AP Calculus BC FRQ Room

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 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

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 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 Piecewise Functions with Parameters

A function is defined piecewise by $$f(x)=\begin{cases}a*x^2+3,& x<1,\\ b*x+1,& x\ge 1.\end{cases}$$

Evaluating a Limit Involving a Radical and Trigonometric Component

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

Evaluating Limits Involving Absolute Value Functions

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

Evaluating Limits Involving Radical Expressions

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

Exploring Removable and Nonremovable Discontinuities

Consider the function $$f(x)=\frac{(x-2)(x+3)}{(x-2)}$$ for $$x\neq2$$ and $$f(2)=7$$. Answer the fo

Exponential Function Limits at Infinity

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

Implicitly Defined Curve and Its Tangent Line

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

Intermediate Value Theorem in Water Tank Levels

The water volume \(V(t)\) in a tank is a continuous function on the interval \([0,10]\) minutes. It

Internet Data Packet Transmission and Error Rates

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

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 Involving Absolute Value

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

Limits Involving Absolute Value Functions

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

Limits with Composite Logarithmic Functions

Consider the function $$t(x)=x*\ln(x)$$ defined for x > 0.

Manufacturing Cost Sequence

A company's per-unit manufacturing cost decreases by $$50$$ dollars each year due to economies of sc

Mixed Function Inflow Limit Analysis

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

Piecewise Function Continuity and Differentiability

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

Removable Discontinuity in a Rational Function

Consider the function given by $$f(x)= \frac{(x+3)*(x-1)}{(x-1)}$$ for $$x \neq 1$$. Answer the foll

Telecommunications Signal Strength

A telecommunications tower emits a signal whose strength decreases by $$20\%$$ for every additional

Water Tank Flow Analysis

A water tank receives water from an inlet and drains water through an outlet. The inflow rate is giv

Water Treatment Plant Discontinuity Analysis

A water treatment plant monitors the inflow to a reservoir. Due to sensor calibration, the inflow ra

Acceleration and Jerk in Motion

The position of a car is given by $$s(t)=t^3-6*t^2+9*t+2$$, where $$t$$ is time in seconds and $$s(t

Analysis of Higher-Order Derivatives

Let $$f(x)=x*e^{-x}$$ model the concentration of a substance over time. Analyze both the first and s

Bacteria Culturing in a Bioreactor

In a bioreactor, the bacterial inflow (growth) rate is given by $$B_{in}(t)=\frac{15}{1+e^{-0.3*(t-5

Calculating Velocity and Acceleration from a Position Function

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

Car Motion and Critical Velocity

The position of a car is described by $$s(t)=t^3 - 12*t^2 + 36*t$$ (with $$s$$ in meters and $$t$$ i

Composite Function Behavior

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

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

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

Continuous Compound Interest Analysis

For an investment, the amount at time $$t$$ (in years) is modeled by $$A(t)=P*e^{r*t}$$, where $$P$$

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

Derivatives of a Rational Function

Consider the function $$g(x)= \frac{2*x^3 - 1}{x^2+4}$$. Use differentiation rules to answer the fol

Differentiation and Linear Approximation for Error Estimation

Let $$f(x) = \ln(x)*x^2$$. Use differentiation and linear approximation to estimate changes in the f

Exploration of Derivative Notation and Higher Order Derivatives

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

Exploration of the Definition of the Derivative as a Limit

Consider the function $$f(x)=\frac{1}{x}$$ for $$x\neq0$$. Answer the following:

Icy Lake Evaporation and Refreezing

An icy lake gains water from melting ice at a rate of $$M_{in}(t)=5+0.2*t$$ liters per hour and lose

Implicit Differentiation: Square Root Equation

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

Instantaneous Rate of Change of a Polynomial Function

Consider the function $$f(x)=2*x^3 - 5*x^2 + 3*x - 7$$ which represents the position (in meters) 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 $$f'(x)$$.

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

Quotient Rule in a Chemical Concentration Model

The concentration of a drug in the bloodstream is modeled by $$C(t)=\frac{t+2}{t^2+1}$$ (in mg/L), w

Rate Function Involving Logarithms

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

River Flow Dynamics

A river experiences seasonal variations. Its inflow is modeled by $$F_{in}(t)=30+10\cos((\pi*t)/12)$

Tangent Line Approximation

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

Tangent Line to a Curve

Consider the function $$f(x)=\sqrt{x+4}$$ modeling a physical quantity. Analyze the behavior at $$x=

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

Temperature Change Rate

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

Using Taylor Series to Approximate the Derivative of sin(x²)

A physicist is analyzing the function $$f(x)=\sin(x^2)$$ and requires an approximation for its deriv

Using the Product Rule in Economics

A company’s revenue function is given by $$R(x)=x*(100-x)$$, where $$x$$ (in hundreds) represents th

Widget Production Rate

A widget manufacturing plant produces widgets according to the function $$P(t)=4*t^2 - 3*t + 10$$ wh

Biological Growth Model Differentiation

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

Chain Rule and 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

Composite Function Rates in a Chemical Reaction

In a chemical reaction, the concentration of a substance at time $$t$$ is given by $$C(t)= e^{-k*(t+

Design Optimization for a Cylindrical Can

A manufacturer wants to design a cylindrical can that holds a fixed volume of $$V = 1000$$ cm³. The

Differentiation of an Arctan Composite Function

For the function $$f(x) = \arctan\left(\frac{3*x}{x+1}\right)$$, differentiate with respect to $$x$$

Differentiation of an Inverse Trigonometric Form

Consider the function $$f(x)=\arcsin\left(\frac{2*x}{1+x^2}\right)$$. Answer the following parts.

Implicit Differentiation and Inverse Functions Combined

Consider the function defined implicitly by the equation $$\sin(y) + y\cos(x) = x.$$ Answer the fo

Implicit Differentiation in a Conic Section

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

Implicit Differentiation in a Logarithmic Equation

Given the equation $$\ln(x*y) + x - y = 0$$, answer the following:

Implicit Differentiation in a Nonlinear Equation

Consider the equation $$x*y + y^3 = 10$$, which defines y implicitly as a function of x.

Implicit Differentiation in Economic Equilibrium

In a market, the relationship between the price $$x$$ (in dollars) and the demand $$y$$ (in thousand

Implicit Differentiation Involving Exponential Functions

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

Implicit Differentiation Involving Logarithms

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

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 of an Ellipse

The ellipse is given 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:

Inverse Analysis of Cubic Plus Linear Function

Consider the function $$f(x)=x^3+x$$ defined for all real numbers. Analyze its inverse function.

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 Radical Context

Let $$f(x)= \sqrt{1+ x^3}$$ and let $$g$$ be its inverse function. Answer the following parts:

Logarithmic and Composite Differentiation

Let $$g(x)= \ln(\sqrt{x^2+1})$$.

Maximizing the Garden Area

A rectangular garden is to be built alongside a river, so that no fence is needed along the river. T

Navigation on a Curved Path: Boat's Eastward Velocity

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

Optimization with Composite Functions - Minimizing Fuel Consumption

A car's fuel consumption (in liters per 100 km) is modeled by $$F(v)= v^2 * e^{-0.1*v}$$, where $$v$

Rainwater Harvesting System

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

Rate of Change in a Biochemical Process Modeled by Composite Functions

The concentration of a biochemical in a cell is modeled by the function $$C(t) = \sin(0.2*t) + 1$$,

Reservoir Levels and Evaporation Rates

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

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

Bacterial Population Growth Analysis

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

Biological Growth Rate

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

Biology: Logistic Population Growth Analysis

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

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

Chemical Reaction Temperature Change

In a laboratory experiment, the temperature T (in °C) of a reacting mixture is modeled by $$T(t)=20+

Continuity in a Piecewise-Defined Function

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

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

Estimation Error with Differentials

Let $$f(x)=x^3$$. Use differentials to estimate the value of $$f(2.05)$$ and determine the approxima

Expanding Circular Ripple

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

Instantaneous vs. Average Speed in a Race

An athlete’s displacement during a 100 m race is modeled by $$s(t)=2*t^3-t^2+1$$, where $$s(t)$$ is

Interpreting Derivatives from Experimental Concentration Data

An experiment records the concentration (in moles per liter) of a substance over time (in minutes).

Limits and L'Hôpital's Rule Application

Consider the function $$f(x)=\frac{5*x^3-4*x^2+1}{7*x^3+2*x-6}$$. Answer the following:

Linear Account Growth in Finance

The amount in a savings account during a promotional period is given by the linear function $$A(t)=1

Marginal Cost and Revenue Analysis

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

Minimizing Travel Time in Mixed Terrain

A hiker travels from point A to point B. On a flat plain the hiker walks at 5 km/h, but on an uphill

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

Optimization of a Rectangular Enclosure

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

Optimization of Material Cost for a Pen

A rectangular pen is to be built against a wall, requiring fencing on only three sides. The area of

Optimizing Factory Production with Log-Exponential Model

A factory's production is modeled by $$P(x)=200x^{0.3}e^{-0.02x}$$, where x represents the number of

Related Rates in a Conical Water Tank

Water is being pumped into a conical tank at a rate of $$2\;m^3/min$$. The tank has a height of 6 m

Related Rates: Expanding Circular Oil Spill

In a coastal region, an oil spill is spreading uniformly and forms a circular region. The area of th

Revenue Function and Marginal Revenue

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

Road Trip Distance Analysis

During a road trip, the distance traveled by a car is given by $$s(t)=3*t^2+2*t+5$$, where $$t$$ is

Series Approximation of a Temperature Function

The temperature in a chemical reaction is modeled by $$T(t)= 100 + \sum_{n=1}^{\infty} \frac{(-1)^n

Series Convergence and Approximation for f(x) Centered at x = 2

Consider the function $$f(x)= \sum_{n=0}^{\infty} \frac{(-1)^n (x-2)^{2*n}}{n+1}$$. Answer the follo

Series Solution of a Drug Concentration Model

The drug concentration in the bloodstream is modeled by $$C(t)= \sum_{n=0}^{\infty} \frac{(-t)^n}{n!

Shadow Lengthening with a Lamp Post

A 2.5 m tall lamp post casts light on a 1.8 m tall man who walks away from the post at a constant sp

Spherical Balloon Inflation

A spherical balloon is being inflated such that its volume increases at a constant rate of $$\frac{d

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

Analysis of a Function with Oscillatory Behavior

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

Application of Rolle's Theorem to a Trigonometric Function

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

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

Area Between Curves and Rates of Change

An irrigation canal has a cross-sectional shape described by \( y=4-x^2 \) for \( |x| \le 2 \). The

Concavity and Inflection Points

Examine the function $$p(x)= x^3-3*x^2-9*x+30$$ to determine its concavity and any inflection points

Convergence and Series Approximation of a Simple Function

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

Curve Sketching Using Derivatives

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

Determining Convergence and Error Analysis in a Logarithmic Series

Investigate the series $$L(x)=\sum_{n=1}^\infty (-1)^{n+1} * \frac{(x-1)^n}{n}$$, which represents a

Error Estimation in Approximating $$e^x$$

For the function $$f(x)=e^x$$, use the Maclaurin series to approximate $$e^{0.3}$$. Then, determine

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 Analysis

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

Extremum Analysis Using the Extreme Value Theorem

Given the function $$f(x)= \cos(x)-x$$ defined on the interval $$\left[0, \frac{\pi}{2}\right]$$, an

Graph Interpretation of a Function's First Derivative

A graph of the derivative function is provided below. Use it to determine the behavior of the origin

Investigation of a Fifth-Degree Polynomial

Consider the function $$f(x)=x^5-5*x^4+10*x^3-10*x^2+5*x-1$$. Answer the following parts:

Investment Portfolio Dividends

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

Linear Approximation of a Radical Function

For the function $$f(x)= \sqrt{x+1}+x$$, find its linear approximation at $$x=3$$ and use it to appr

Population Growth Model Analysis

A population of organisms is modeled by the function $$P(t)= -2*t^2+20*t+50$$, where $$t$$ is measur

Projectile Motion Analysis

A projectile is launched vertically with its height given by $$s(t) = -16*t^2 + 64*t + 80$$ (in feet

Related Rates: Changing Shadow Length

A 2-meter tall lamppost casts a shadow of a 1.6-meter tall person who is walking away from the lampp

Related Rates: Expanding Balloon

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

Retirement Savings with Diminishing Deposits

Alex starts a retirement savings plan where the deposit each month forms a geometric sequence. In th

Series Convergence and Differentiation in Thermodynamics

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

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 $$\cos(2*x)$$

Consider the function $$f(x)=\cos(2*x)$$. Construct its 4th degree Maclaurin polynomial, determine t

Temperature Change in a Weather Balloon

A weather balloon’s temperature and altitude are related by the implicit equation $$T*e^{z} + z = 50

Volume Using Cylindrical Shells

The region bounded by $$y=x$$, $$y=0$$, and $$x=4$$ is revolved about the y-axis to form a solid.

Antiderivatives and the Fundamental Theorem

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

Area Estimation with Riemann Sums

Consider the function $$f(x)=x^2-4*x+3$$ on the interval $$[1,5]$$. Using a partition of 4 equal sub

Area Under a Piecewise Function

A function is defined piecewise as follows: $$f(x)=\begin{cases} x & 0 \le x \le 2,\\ 6-x & 2 < x \

Car Motion: From Acceleration to Distance

A car has an acceleration given by $$a(t)= 3 - 0.5*t$$ m/s² for time t in seconds. The initial velo

Charging a Battery

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

Convergence of an Improper Integral

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

Cost and Inverse Demand in Economics

Consider the cost function representing market demand: $$f(x)= x^2 + 4$$ for $$x\ge0$$. Answer the f

Drug Absorption Modeling

The rate of drug absorption into the bloodstream is modeled by $$C'(t)= 2*e^{-0.5*t}$$ mg/hr, with a

Error Bound Analysis for the Trapezoidal Rule

For the function $$f(x)=\ln(x)$$ on the interval $$[1,2]$$, the error bound for the trapezoidal rule

Flow of Traffic on a Bridge

Cars cross a bridge at a rate modeled by $$R(t)=300+50*\cos\left(\frac{\pi*t}{6}\right)$$ vehicles p

Integration of a Rational Function

Consider the function $$f(x)=\frac{1}{x^2+4}$$ on the interval $$[0,2]$$. Evaluate the area under th

Integration of a Rational Function via Partial Fractions

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

Integration via U-Substitution for a Composite Function

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

Investigating Partition Sizes

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

Midpoint Approximation Analysis

Let $$f(x)=\sqrt{x}$$ on the interval [0, 9]. Answer the following:

Non-Uniform Subinterval Riemann Sum

A function $$f(t)$$ is measured at non-uniform time intervals as recorded in the table below: | t (

Particle Motion and the Fundamental Theorem of Calculus

A particle moves along a straight line with its velocity given by $$v(t)=3*t^2-12*t+9$$ (in m/s) for

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

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

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,

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

Autonomous ODE: Equilibrium and Stability

Consider the autonomous differential equation $$\frac{dy}{dx}= y*(2-y)*(y+1)$$. Answer the following

Bank Account Growth with Continuous Compounding

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

City Population with Migration

The population $$P(t)$$ of a city changes as individuals migrate in at a constant rate of $$500$$ pe

Cooling Coffee Data Analysis

A cup of coffee cools down in a room according to Newton's Law of Cooling. The temperature $$T(t)$$

Differential Equation Involving Logarithms

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

Direction Fields and Isoclines

Examine the differential equation $$\frac{dy}{dx} = \frac{x+y}{x-y}$$. A direction field displaying

Disease Spread Model

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

Economic Model: Differential Equation for Cost Function

A company’s marginal cost is represented by $$MC(x)=\frac{dC}{dx}=3+2*x$$, where $$x$$ is the number

Epidemic Spread Modeling

In a simplified epidemic model, the number of infected individuals $$I(t)$$ is modeled by the logist

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 Equation

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

Exponential Growth and Decay

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

FRQ 10: Cooling of a Metal Rod

A metal rod cools in a room according to Newton's Law of Cooling: $$\frac{dT}{dt} = -k (T - A)$$. Th

Integrating Factor Method

Solve the differential equation $$\frac{dy}{dx} + \frac{2}{x} y = \frac{\sin(x)}{x}$$ for $$x>0$$.

Investment Account Growth with Fees

An investment account with balance $$A(t)$$ grows at a continuously compounded annual rate of $$6\%$

Newton's Law of Cooling

A cup of coffee at an initial temperature of $$90^\circ C$$ is placed in a room maintained at a cons

Particle Motion in the Plane with Non-constant Acceleration

A particle moves in the $$xy$$-plane with an acceleration vector given by $$a(t)=\langle 2, e^t \ran

Piecewise Differential Equation with Discontinuities

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

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 Saturation Model

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

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 Maclaurin Series Approximation

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

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 Initial Condition

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

Separation of Variables with Trigonometric Functions

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

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

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

Solution and Analysis of a Linear Differential Equation with Equilibrium

Consider the differential equation $$\frac{dy}{dx} = 3*y - 2$$, with the initial condition $$y(0)=1$

Tank Draining Problem

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

Temperature Control in a Chemical Reaction Vessel

In a chemical reactor, the temperature $$T(t)$$ is regulated by both natural cooling and an external

Water Tank Inflow-Outflow Model

A water tank is subject to an inflow and outflow. The inflow rate is given by $$I(t)=3*t+2$$ liters

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 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 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 Economic Curves

In an economic model, two cost functions are given by $$C_1(x)=100-2*x$$ and $$C_2(x)=60-x$$, where

Average Concentration of a Drug in Bloodstream

The concentration of a drug in the bloodstream is modeled by $$C(t)=3e^{-0.9*t}+2$$ mg/L, where $$t$

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 Analysis

A meteorological station recorded the temperature in a region as a function of time given by $$T(t)

Average Temperature of a Day

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

Average Temperature Over a Day

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

Average Value of a Population Growth Rate

The instantaneous growth rate of a bacterial population is modeled by the function $$r(t)=0.5*\cos(0

Average Value of a Velocity Function

The velocity of a car is modeled by $$v(t)=3*t^2-12*t+9$$ (m/s) for $$t\in[0,5]$$ seconds. Answer th

Cost Analysis: Area Between Production Cost Curves

Suppose two cost functions for producing goods are given by $$f(x)=20+2*x$$ and $$g(x)=5*x-\frac{1}{

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[

Integral Approximation Using Taylor Series

Approximate the integral $$\int_{0}^{0.2} \sin(2*x)\,dx$$ by using the Taylor series expansion of $$

Integration in Cost Analysis

In a manufacturing process, the cost per minute is modeled by $$C(t)=t^2 - 4*t + 7$$ (in dollars per

Medical Imaging: Reconstruction of a Cross-Section

In a medical imaging technique, the cross-sectional area of a tumor is modeled by $$A(x)=5*e^{-0.5*x

Optimization and Integration: Maximum Volume

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

Pollution Concentration in a Lake

A lake has a pollution concentration modeled by $$C(x) = 16 - x^2$$ (in mg/L), where $$x$$ (in meter

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

Position and Velocity from Tabulated Data

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

Profit-Cost Area Analysis

A company’s profit (in thousands of dollars) is modeled by $$P(x) = -x^2 + 10*x$$ and its cost by $$

Savings Account with Decreasing Deposits

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

Series and Integration Combined: Error Bound in Integration

Consider the integral $$\int_{0}^{0.5} \frac{1}{1+x^2} dx$$. Use the Taylor series expansion of the

Volume by Revolution: Washer Method

Consider the region bounded by the curves $$y=x+2$$ and $$y=x^2$$. When this region is rotated about

Volume by the Shell Method

Consider the region bounded by $$y=\sqrt{x}$$, $$y=0$$, and $$x=4$$. This region is revolved about t

Volume of a Solid by the Disc Method

Consider the region bounded by the curve $$y=\sqrt{x}$$ and the x-axis for $$x\in[0,4]$$. This regio

Volume of a Solid using the Shell Method

Consider the region bounded by $$y=\sqrt{x}$$, $$y=0$$, $$x=1$$, and $$x=4$$. When this region is ro

Work Done by a Variable Force

A variable force acting along the x-axis is given by $$F(x) = 2 * x + 3$$ (in Newtons). An object mo

Analysis of a Polar Rose

Examine the polar curve given by $$ r=3*\cos(3\theta) $$.

Arc Length of a Parametric Curve

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

Arc Length of a Parametric Curve

Consider the parametric curve defined by $$x(t)= t^2$$ and $$y(t)= t^3$$ for $$0 \le t \le 1$$. Anal

Arc Length of a Polar Curve

Consider the polar curve given by $$r(\theta)=1+\frac{\theta}{\pi}$$ for $$0 \le \theta \le \pi$$. A

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 of a Region in Polar Coordinates with an Internal Boundary

Consider a region bounded by the outer polar curve $$R(\theta)=5$$ and the inner polar curve $$r(\th

Average Position from a Vector-Valued Function

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

Conversion of Polar to Parametric Form

A particle’s motion is given in polar form by the equations $$r = 4$$ and $$\theta = \sqrt{t}$$ wher

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

Enclosed Area of a Parametric Curve

A closed curve is given by the parametric equations $$x(t)=3*\cos(t)-\cos(3*t)$$ and $$y(t)=3*\sin(t

Equivalence of Parametric and Polar Circle Representations

A circle is represented by the parametric equations $$x(t)=3\cos(t)$$ and $$y(t)=3\sin(t)$$ for $$0\

Integration of Vector-Valued Acceleration

A particle's acceleration is given by the vector function $$\mathbf{a}(t)=\langle 2*t,\; 6-3*t \rang

Intersection of Polar and Parametric Curves

Consider the polar curve $$r=4\cos(\theta)$$ and the parametric line given by $$x=1+t$$, $$y=2*t$$,

Motion Along a Parametric Curve

Consider the parametric equations $$x(t)= 2*t - 3$$ and $$y(t)= t^2$$, where $$t$$ represents time i

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 Curve: Intersection with a Line

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

Parametric Equations of a Cycloid

A cycloid is generated by a point on the circumference of a circle of radius $$r$$ rolling along a s

Particle Motion with Logarithmic Component

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

Particle Motion with Uniform Angular Change

A particle moves in the polar coordinate plane with its distance given by $$r(t)= 3*t$$ and its angl

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

Reparameterization between Parametric and Polar Forms

A curve is described by the parametric equations $$x(t)=a*t$$ and $$y(t)=b*t^2$$, where $$a$$ and $$

Spiral Motion with a Damped Vector Function

An object moves according to the spiral vector function $$\mathbf{r}(t)=\langle e^{-t}\cos(t),\; e^{

Symmetry and Self-Intersection of a Parametric Curve

Consider the parametric curve defined by $$x(t)= \sin(t)$$ and $$y(t)= \sin(2*t)$$ for $$t \in [0, \

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 Function and Particle Motion

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

Vector-Valued Functions in 3D

A space curve is described by the vector function $$\mathbf{r}(t)=\langle e^t,\cos(t),\ln(1+t) \rang

Vector-Valued Functions: Velocity and Acceleration

A particle's position is given by the vector-valued function $$\vec{r}(t) = \langle \sin(t), \ln(t+1

Work Done Along a Path in a Force Field

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