AP Calculus BC FRQ Room

Algebraic Simplification and Limit Evaluation

Consider the function $$f(x)= \frac{x^2-4}{x-2}$$ defined for $$x \neq 2$$ and undefined at $$x=2$$.

Application of the Squeeze Theorem

Let $$f(x)=x^2 * \sin(\frac{1}{x})$$ for $$x \neq 0$$. Answer the following:

Caffeine Metabolism in the Human Body

A person consumes a cup of coffee containing 100 mg of caffeine at the start, and then drinks one cu

Continuity of an Integral-Defined Function

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

Evaluating Limits via Rationalizing Techniques

Let $$f(x)=\frac{\sqrt{2*x+9}-3}{x}.$$ Answer the following parts.

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

Graph Analysis of a Discontinuous Function

Examine a function $$f(x)=\frac{x^2-4}{x-2}$$. A graph of the function is provided in the attached s

Graph Analysis of Discontinuities

A function $$q(x)$$ is defined piecewise as follows: $$q(x)=\begin{cases} x+2, & x<1, \\ 4, & x=1,

Horizontal and Vertical Asymptotes of a Rational Function

Let $$h(x)=\frac{2*x^2-3*x+1}{x^2-1}.$$ Answer the following:

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

Limits and Continuity of Radical Functions

Examine the function $$f(x)= \frac{\sqrt{x+4}-2}{x}$$ defined for $$x \neq 0$$.

Limits at Infinity and Horizontal Asymptotes

Consider the rational function $$g(x)= \frac{4*x^3-x+2}{2*x^3+3*x^2-5}$$.

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

Oscillatory Functions and Discontinuity

Consider the function $$f(x)= \begin{cases} \sin\left(\frac{1}{x}\right), & x\neq0 \\ 0, & x=0 \end{

Piecewise Function Continuity

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

Population Growth and Limits

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

Rational Function Limit and Continuity

Consider the function $$f(x)=\frac{x^2+5*x+6}{x+3}$$ defined for $$x\neq -3$$. Notice that the funct

Real-World Temperature Sensor Analysis

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

Resistor Network Convergence

A resistor network is constructed by adding resistors in a ladder configuration. The resistance adde

Trigonometric Limits Analysis

Evaluate the following limits involving trigonometric functions.

Analyzing Motion Through Derivatives

A car’s position as a function of time is given by $$s(t) = 4*t^3 - 12*t^2 + 9*t + 5,$$ where $$s

Application of Derivative to Relative Rates in Related Variables

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

Applying the Quotient Rule

Let the function $$R(x)=\frac{x^2+1}{2*x-1}$$ represent a ratio used to gauge the rate of return on

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

Chain Rule Verification with a Power Function

Let $$f(x)= (3*x+2)^4$$.

Composite Function and Chain Rule Application

Consider the function $$h(x)=\sin(2*x^2+3)$$. Using the chain rule, answer the following parts:

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

Compound Exponential Rate Analysis

Consider the function $$f(t)=\frac{e^{2*t}}{1+t}$$, which arises in compound growth models. Analyze

Cooling Model Rate Analysis

The temperature of a cooling object is modeled by $$T(t)=e^{-2*t}+\ln(t+3)$$, where $$t$$ is time in

Derivative Estimation from a Graph

A graph of a function $$f(x)$$ is provided in the stimulus. Using the graph, answer the following pa

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 Piecewise-Defined Function

Consider the piecewise function $$f(x)=\begin{cases}x^2+2*x, & x \le 3 \\ 4*x-5, & x > 3 \end{cases}

Differentiation of Inverse Functions

Let $$f(x)=3*x+2$$ and let $$f^{-1}(x)$$ denote its inverse function. Answer the following:

Epidemic Spread Rate: Differentiation Application

The number of infected individuals in an epidemic is modeled by $$I(t)= \frac{200}{1+e^{-0.5(t-5)}}$

Exponential Population Growth in Ecology

A certain species in a reserve is observed to grow according to the function $$P(t)=1000*e^{0.05*t}$

Graph Behavior of a Log-Exponential Function

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

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: Square Root Equation

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

Instantaneous vs. Average Rate of Change

Consider the trigonometric function $$f(x)= \sin(x)$$.

Logarithmic Differentiation in Temperature Modeling

The temperature distribution along a rod is modeled by the function $$T(x)=\ln(5*x^2+1)*e^{-x}$$. He

Oil Spill Containment

Following an oil spill, containment efforts recover oil at a rate of $$O_{in}(t)=40-2*t$$ (accumulat

Population Dynamics: Derivative and Series Analysis

A town's population is modeled by the continuous function $$P(t)= 1000e^{0.04t}$$, where t is in yea

Product and Quotient Rule Application

Consider the function $$f(x)=\frac{x*\ln(x)}{e^{x}+2}$$, defined for $$x>0$$. Analyze its behavior u

Related Rates: Expanding Balloon

A spherical balloon is being inflated so that its volume $$V$$ (in m³) and radius $$r$$ (in m) satis

Renewable Energy Storage

A battery storage system experiences charging at a rate of $$C(t)=50+10\sin(0.5*t)$$ kWh and dischar

Tangent Line to a Logarithmic Function

Consider the function $$f(x)= \ln(x+1)$$.

Testing Differentiability at a Junction Point

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

Tracking a Car's Velocity

A car moves along a straight road according to the position function $$s(t)=t^3 - 6*t^2 + 9*t + 2$$,

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

Urban Population Flow

A city’s population changes due to migration. The inflow of people is modeled by $$M_{in}(t)=8-0.5*t

Using the Limit Definition for a Non-Polynomial Function

Consider the function $$f(x)= \sqrt{x+4}$$ for $$x\ge -4$$. Answer the following:

Water Tank: Inflow-Outflow Dynamics

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

Analysis of a Piecewise Function with Discontinuities

Consider the piecewise function $$ f(x) = \begin{cases} 2*x+1, & x < 1, \\ 3, & 1 \le x \le 2, \\ \s

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 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 and Implicit Differentiation with Trigonometric Functions

Consider the equation $$\sin(x*y)+x^2=y^2$$ which relates x and y. Answer the following parts:

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 Function: Polynomial Exponent

Consider the function $$ f(x)= (2*x^2+3*x-5)^3 $$. Analyze the function's derivative and behavior.

Composite Functions in a Biological Model

In a biological model, the concentration of a substance is given by $$P(x)=e^{-\sqrt{x^2+1}}$$, wher

Derivative of an Inverse Function with a Quadratic

Consider the function $$f(x) = x^2 + 6*x + 9$$ defined on $$x \ge -3$$. Let $$g$$ be the inverse of

Differentiation Involving Absolute Values and Composite Functions

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

Differentiation of an Inverse Function

Let f be a differentiable and one-to-one function with inverse $$f^{-1}$$. Suppose that $$f(3)=7$$ a

Differentiation of an Inverse Trigonometric Function

Define $$h(x)= \arctan(\sqrt{x})$$. Answer the following:

Differentiation of Composite Exponential and Trigonometric Functions

Let $$f(x) = e^{\sin(x^2)}$$ be a composite function. Differentiate $$f(x)$$ and simplify your answe

Financial Flow Analysis: Investment Rates

An investment fund experiences deposits at a rate modeled by the composite function $$D(t)=g(h(t))$$

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 in a Non-Standard Function

Consider the equation $$x^2*y + \sin(y) = x$$, which implicitly defines $$y$$ as a function of $$x$$

Implicit Differentiation in a Radical Equation

The relationship between $$x$$ and $$y$$ is given by $$\sqrt{x} + \sqrt{y} = 6$$.

Implicit Differentiation in an Economic Model

A company’s production is modeled by the implicit relationship $$x*y^2 + \ln(x+y) = 10$$, where $$x$

Implicit Differentiation of a Circle

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

Implicit Differentiation on an Ellipse

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

Implicit Differentiation with Exponential and Trigonometric Mix

Consider the equation $$e^{x*y} + \sin(x) - y = 0$$. Differentiate implicitly with respect to $$x$$

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

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 Derivative in a Cubic Function

Let $$f(x)= x^3+ 2*x - 1$$, a one-to-one differentiable function. Its inverse function is denoted as

Inverse Function Differentiation for a Quadratic Function

Let $$ f(x)= (x+1)^2 $$ with the domain $$ x\ge -1 $$. Consider its inverse function $$ f^{-1}(y) $$

Inverse Function Differentiation in a Trigonometric Context

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

Inverse Function Differentiation in Economics

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

Inverse Function Differentiation in Thermodynamics

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

Power Series Representation and Differentiation of a Composite Function

Let $$f(x)= \sin(x^2)$$ and consider its Maclaurin series expansion.

Rainwater Harvesting System

A rainwater harvesting system collects water in a reservoir. The inflow rate is given by the composi

Shadow Length and Related Rates

A 1.8 m tall person walks away from a 4 m tall lamppost at a speed of 1.2 m/s. Let $$x$$ be the dist

Taylor Polynomial and Error Bound for a Trigonometric Function

Let $$f(x) = \cos(2*x)$$. Develop a second-degree Taylor polynomial centered at 0, and analyze the a

Analysis of a Piecewise Function with Discontinuities

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

Analyzing a Production Cost Function

A company's cost function for producing goods is given by $$C(x)=x^3-12x^2+40x+100$$, where x repres

Application of L’Hospital’s Rule

Evaluate the limit $$\lim_{x \to \infty} \frac{5x^3 - 2x + 1}{3x^3 + 7}$$ using L’Hospital’s Rule.

Area Under a Curve: Definite Integral Setup

Consider the function $$f(x) = x^3 - 4x + 1$$ on the interval $$[0, 3]$$. Explore the area between t

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

Conical Tank Water Flow

Water is pumped into a conical tank at a rate of $$\frac{dV}{dt}=9\text{ ft}^3/\text{min}$$. The tan

Cooling Coffee Temperature Change

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

Cooling Coffee: Temperature Change

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

Cycloid Tangent Line

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

Differentials in Engineering: Beam Stress Analysis

The stress S (in Pascals) experienced by an engineering beam under load is modeled by $$S(x)=0.02*x^

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

Economic Marginal Cost Analysis

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

Economic Model: Revenue and Cost Rates

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

Ellipse Tangent Line Analysis

Consider the ellipse given by the implicit equation $$9*x^2 + 4*y^2 = 36$$. Answer the following par

Estimating Rate of Change from Table Data

The following table shows the velocity (in m/s) of a car at various times recorded during an experim

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 Rectangle: Related Rates

A rectangle has a length $$l$$ and width $$w$$ that are changing with time. At a certain moment, the

Exponential Cooling Rate Analysis

The temperature of a cup of coffee is modeled by $$T(t)=70+50e^{-0.05t}$$ (in °F), where t is in min

Implicit Differentiation in a Tank Filling Problem

A tank's volume and liquid depth are related by $$V=10y^3$$, where y (in meters) is the depth. Water

Industrial Mixer Flow Rates

In an industrial mixer, an ingredient is added at a rate of $$I(t)=7t$$ (kg per minute) and is consu

Inflating Spherical Balloon

A spherical balloon is being inflated so that its volume increases at a constant rate of $$\frac{dV}

Infrared Sensor Distance Analysis

An infrared sensor measures the distance to a moving target using the function $$d(t)=50*e^{-0.2*t}+

L'Hospital's Rule for Indeterminate Limits

Evaluate the limit $$\lim_{x \to \infty} \frac{5*x^3 - 4*x^2 + 1}{7*x^3 + 2*x - 6}$$ using L'Hospita

Linearization to Estimate Change in Electrical Resistance

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

Marginal Cost Analysis

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

Parametric Motion in the Plane

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

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)=

Particle Motion Analysis

A particle's position along a line is given by $$s(t) = t^3 - 6*t^2 + 9*t + 2$$, where $$t$$ is meas

Particle on Implicit Curve

A particle moves so that its coordinates $$(x(t), y(t))$$ always satisfy the equation $$x^2 + x*y +

Savings Account Dynamics

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

Vertical Projectile Motion

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

Analysis of a Piecewise Function's Differentiability and Extrema

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

Analysis of an Exponential Function

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

Analysis of Relative Extrema and Increasing/Decreasing Intervals

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

Average vs. Instantaneous Profit Rate

A company’s profit is modeled by the function $$P(t)= 0.2*t^3 - 3*t^2 + 10*t$$, where $$t$$ is the t

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

Car Motion: Velocity and Total Distance

A car’s position along a straight road is modeled by $$s(t) = t^3 - 6*t^2 + 9*t + 15$$ (in meters),

Concavity Analysis in a Revenue Model

A company’s revenue (in thousands of dollars) is modeled by the function $$R(x) = -0.5*x^3 + 6*x^2 -

Concavity and Inflection Points

The function $$f(x) = x^3 - 6*x^2 + 9*x + 2$$ models a certain process. Use the second derivative to

Concavity and Inflection Points

Let $$f(x)=x^3-6x^2+9x+2.$$ Answer the following parts:

Economic Equilibrium and Implicit Differentiation

An economic equilibrium is modeled by the implicit equation $$e^{p}*q + p^2 = 100$$, where \( p \) r

Energy Consumption Rate Model

A household's energy consumption rate (in kW) is modeled by $$E(t) = 2*t^2 - 8*t + 10$$, where t is

Epidemic Infection Model

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

Extreme Value Analysis of a Logarithmic-Exponential-Polynomial Function

Examine the function $$f(x)= x^2\,e^{-x} + \ln(x)$$ defined for $$x > 0$$. Apply methods to find its

Extreme Value Theorem in a Polynomial Function

Consider the function $$h(x)=x^4-8*x^2+16$$ defined on the closed interval $$[-3,3]$$. Answer the fo

Finding Local Extrema for an Exponential-Logarithmic Function

The function $$g(x)= e^x\ln(x)$$ is defined for $$x > 0$$. Analyze the function as follows:

Fractal Tree Branch Lengths

A fractal tree is constructed as follows: The trunk has a length of 10 meters. At each generation, e

Graph Analysis of a Logarithmic Function

Consider the function $$g(x)= \ln(x) - \frac{1}{x}$$ defined for $$x>0$$. Analyze its behavior and g

Graph Analysis of Experimental Data

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

Investment Portfolio Dividends

A company pays annual dividends that form an arithmetic sequence. The dividend in the first year is

Investment with Increasing Contributions and Interest

An investor begins with an account balance of $$5000$$ dollars which earns an annual interest rate o

Logarithmic Function Derivative Analysis

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

Optimization in a Geometric Setting: Garden Design

A farmer is designing a rectangular garden adjacent to a river. No fence is needed along the river s

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

Optimization: Maximizing Rectangular Area with a Fixed Perimeter

A farmer has 300 meters of fencing to enclose a rectangular field that borders a straight river (no

Related Rates: Draining Conical Tank

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

Roller Coaster Height Analysis

A roller coaster's height (in meters) as a function of time (in seconds) is given by $$h(t) = -0.5*t

Ski Resort Snow Accumulation and Melting

At a ski resort, snow accumulates naturally at a rate given by $$S(t)=50*\exp(-0.1*t)$$ cm/hour due

Stress Analysis in Engineering Structures

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

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

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

Taylor Series in Differential Equations: $$y'(x)=y(x)\cos(x)$$

Consider the initial value problem $$y'(x)= y(x)\cos(x)$$ with $$y(0)=1$$. Assume a power series sol

Accumulated Displacement from a Piecewise Velocity Function

A particle moves along a line with velocity given by $$ v(t)= \begin{cases} t^2-1, & 0 \le t < 2, \\

Analyzing an Invertible Cubic Function

Consider the function $$f(x) = x^3 + 2*x + 1$$ defined for all $$x$$. Answer the following questions

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

Area Between Curves

Consider the curves $$f(x)=x^2$$ and $$g(x)=4*x - 3$$. These curves intersect and enclose a region.

Area Under a Parametric Curve

A curve is defined parametrically by $$x(t)=t^2$$ and $$y(t)=t^3-3*t$$ for $$t \in [-2,2]$$.

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 \

Average Value of an Exponential Function

For the function $$f(x)= x*e^{-x}$$, determine the average value on the interval $$[0,2]$$. Answer t

Cost Accumulation via Integration

A manufacturing process has a marginal cost function given by $$MC(x)= 4 + 3*x$$ dollars per item, w

Definite Integration of a Polynomial Function

For the function $$f(x)=5*x^{3}$$ defined on the interval $$[1,2]$$, determine the antiderivative an

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 \

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

Economic Surplus: Area between Supply and Demand Curves

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

Error Estimation in Riemann Sum Approximations

Consider the function $$f(x)=\sqrt{x}$$ on the interval $$[1,9]$$. When approximating the definite i

Evaluating a Trigonometric Integral

Evaluate the integral $$\int_{0}^{\pi/2} \cos(3*x)\,dx$$.

Evaluating an Integral Involving an Exponential Function

Evaluate the definite integral $$\int_{0}^{\ln(4)} e^{x}\,dx$$.

Integration Involving Inverse Trigonometric Functions

Consider the function $$f(x)= \tan^{-1}(x)$$. Answer the following questions regarding its inverse a

Integration Using U-Substitution

Evaluate the indefinite integral $$\int (4*x+2)^5\,dx$$ using u-substitution.

Population Growth: Rate to Accumulation

A population's growth rate (in thousands of individuals per year) is modeled by $$P'(t)=2*t - 1$$ fo

Population Model Inversion and Accumulation

Consider the logistic-type function $$f(t)= \frac{8}{1+e^{-t}}$$, representing a population model, d

Radioactive Decay: Accumulated Decay

A radioactive substance decays according to $$m(t)=50 * e^(-0.1*t)$$ (in grams), with time t in hour

Related Rates: Expanding Circular Ripple

A stone is dropped into a still pond, producing a circular ripple. The radius $$r$$ of the ripple (i

Riemann Sum from a Table: Plant Growth Data

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

Trapezoidal Approximation of a Definite Integral from Tabular Data

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

Volume Accumulation in a Reservoir

A reservoir is being filled at a rate given by $$R(t)= e^{2*t}$$ liters per minute. Determine the t

Water Accumulation in a Reservoir

A reservoir receives water at an inflow rate modeled by $$r(t)=\frac{20}{t+1}$$ (in cubic meters per

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

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

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

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

Drug Concentration in the Bloodstream

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

Dye Dilution in a Stream

A river has dye added at a constant rate of $$0.5$$ kg/min, and the dye is removed at a rate proport

Euler's Method Approximation

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

Exact Differential Equation

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

Exact Differential Equations and Integrating Factors

Consider the differential equation $$ (2*x*y + y^2)\,dx + (x^2 + 2*x*y - 1)\,dy = 0$$. Answer the fo

Exponential Growth via Slope Field Analysis

Consider the differential equation $$\frac{dy}{dx} = x * y$$ with the initial condition $$y(0)=2$$.

FRQ 8: RC Circuit Analysis

In an RC circuit, the voltage across the capacitor, $$V(t)$$, satisfies the differential equation $$

Implicit Differentiation and Homogeneous Equation

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

Investment Growth with Nonlinear Dynamics

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

Logistic Model in Population Dynamics

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

Modeling Exponential Growth

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

Newton's Law of Cooling

A hot liquid is cooling in a room. The temperature $$T(t)$$ (in degrees Celsius) of the liquid at ti

Particle Motion with Variable Acceleration

A particle moving along a straight line has an acceleration given by $$a(t) = 6 - 4*t$$ (in m/s²). A

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 in Ecology

Consider the differential equation that models the growth of a fish population in a lake: $$\frac{dP

Population Growth with Harvesting

A fish population in a lake is modeled by the differential equation $$\frac{dP}{dt}= rP - H$$, where

Population Growth with Logistic Differential Equation

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

Radio Signal Strength Decay

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

Radioactive Decay

A radioactive substance decays according to the law $$\frac{dN}{dt} = -k*N$$. The half-life of the s

Separable DE with Trigonometric Component

Solve the differential equation $$\frac{dy}{dx}=\sin(x)*\cos(y)$$ with the initial condition $$y(0)=

Separable Differential Equation and Slope Field Analysis

Consider the differential equation $$\frac{dy}{dx} = x*y$$ with the initial condition $$y(0)=2$$. A

Separation of Variables with Trigonometric Functions

Solve the differential equation $$\frac{dy}{dx} = \frac{\sin(x)}{1+y^2}$$ by using separation of var

Series Solution for a Differential Equation

Consider the differential equation $$\frac{dy}{dx}= y^2 \sin(x)$$ with the initial condition $$y(0)=

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

Variable Carrying Capacity in Population Dynamics

In a modified logistic model, the carrying capacity of a population is time-dependent and given by $

Analysis of Particle Motion in the Plane

A particle’s position is given by the vector function $$\mathbf{r}(t)=\langle e^{-t},\,\sin(t)\rangl

Analyzing a Motion Graph from Data

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

Arc Length of a Cable

A cable hanging in a particular configuration follows the curve $$y=\ln(x+1)$$ for $$x\in[0,4]$$. De

Arc Length of a Logarithmic Curve

Consider the curve defined by $$y = \ln(\sec(t))$$ for $$t$$ in the interval $$[0,\pi/4]$$. Determin

Area Between Exponential Curves

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

Area Between Two Curves in a Water Channel

A channel cross‐section is defined by two curves: the upper boundary is given by $$f(x)=12-0.8*x$$ a

Average Population in a Logistic Model

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

Average Temperature Analysis

A research team models the ambient temperature in a region over a 24‐hour period with the function $

Average Temperature Over a Day

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

Average Value of a Trigonometric Function

Let $$f(x)=C+\cos(2*x)$$ be defined on the interval $$[0,\pi]$$. Answer the following:

Economic Analysis: Consumer and Producer Surplus

In a market analysis, the demand and supply for a product are modeled by the equations: Demand: $$D(

Inverse Function Analysis

Consider the function $$f(x)=3*x^3+2$$ defined for all real numbers.

Logarithmic and Exponential Equations in Integration

Let $$f(x)=\ln(x+2)$$. Consider the expression $$\frac{1}{6}\int_0^6 [f(x)]^2dx=k$$, where it is giv

Particle Motion Analysis with Variable Acceleration

A particle moving along a straight line has an acceleration given by $$a(t)=4*e^{-t}-\sin(t)$$ (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.

Total Change in Temperature Over Time (Improper Integral)

An object cools according to the function $$\Delta T(t) = e^{-0.5*t}$$, where $$t\ge 0$$ is time in

Total Distance Traveled with Changing Velocity

A runner’s velocity is given by $$v(t)=3*(t-1)*(t-4)$$ m/s for $$0 \le t \le 5$$ seconds. Note that

Volume about a Vertical Line using Two Methods

A region in the first quadrant is bounded by $$y=x$$, $$y=0$$, and $$x=2$$. This region is rotated a

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 Square Cross Sections

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

Volume of a Solid with Variable Cross Sections

A solid has a cross-sectional area perpendicular to the x-axis given by $$A(x)=4-x^2$$ for $$x\in[-2

Volume with Equilateral Triangle Cross Sections

The region bounded by $$y=4-x^2$$ and $$y=0$$ for $$x\in[-2,2]$$ serves as the base of a solid. Cros

Work Done by a Variable Force

A force acting on an object is given by the function $$F(x)=3*x^2$$ (in Newtons). The object moves a

Acceleration Analysis in a Vector-Valued Function

Consider a particle whose position is given by $$ r(t)=\langle \sin(2*t),\; \cos(2*t) \rangle $$ for

Analyzing Oscillatory Motion in Parametric Form

The motion of an oscillating particle is given by $$x(t)=e^{-t}\cos(2t)$$ and $$y(t)=e^{-t}\sin(2t)$

Arc Length of a Vector-Valued Curve

A vector-valued function is given by $$\mathbf{r}(t)=\langle e^t,\, \sin(t),\, \cos(t) \rangle$$ for

Area Between Polar Curves

Consider the polar curves defined by $$r_1= 4$$ and $$r_2= 2+2\cos(\theta)$$. Find the area of the r

Area Between Two Polar Curves

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

Area Between Two Polar Curves

Consider the two polar curves $$r_1(θ)= 3+\cos(θ)$$ and $$r_2(θ)= 1+\cos(θ)$$. Answer the following:

Area Enclosed by a Polar Curve

Consider the polar curve defined by $$r=2+2\sin(\theta)$$. This curve is a cardioid. Answer the foll

Area Enclosed by a Polar Curve: Lemniscate

The lemniscate is defined by the polar equation $$r^2=8\cos(2\theta)$$.

Component-Wise Integration of a Vector-Valued Function

Given the acceleration vector $$\textbf{a}(t)= \langle 3\cos(t), -3\sin(t) \rangle$$, answer the fol

Comprehensive Motion Analysis Using Parametric and Vector-Valued Functions

A particle moves with position given by $$ r(t)=\langle t*e^{-t},\;\ln(1+t) \rangle $$ for $$ t\ge0

Curvature and Oscillation in Vector-Valued Functions

Given the vector function $$\vec{r}(t)=\langle \ln(t+1), \cos(t) \rangle$$ for $$t \ge 0$$, answer t

Designing a Race Track with Parametric Equations

An engineer designs a race track whose left and right boundaries are described by the curves $$C_1:

Double Integration in Polar Coordinates for Mass Distribution

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

Exponential Decay in Vector-Valued Functions

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

Intersection of Parametric Curves

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

Kinematics in Polar Coordinates

An object moves so that its position in polar coordinates is given by $$r(t)= 4 - t$$ and $$\theta(t

Kinematics on a Circular Path

A particle moves along a circle given by the parametric equations $$x(t)= 5*\cos(t)$$ and $$y(t)= 5*

Modeling with Polar Data

A researcher collects the following polar coordinate data for a phenomenon.

Optimization of Walkway Slope with Fixed Arc Length

A walkway is designed with its shape given by the parametric equations $$x(t)= t$$ and $$y(t)= c*t*(

Parameter Values from Tangent Slopes

A curve is defined parametrically by $$x(t)=t^2-4$$ and $$y(t)=t^3-3t$$. Answer the following:

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

Parametric Slope and Arc Length

Consider the parametric curve defined by $$x(t)= t-\ln(t)$$ and $$y(t)= t\cdot\ln(t)$$ for $$t > 1$$

Polar Coordinates and Area Computation

Examine the polar curve $$r = 2 + \sin(2\theta)$$ and determine the area of the region it encloses.

Self-Intersection in a Parametric Curve

Consider the parametric curve defined by $$ x(t)=t^2-t $$ and $$ y(t)=t^3-3*t $$. Investigate whethe

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

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 3D Projectile Motion

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