $$y=\cos(x)$$ for $$0 \le x \le \frac{\pi}{4}$$. Answer the fo

Area Under the Curve for a Quadratic Function

Consider the quadratic function $$h(x)= x^2 + 2*x$$. Find the area between the curve and the $$x$$-a

Car Acceleration, Velocity, and Distance

In a physics experiment, the acceleration of a car is modeled by the function $$a(t)=4*t-1$$ (in m/s

Composite Functions and Inverses

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

Continuous Antiderivative for a Piecewise Function

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

Determining the Average Value via Integration

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

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 \

Estimating Rainfall Accumulation

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

Improper Integral and the p-Test

Determine whether the improper integral $$\int_1^{\infty} \frac{1}{x^2}\,dx$$ converges, and if it c

Integration by U-Substitution and Evaluation of a Definite Integral

Evaluate the definite integral $$\int_{0}^{1} \frac{2*t}{(t^2+1)^2}\, dt$$ by applying U-substitut

Integration via U-Substitution for a Composite Function

Evaluate the integral of a composite function and its definite form. In particular, consider the fun

Logistic Growth and Population Integration

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

Motion and Accumulation: Particle Displacement

A particle moving along a straight line has a velocity function given by $$v(t)=3*t^{2} - 12*t + 5$$

Rate of Production in a Factory

A factory has a production rate given by $$R(t)=100+20*\cos\left(\frac{\pi*t}{4}\right)$$ units per

Riemann Sum Approximations: Midpoint vs. Trapezoidal

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

Total Cost from a Marginal Cost Function

A company’s marginal cost function is given by $$MC(x)= 4*x+7$$ (in dollars per unit), where x repre

Trapezoidal Approximation for a Curved Function

Consider the function $$f(x)=x^2+2$$ on the interval [1, 5]. Answer the following:

Trapezoidal Sum Approximation for $$f(x)=\sqrt{x}$$

Consider the function $$f(x)=\sqrt{x}$$ on the interval $$[0,4]$$. Use a trapezoidal sum with 4 equa

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 by Cross-Section: Squares on a Parabolic Base

A solid has a base in the xy-plane bounded by the curves $$y=x^2$$ and $$y=4$$. Cross-sections perpe

Water Volume Accumulation with a Faulty Sensor Reading

Water flows into a container at a rate given by $$ r(t)= \begin{cases} 4t, & 0 \le t < 3, \\ 10, & t

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 a Piecewise Function with Potential Discontinuities

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

Chain Reaction in a Nuclear Reactor

A simplified model for a nuclear chain reaction is given by the logistic differential equation $$\fr

Chemical Reaction Rate and Series Approximation

A chemical reaction is modeled by the differential equation $$\frac{dC}{dt} = -0.2 * C^2$$ with the

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

Differential Equation in a Gravitational Context

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

Euler's Method Approximation

Let the differential equation be $$\frac{dy}{dx} = x+y$$ with the initial condition $$y(0)=1$$. Usin

Exact Differential Equations

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

Exponential Growth and Decay

A bacterial population grows according to the differential equation $$\frac{dy}{dt}=k\,y$$ with an i

Exponential Growth with Shifted Dependent Variable

The differential equation $$\frac{dy}{dx} = e^{x}*(y+2)$$ is used to model a growth process where th

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 13: Cooling of a Planetary Atmosphere

A planetary atmosphere cools according to Newton's Law of Cooling: $$\frac{dT}{dt}=-k(T-T_{eq})$$, w

FRQ 18: Enzyme Reaction Rates

A chemical concentration $$C(t)$$ in a reaction decreases according to the differential equation $$\

Implicit Differentiation in a Differential Equation Context

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

Logistic Growth in Populations

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

Logistic Population Growth Model

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

Maclaurin Series Solution for a Differential Equation

Given the differential equation $$\frac{dy}{dx} = y * \cos(x)$$ with initial condition $$y(0)=1$$, f

Modeling Exponential Growth

A population follows the differential equation $$\frac{dP}{dt} = k*P$$. Given that the population do

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: Temperature Change

A hot object is cooling in a room with an ambient temperature of 20°C. Measurements of the object's

Nonlinear Differential Equation with Implicit Solution

Consider the nonlinear differential equation $$\frac{dy}{dx} = \frac{x^2+y^2}{2xy}$$. An implicit so

Parameter Identification in a Cooling Process

The temperature of an object cooling in an environment at $$20^\circ C$$ is modeled by Newton's Law

Parametric Equations and Differential Equations

A particle moves in the plane along a curve defined by the parametric equations $$x(t)=\ln(t)$$ and

Pollutant Concentration in a Lake

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

Population Dynamics with Harvesting

A species population is modeled by the differential equation $$\frac{dP}{dt} = 0.2*P\left(1-\frac{P}

Population Dynamics with Harvesting

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

Radio Signal Strength Decay

A radio signal's strength $$S$$ decays with distance r according to the differential equation $$\fra

Radioactive Decay Data Analysis

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

Second-Order Differential Equation in a Mass-Spring System

A mass-spring system without damping is modeled by the differential equation $$m\frac{d^2x}{dt^2}+kx

Separable Differential Equation with Initial Condition

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

Separable Differential Equation with Parameter Identification

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

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 Equilibrium Analysis for $$\frac{dy}{dx}= y*(1-y)$$

Consider the autonomous differential equation $$\frac{dy}{dx}= y*(1-y)$$. Answer the following:

Solution Curve Sketching Using Slope Fields

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

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

Spring-Mass System with Damping

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

Water Pollution with Seasonal Variation

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

Area Between a Function and Its Tangent Line

Let $$f(x)=x^3-x$$. At the point $$x=1$$, find the tangent line to the curve and determine the area

Area Between a Parabola and a Line

Let $$f(x)= x^2$$ and $$g(x)= 2*x + 3$$. Determine the area of the region bounded by these two curve

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 Under a Parametric Curve

Consider the parametric equations $$x= t^2$$ and $$y= t^3 + t$$ for $$t \in [0,2]$$. Find the area u

Average Temperature Computation

Consider a scenario in which the temperature (in °C) in a region is modeled by the function $$T(t)=

Average Temperature Over a Day

A function modeling the temperature (in °F) throughout a day is given by $$T(t)= 3*\sin\left(\frac{\

Balloon Inflation Related Rates

A spherical balloon is being inflated such that its radius $$r(t)$$ (in centimeters) increases at a

Car Braking and Stopping Distance

A car decelerates with an acceleration given by $$a(t)=-2*t$$ (in m/s²) and has an initial velocity

Complex Integral Evaluation with Exponential Function

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

Consumer Surplus Analysis

The demand function for a product is given by $$D(p)=120-2*p$$, where \(p\) is the price in dollars.

Cost Analysis of a Water Channel

A water channel has a cross-sectional shape defined by the region bounded by $$y=\sqrt{x}$$ and $$y=

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 $

Distance Traveled from a Velocity Function

A car has a velocity function given by $$v(t)=t^2-4*t+3$$ (in m/s) for $$t$$ in seconds from 0 to 5.

Distance Traveled versus Displacement

A particle moves along a line with velocity given by $$v(t)=t^3 - 6*t^2 + 9*t$$ (in m/s) for $$t\in[

Fluid Flow in a River

The rate of water flow in a river is given by $$Q(t)=50+10*\sin\left(\frac{\pi}{6}*t\right)$$ cubic

Fluid Force on a Submerged Plate

A vertical plate submerged in water experiences a force due to fluid pressure given by $$F(y)=\rho*g

Moment of Inertia of a Thin Plate

A thin plate occupies the region bounded by the curves $$y= x$$ and $$y= x^2$$ for $$0 \le x \le 1$$

Optimization and Integration: Maximum Volume

A company designs open-top cylindrical containers to hold $$500$$ liters of liquid. (Recall that $$1

Particle Motion: Position, Velocity, and Acceleration

A particle moves along a straight line with acceleration given by $$a(t)=3*t-4$$ (in m/s²). At time

Population Growth: Cumulative Increase

A bacterial culture grows at a rate given by $$r(t)=3*e^{0.2*t}$$ (in thousands per hour), where $$t

Series Convergence and Approximation

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

Surface Area of a Solid of Revolution

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

Volume by the Shell Method: Rotating a Region

Consider the region bounded by the curve $$y=\sqrt{x}$$, the line $$y=0$$, and the vertical line $$x

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 Rotated about y = -1

The region bounded by $$y=\sqrt{x}$$ and $$y=0$$ for $$x\in[0,9]$$ is rotated about the line $$y=-1$

Volume of a Solid with Equilateral Triangle Cross Sections

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

Volume Using Washer Method

Consider the region bounded by the curves $$y=\sqrt{x}$$ and $$y=\frac{x}{2}$$. This region is rotat

Work Done by a Variable Force

A force acting on an object moving along a straight line is given by $$F(x)= 6 - x$$ (in Newtons) as

Acceleration in Polar Coordinates

An object moves in the plane with its position expressed in polar coordinates by $$r(t)= 4+\sin(t)$$

Analysis of a Cycloid

A cycloid is defined by the parametric equations $$x(t)=2*(t-\sin(t))$$ and $$y(t)=2*(1-\cos(t))$$ f

Arc Length of a Cycloid

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

Arc Length of a Parabolic Curve

The parametric curve defined by $$x(t)=t$$ and $$y(t)=t^2$$ models a portion of a parabolic path for

Arc Length of a Parametric Curve

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

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 Polar Curve

Consider the polar curve given by $$r(θ)= 1+\sin(θ)$$ for $$0 \le θ \le \pi$$. Answer the following:

Arc Length of a Quarter-Circle

Consider the circle defined parametrically by $$x(t)=\cos(t)$$ and $$y(t)=\sin(t)$$ for $$0 \le t \l

Conversion Between Polar and Cartesian Coordinates

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

Conversion of Polar to Cartesian Coordinates

Consider the polar curve $$ r=4*\cos(\theta) $$. Analyze its Cartesian equivalent and some of its pr

Converting Polar to Cartesian and Computing Slope

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

Designing a Parametric Curve for a Cardioid

A cardioid is described by the polar equation $$r(\theta)=1+\cos(\theta)$$.

Intersection of Polar Curves

Consider the polar curves given by $$r=2\sin(\theta)$$ and $$r=1+\cos(\theta)$$. Answer the followin

Logarithmic Exponential Transformations in Polar Graphs

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

Motion Along a Helix

A particle moves along a helix described by the vector-valued function $$\vec{r}(t)=<\cos(t),\, \sin

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 Egg Curve Analysis

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

Parametric Intersection of Curves

Consider the curves $$C_1: x(t)=\cos(t),\, y(t)=\sin(t)$$ for $$0 \le t \le 2\pi$$ and $$C_2: x(s)=1

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 using Parametric Equations

A projectile is launched with an initial speed of $$v_0 = 20\,\text{m/s}$$ at an angle of $$30^\circ

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 Integrals in Motion

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

Weather Data Analysis from Temperature Table

A meteorologist records the temperature (in $$^\circ C$$) at a weather station at various times (in