AP Calculus BC FRQ Room

Applying the Squeeze Theorem

Let $$f(x)=x^2\sin(1/x)$$ for $$x \neq 0$$ and define $$f(0)=0$$. Use the Squeeze Theorem to evaluat

Approaching Vertical Asymptotes

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

Calculating Tangent Line from Data

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

Composite Function in Water Level Modeling

Suppose the water volume in a tank is given by a composite function \(V(t)=f(g(t))\) where $$g(t)=\f

Continuity in Piecewise Defined Functions

Consider the piecewise function $$f(x)= \begin{cases} x^2+1, & \text{if } x \leq 3 \\ 2*x+k, & \text

Continuous Extension of a Log‐Ratio Function

Define $$g(x)= \frac{\ln(1+e^x)}{x}$$ for $$x \neq 0$$ and let $$g(0)=m$$ be chosen for continuity.

Determining Limits for a Function with Absolute Values and Parameters

Consider the function $$ f(x)= \begin{cases} \frac{|x-2|}{x-2}, & x \neq 2 \\ c, & x = 2 \end{cases

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:

Exponential Inflow with a Shift in Outflow Rate

A water tank receives water at a rate given by $$R_{in}(t)=20\,e^{-t}$$ liters per minute. The water

Implicitly Defined Curve and Its Tangent Line

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

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 and Asymptotic Behavior of Rational Functions

Let $$k(x)=\frac{5*x^2-2*x+7}{x^2+4}.$$ Answer the following:

Limits Involving Exponential Functions

Consider the function $$f(x)= \frac{e^{2*x}-1}{x}$$ defined for $$x\neq0$$.

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 Discontinuity Analysis

Consider the piecewise function $$ f(x)= \begin{cases} x+2, & x < 1 \\ 3-x, & x \ge 1 \end{cases} $

Oscillatory Behavior and Limits

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

Rate of Change in a Chemical Reaction (Implicit Differentiation)

In a chemical reaction the concentration C (in M) of a reactant is related to time t (in minutes) by

Squeeze Theorem Application

Consider the function $$f(x)=\frac{\sin(3*x)}{x}$$ defined for x ≠ 0.

Squeeze Theorem in Oscillatory Functions

Consider the function $$f(x)= x\sin(1/x)$$ for $$x \neq 0$$ and define $$f(0)=0$$.

Squeeze Theorem with an Oscillating Function

Let $$f(x)=x * \cos(\frac{1}{x})$$ for $$x \neq 0$$, and define $$f(0)=0$$. Answer the following:

Water Flow Measurement Analysis

A water flow sensor measures the flow rate $$Q(t)$$ (in cubic meters per second) of a stream at vari

Zeno’s Maze Runner

A runner attempts to reach a wall 100 meters away by covering half of the remaining distance with ea

Analysis of a Quadratic Function

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

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

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

Derivative from the Limit Definition: Function $$f(x)=\sqrt{x+2}$$

Consider the function $$f(x)=\sqrt{x+2}$$ for $$x \ge -2$$. Using the limit definition of the deriva

Derivative of Inverse Functions

Let $$f(x)=3*x+\sin(x)$$, which is assumed to be one-to-one with an inverse function $$f^{-1}(x)$$.

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

Differentiating Composite Functions using the Chain Rule

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

Differentiation and Linear Approximation for Error Estimation

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

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

Higher-Order Derivatives

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

Hot Air Balloon Altitude Analysis

A hot air balloon’s altitude is modeled by the function $$h(t)=5*\sqrt{t+1}$$, where $$h$$ is in met

Implicit Differentiation in a Geometric Context

Consider the circle defined by the equation $$x^2+y^2=25.$$ (a) Use implicit differentiation to f

Implicit Differentiation on an Ellipse

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

Implicit Differentiation with Exponential and Trigonometric Functions

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

Instantaneous Versus Average Rates: A Comparative Study

Examine the function $$f(x)=\ln(x)$$. Analyze its average and instantaneous rates of change over a g

Logarithmic Differentiation Simplification

Consider the function $$h(x)=\ln\left( \frac{(x^2+1)^{3}*e^{2*x}}{\sqrt{x+2}} \right)$$.

Maclaurin Polynomial for √(1+x)

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

Particle Motion in the Plane

A particle moves in the plane with its position given by $$x(t)=t^2-4*t+1$$ and $$y(t)=3*t-2.5$$, wh

Profit Optimization via Derivatives

A company's profit function is given by $$P(x)=-2*x^2 + 40*x - 100$$, where $$x$$ represents the num

Testing Differentiability at a Junction Point

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

Calculating an Inverse Trigonometric Derivative in a Physics Context

A pendulum's angle is modeled by $$\theta = \arcsin(0.5*t)$$, where $$t$$ is time in seconds and $$\

Chain Rule and Taylor/Maclaurin Series for an Exponential Function

Consider the function $$h(x) = e^{\sin(2*x)}$$, which is a composite of the exponential and sine fun

Chain Rule for Inverse Trigonometric Functions in Optics

In an optics experiment, the angle of incidence $$\theta(t)$$ (in radians) is modeled by $$\theta(t)

Chain Rule in a Power Function

Consider the function $$f(x)= (3*x^2 + 2*x + 1)^5$$. Use the chain rule to find its derivative, eval

Composite Differentiation in Biological Growth

A biologist models the temperature $$T$$ (in °C) of a culture over time $$t$$ (in hours) by the func

Composite Exponential Logarithmic Function Analysis

Consider the function $$f(x)=\ln(2*e^{3*x}+5)$$ which models a logarithmic transformation of an expo

Composite Function with Exponential and Radical

Consider the function $$ f(x)= \sqrt{e^{5*x}+x^2} $$.

Composite Function with Implicitly Defined Inner Function

Let the function $$h(x)$$ be defined implicitly by the equation $$h(x) - \ln(h(x)) = x$$, and consid

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 of an Inverse Trigonometric Form

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

Drug Concentration in the Bloodstream

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

Enzyme Kinetics in a Biochemical Reaction

In an enzymatic reaction, the substrate concentration $$S(t)$$ and the product concentration $$P(t)$

Implicit Differentiation in a Radical Equation

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

Implicit Differentiation of a Circle

Consider the circle defined by $$x^2+y^2=25$$. Answer the following parts:

Implicit Differentiation on an Elliptical Curve

Consider the ellipse $$\frac{x^2}{9}+\frac{y^2}{4}=1$$ representing an object’s cross-section. Answe

Implicit Differentiation with Product and Chain Rule in a Thermal Expansion Model

A material's length $$L$$ (in meters) under thermal expansion satisfies the equation $$L - \sin(L *

Inverse Function Derivatives in a Sensor Model

An instrument outputs a reading defined by $$f(x)= x^3 + 2$$, where $$x$$ represents the voltage inp

Inverse Function Differentiation for a Cubic Function

Let $$ f(x)= x^3+x $$. This function is invertible over all real numbers.

Inverse Function Differentiation for Cubic Functions

Let $$f(x)= x^3 + 2*x$$, and let $$g(x)$$ be its inverse function. Answer the following:

Inverse Trigonometric Function in a Navigation Problem

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

Inverse Trigonometric Functions: Analysis and Application

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

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 in Manufacturing Material

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

Polar and Composite Differentiation: Arc Slope for a Polar Curve

Consider the polar curve $$r(\theta)=2+\cos(\theta)$$. Answer the following parts:

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

Tangent Line to an Ellipse

Consider the ellipse given by $$\frac{x^2}{16} + \frac{y^2}{9} = 1$$. Determine the slope of the tan

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 $

Approximating Function Values Using Linearization

Consider the function $$f(x)=x^4$$. Use linearization at x = 4 to approximate the value of $$f(3.98)

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

Deceleration of a Vehicle on a Straight Road

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

Draining Hemispherical Tank

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

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 $$x$$

Expanding Rectangle: Related Rates

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

Exponential and Trigonometric Bounded Regions

Let the region in the xy-plane be bounded by $$y = e^{-x}$$, $$y = 0$$, and the vertical line $$x =

Financial Model Inversion

Consider the function $$f(x)=\ln(x+2)+x$$ which models a certain financial indicator. Although an ex

Fuel Consumption Rate Analysis

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

Graphical Data and Derivatives

A set of experimental data is provided below, showing the concentration (in moles per liter) of a ch

Graphical Interpretation of Slope and Instantaneous Rate

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

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

Implicit Differentiation in Astronomy

The trajectory of a comet is given by the ellipse $$x^2 + 4*y^2 = 16$$, where \(x\) and \(y\) (in as

Implicit Differentiation on a Circle

Consider the circle defined by $$x^2+y^2=25$$, where both $$x$$ and $$y$$ are functions of time $$t$

Integration of Flow Rates Using the Trapezoidal Rule

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

Linearization Approximation Problem

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

Linearization in Finance

The value of an investment is modeled by $$V(x)=1000x^{0.5}$$ dollars, where x represents a market i

Linearization of a Power Function

Let $$f(x)=x^4$$. Use linearization at $$x=4$$ with $$\Delta x=-0.02$$ to approximate $$(3.98)^4$$.

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

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.

Motion with Non-Uniform Acceleration

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

Polar Curve: Slope of the Tangent Line

Consider the polar curve defined by $$r(\theta)=10e^{-0.1*\theta}$$.

Pollution Accumulation in a Lake

A lake is subject to pollution with pollutants entering at a rate of $$I(t)=3e^{0.1t}$$ (kg per day)

Population Growth Analysis

A certain bacterial population in a lab grows according to the model $$P(t)=100\cdot e^{0.03*t}$$, w

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

Revenue Concavity Analysis

A company’s revenue from sales is modeled by the function $$R(x)= 300*x - 2*x^2$$, where \(x\) repre

Series Analysis in Acoustics

The sound intensity at a distance is modeled by $$I(x)= I_0 \sum_{n=0}^{\infty} \frac{(-1)^n (x-10)^

Temperature Change of Cooling Coffee

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

Water Filtration Plant Analysis

A water filtration plant processes water entering at a rate of $$I(t)=60-2t$$ (liters per minute) an

Aircraft Climb Analysis

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

Analyzing Inverses in a Rate of Change Scenario

Consider the function $$f(x)= \ln(x+5) + x$$ defined for $$x > -5$$. This function models a system's

Bouncing Ball with Energy Loss

A ball is dropped from a height of 100 meters. Each time it bounces, it reaches 60% of the height fr

Composite Functions and Derivatives

Let $$h(x)=f(g(x))$$ where $$f(u)=u^2+3$$ and $$g(x)=\sin(x)$$. Analyze the composite function on th

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

Differentiability and Critical Points of a Piecewise Function

Consider the piecewise function $$ f(x)= \begin{cases} x^2 & \text{if } x \le 2, \\ 4*x-4 & \text{i

Discounted Cash Flow Analysis

A project is expected to return cash flows that decrease by 10% each year from an initial cash flow

Error Estimation in Approximating $$e^x$$

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

Expanding Oil Spill - Related Rates

A circular oil spill is expanding such that its area is given by $$A(t) = \pi*[r(t)]^2$$. The radius

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

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

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.

Investigation of a Series with Factorials and Its Operational Calculus

Consider the series $$F(x)=\sum_{n=0}^\infty \frac{x^n}{n!}$$, which represents an exponential funct

Lake Ecosystem Nutrient Dynamics

In a lake, nutrients (phosphorus) enter at a rate given by $$N_{in}(t)=5*\sin(t)+10$$ mg/min and are

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

Logistic Growth in Biology

The logistic growth of a species is modeled by $$P(t) = \frac{1}{1 + e^{-0.5*(t-4)}}$$, where t is i

Manufacturing Optimization in Production

A company’s profit (in thousands of dollars) from producing x (in thousands of units) is given by $$

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

Piecewise Function with Absolute Value

Consider the function defined by $$ g(x)=\begin{cases} |x-1| & \text{if } x<2, \\ 3x-5 & \text{if }

Radiocarbon Dating in Artifacts

An archaeological artifact contains a radioactive isotope with an initial concentration of 100 units

Radius of Convergence and Series Manipulation in Substitution

Let $$f(x)=\sum_{n=0}^\infty c_n * (x-2)^n$$ be a power series with radius of convergence $$R = 4$$.

Rate of Change in a Logarithmic Temperature Model

A cooling process is modeled by the temperature function $$T(t)= 100 - 20\,\ln(t+1)$$, where t is me

Retirement Savings with Diminishing Deposits

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

Stock Price Analysis

The daily closing price of a stock (in dollars) is recorded at various days. Use the stock price dat

Taylor Series for $$e^{-x^2}$$

Consider the function $$f(x)=e^{-x^2}$$. In this problem, you will derive its Maclaurin series up to

Travel Distance from Speed Data

A traveler’s speed (in km/h) is recorded at various times during a trip. Use the data to approximate

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, \\

Accumulated Displacement from a Velocity Function

A car’s velocity is given by the function $$v(t)=4 + t$$ (in m/s) over the interval [0, 8] seconds.

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 the Curves f(x)=x² and g(x)=2x+3

Given the two functions $$f(x)= x^2$$ and $$g(x)= 2*x+3$$ on the interval where they intersect, dete

Area Under a Parametric Curve

Consider the parametric curve defined by $$x(t)=t$$ and $$y(t)=t^2$$ for $$t \in [0,3]$$. The area u

Bacteria Population Accumulation

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

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 $$x\ge2$$,

Displacement and Distance from a Velocity Function

A particle moves along a straight line with its velocity given by $$v(t)=3\sin(t)$$ (in m/s) for $$t

Estimating Area Under a Curve Using Riemann Sums

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

Estimating Chemical Production via Riemann Sums

In a laboratory experiment, the reaction rate of a chemical process is recorded at various times. Th

Estimating Rainfall Accumulation

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

Evaluating a Complex Integral

Evaluate the integral $$\int_{0}^{2} 2*x*(x+1)^3\,dx$$ using an appropriate substitution method.

Evaluating an Integral via U-Substitution

Evaluate the integral $$\int_{1}^{5} (x-4)^{10}\,dx$$ using u-substitution.

Graphical Analysis of Riemann Sums

A graph titled 'Graph of Experimental Data' shows a curve representing the height function $$h(t)$$

Graphical Transformations and Inverse Functions

Consider the linear function $$f(x)= \frac{1}{2}*x + 5$$ defined for all real $$x$$. Answer the foll

Integration by Parts: Logarithmic Function

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

Integration Involving Inverse Trigonometric Functions

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

Interpreting the Constant of Integration in Cooling

An object cools according to the differential equation $$\frac{dT}{dt}=-k*(T-20)$$ where $$T(t)$$

Limit of a Riemann Sum as a Definite Integral

Consider the limit of the Riemann sum given by $$\lim_{n \to \infty} \sum_{i=1}^{n} \left( \frac{6}{

Mechanical Systems: Total Change and Inverse Analysis

Consider the function \(f(x)= x^3 + 3*x\) defined for all real \(x\), modeling a mechanical system.

Midpoint Approximation Analysis

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

Numerical Approximation: Trapezoidal vs. Simpson’s Rule

The function $$f(x)=\frac{1}{1+x^2}$$ is to be integrated over the interval [-1, 1]. A table of valu

Parameter-Dependent Integral Function Analysis

Define the function $$F(x)=\int_(1)^(x) \frac{\ln(t)}{t} dt$$ for x > 1. This function accumulates 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

Power Series Approximation of an Integral Function

The function $$f(x)=e^{-x^2}$$ does not have an elementary antiderivative. Its definite integral can

Riemann and Trapezoidal Sums with Inverse Functions

Consider the function $$f(x)= 3*\sin(x) + 4$$ defined on the interval \( x \in [0, \frac{\pi}{2}] \)

Riemann Sum Approximations: Midpoint vs. Trapezoidal

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

Taylor/Maclaurin Series Approximation and Error Analysis

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

Total Work Done by a Variable Force

A variable force $$F(x)$$ (in Newtons) is applied along a displacement, and its values are recorded

Trapezoidal Approximation for a Curved Function

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

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

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

Variable Interest Rate and Continuous Growth

An investment grows continuously with a variable interest rate given by $$r(t)=0.05+0.01*t$$. The in

Volume of a Solid by the Shell Method

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

Water Tank Inflow and Outflow

A water tank begins operation at t = 0 with an initial volume of 0 liters. Water flows in through an

Work Done by a Variable Force

A variable force given by $$F(x)= 3*x^2$$ (in Newtons) acts on an object as it moves along a straigh

Autocatalytic Reaction Dynamics

Consider an autocatalytic reaction described by the differential equation $$\frac{dy}{dt} = k*y*\ln|

Bacterial Growth with Time-Dependent Growth Rate

A bacterial culture grows according to the differential equation $$\frac{dP}{dt}=(r_0+r_1*t)P$$, whe

Capacitor Discharge in an RC Circuit

In an RC circuit, the voltage $$V(t)$$ across a discharging capacitor obeys the differential equatio

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

Electrical Circuit Analysis Using an RL Circuit

An RL circuit is described by the differential equation $$L\frac{di}{dt}+R*i=E$$, where $$L$$ is the

Exact Differential Equations

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

Forced Oscillation in a Damped System

Consider the differential equation $$\frac{dx}{dt}=-0.2*x+\sin(t)$$ with initial condition $$x(0)=1$

FRQ 2: Separable Differential Equation with Initial Condition

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

FRQ 11: Linear Differential Equation via Integrating Factor

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

FRQ 14: Dynamics of a Car Braking

A car braking is modeled by the differential equation $$\frac{dv}{dt} = -k*v$$, where the initial ve

FRQ 17: Slope Field Analysis and Particular Solution

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

Implicit Solution of a Separable Differential Equation

Solve the differential equation $$\frac{dy}{dx}=\frac{y+1}{x}$$ with the initial condition $$y(1)=2$

Logistic Equation with Harvesting

A fish population in a lake follows a logistic growth model with the addition of a constant harvesti

Maclaurin Series Solution for a Differential Equation

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

Mixing Problem in a Tank

A tank initially contains $$100$$ liters of water with $$5$$ kg of dissolved salt. Brine with a salt

Modeling Cooling in a Variable Environment

Suppose the cooling of a heated object is modeled by the differential equation $$\frac{dT}{dt} = -k*

Population Dynamics in Ecology

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

RL Circuit Analysis

An RL circuit is described by the differential equation $$L\frac{di}{dt} + R*i = V$$, where $$L=0.5\

Separable Differential Equation and Maclaurin Series Approximation

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

Sketching a Solution Curve from a Slope Field

A slope field for the differential equation $$\frac{dy}{dt}=y(1-y)$$ is provided. Use the slope fiel

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 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$

Temperature Regulation in Biological Systems

In a biological system, the temperature \(T(t)\) (in °C) of an organism is modeled by the differenti

Verification of Integral Representation of Solutions

Let $$y(x)=\int_0^x e^{-(x-t)} f(t)\,dt$$, where $$f(t)$$ is a continuous function. Answer the follo

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

Accumulated Interest in a Savings Account

An investor’s savings account experiences continuous deposits and withdrawals. The deposit rate is g

Analysis of a Function with a Removable Discontinuity

Consider the function $$f(x)=\frac{x^2-4}{x-2}$$ for $$x\neq2$$, with an assigned value of $$f(2)=3$

Analyzing a Reservoir's Volume Over Time

Water flows into a reservoir at a variable rate given by $$R(t)=50e^{-0.1*t}$$ m³/hour and simultane

Analyzing Acceleration Data from Discrete Measurements

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

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 from Experimental Data

In an experiment, researchers recorded measurements for two functions, $$f(t)$$ and $$g(t)$$, repres

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 Under an Exponential Decay Curve

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

Average Car Speed Analysis from Discrete Data

A car's speed (in km/h) is recorded at equal time intervals over a 1-hour journey. Analyze the car's

Average Temperature Analysis

A weather station records the temperature throughout a day. The temperature, in degrees Celsius, is

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

Center of Mass of a Non-uniform Rod

A thin rod of length 10 m has a linear density given by $$\lambda(x)= 3 + 0.5*x$$ (in kg/m) for $$0

Cyclist's Journey: Displacement versus Total Distance

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

Displacement vs. Distance: Analysis of Piecewise Velocity

A particle moves along a line with velocity given by $$v(t)=\begin{cases} t^2, & 0 \le t < 2,\\ 8-t^

Electric Current and Charge

An electric current in a circuit is defined by $$I(t)=4*\cos\left(\frac{\pi}{10}*t\right)$$ amperes,

Electrical Charge Distribution

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

Inflow Rate to a Reservoir

The inflow rate of water into a reservoir is given by $$R(t)=\frac{100*t}{5+t}$$ (in cubic meters pe

Motion Analysis on a Particle with Variable Acceleration

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

Net Cash Flow Analysis

A company’s net cash flow is modeled by $$N(t)=50*\ln(t+1) - 2*t$$ (in thousands of dollars per mont

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

Particle on a Line with Variable Acceleration

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

Projectile Motion Analysis

A projectile is launched vertically upward with an initial velocity of $$20$$ m/s. The only accelera

Rainfall Accumulation Analysis

A meteorological station recorded the following rainfall data (in mm) over a 24‐hour period. The rai

Solid of Revolution using Washer Method

The region bounded by the curves $$y = x^2$$ and $$y = 2 * x$$ is rotated about the x-axis. Answer t

Total Charge in an Electrical Circuit

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

Volume by Cross‐Sectional Area in a Variable Tank

A tank has a variable cross‐section. For a water level at height $$y$$ (in cm), the width of the tan

Volume of a Solid with Elliptical Cross Sections

Consider a solid whose base is the region bounded by $$y=x^2$$ and $$y=4$$. Cross sections perpendic

Volume Using the Shell Method

Consider the region bounded by $$y=x^2$$ and $$y=4$$ with $$x\ge0$$. This region is rotated about th

Volume with Square Cross Sections

The region in the $$xy$$-plane is bounded by $$y=\sqrt{x}$$, $$y=0$$, and $$x=4$$. A solid is formed

Work Done by a Variable Force

A force acting on an object along a displacement is given by $$F(x)=3*x^2 -2*x+1$$ (in Newtons), whe

Work Done by a Variable Force

A variable force given by $$F(x)= 2*x + 3$$ (in Newtons) is applied to an object as it moves along a

Analyzing a Cycloid

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

Arc Length of a Decaying Spiral

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

Arc Length of a 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 Polar Curve

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

Area between Two Polar Curves

Given the polar curves $$R(\theta)=3$$ and $$r(\theta)=2$$ for $$0 \le \theta \le 2\pi$$, find the a

Combined Motion Analysis

A particle’s path is described by the parametric equations $$x(t)= \ln(1+ t^2)$$ and $$y(t)= \sqrt{t

Comparing Representations: Parametric and Polar

A curve is represented by the parametric equations $$x(t)=3\cos(t)-\sin(t)$$ and $$y(t)=3\sin(t)+\co

Conversion and Tangents in Polar Coordinates

Consider the polar curve $$r=\sec(\theta)$$ for $$\theta \in \left[0, \frac{\pi}{4}\right]$$.

Exponential-Logarithmic Particle Motion

A particle moves in the plane with its position given by the parametric equations $$x(t)=e^{t}+\ln(t

Intersection Analysis with the Line y = x

Given the parametric equations $$x(t)=\ln(t+2)$$ and $$y(t)=t^2-1$$ for $$t \ge 0$$, answer the foll

Intersection of Two Parametric Curves

Two curves are represented parametrically as follows: Curve A is given by $$x(t)=t^2, \; y(t)=2*t+1$

Kinematics in Polar Coordinates

A particle’s position in polar coordinates is given by $$r(t)= \frac{5*t}{1+t}$$ and $$\theta(t)= \f

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 Equations from Real-World Data

A moving vehicle’s position is modeled by the parametric equations $$ x(t)=3*t+1 $$ and $$ y(t)=t^2-

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

Parametric Motion with Damping

A particle's motion is modeled by the parametric equations $$x(t)= e^{-t}\cos(t)$$ and $$y(t)= e^{-t

Parametric Plotting and Cusps

Let the parametric equations be $$ x(t)=t-\sin(t) $$ and $$ y(t)=1-\cos(t) $$ for $$ 0 \le t \le 2\p

Parametric Slope and Arc Length

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

Parametric to Polar and Integration

The spiral curve is given in parametric form by $$x(t)=t*\cos(t)$$ and $$y(t)=t*\sin(t)$$ for $$t\ge

Particle Motion in the Plane

Consider a particle whose motion in the plane is defined by the parametric equations $$x(t) = t^2 -

Polar Equations and Slope Analysis

Given the polar curve $$r = 4\sin(\theta)$$, analyze its Cartesian form and tangent properties.

Polar Spiral: Area and Arc Length

Consider the polar spiral defined by $$r(\theta)=1+\frac{\theta}{2}$$ for $$0\le\theta\le 2\pi$$. An

Sensitivity Analysis and Linear Approximation using Implicit Differentiation

The variables $$x$$ and $$y$$ satisfy the equation $$xy+\ln(y)=5$$.

Spiral Path Analysis

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

Synthesis of Parametric, Polar, and Vector Concepts

A drone's flight path is given in polar coordinates by $$r(\theta)= 5+ 2\sin(\theta)$$. It is parame

Tangent Line to a Parametric Curve

Consider the circle parametrized by $$x(t)=3\sin(t)$$ and $$y(t)=3\cos(t)$$ for $$0\le t\le 2\pi$$.

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 Function of Particle Trajectory

A particle in space follows the vector function $$\mathbf{r}(t)=\langle t, t^2, \sqrt{t} \rangle$$ f

Weather Data Analysis from Temperature Table

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