AP Calculus BC FRQ Room

Analysis of a Jump Discontinuity

Consider the function $$f(x)=\begin{cases} 3*x+1, & x<4 \\ 2*x-3, & x\geq4 \end{cases}$$.

Analyzing Limits of a Combined Exponential‐Log Function

Consider $$f(x)= e^{-x}\,\ln(1+\sqrt{x})$$ for $$x \ge 0$$. Analyze the limits as $$x \to 0^+$$ and

Application of the Squeeze Theorem with Trigonometric Oscillations

Consider the function $$f(x)= x^2*\sin(1/x)$$ for $$x \neq 0$$ and $$f(0)=0$$. Answer the following

Approximating Limits Using Tabulated Values

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

Asymptotic Behavior and Horizontal Limits

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

Car Braking Distance and Continuity

A car decelerates to a stop, and its velocity $$v(t)$$ in m/s is recorded in the following table, wh

Computing a Limit Using Algebraic Manipulation

Evaluate the limit $$\lim_{x\to2} \frac{x^2-4}{x-2}$$ using algebraic manipulation.

Continuity Analysis in Road Ramp Modeling

A highway ramp is modeled by the function $$y(x)= \frac{(x-3)(x+2)}{x-3}$$ for $$x\neq3$$, where x i

Continuity Assessment of a Rational Function with a Redefined Value

Consider the function $$r(x)= \begin{cases}\frac{x^2-9}{x-3}, & x \neq 3 \\ 7, & x=3\end{cases}$$.

Continuity in Piecewise Functions with Parameters

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

Determining Continuity via Series Expansion

Consider the function $$f(x)= \frac{e^x - \ln(1+x) - x - 1}{x^2}$$ for $$x \neq 0$$ with $$f(0)=L$$.

Epsilon-Delta Proof for a Polynomial Function

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

Graph Analysis of Discontinuities

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

Graphical Analysis of Volume with a Jump Discontinuity

A graph of the water volume \(V(t)\) in a tank shows a jump discontinuity at \(t=6\) minutes. Answer

Implicitly Defined Function and Differentiation

Consider the curve defined implicitly by the equation $$x*y + \sin(x) + y^2 = 10$$. Answer the follo

Intermediate Value Theorem in Engineering Context

In a structural analysis, the stress on a beam is modeled by a continuous function $$S(x)$$ on the i

Inverse Function Analysis and Continuity

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

Left-Hand and Right-Hand Limits for a Sign Function

Consider the function $$f(x)= \frac{x-2}{|x-2|}$$.

Limit and Continuity with Parameterized Functions

Let $$ f(x)= \frac{e^{3x} - 1 - 3x}{\ln(1+4x) - 4x}, $$ for $$x \neq 0$$ and define \(f(0)=L\) for c

Limit Evaluation Involving Radicals and Rationalization

Evaluate the limit $$\lim_{x \to 4} \frac{\sqrt{x}-2}{x-4}$$.

Limits and Continuity of Radical Functions

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

Modeling Temperature Change with Continuity

A city’s temperature throughout the day is modeled by the continuous function $$T(t)=\frac{1}{2}t^2-

Pendulum Oscillations and Trigonometric Limits

A pendulum’s angular displacement from the vertical is given by $$\theta(t)= \frac{\sin(2*t)}{t}$$ f

Rational Function Analysis of a Drainage Rate

A drain’s outflow rate is given by $$R_{out}(t)=\frac{3\,t^2-12\,t}{t-4}$$ for \(t\neq4\). Answer th

Rational Functions and Limit at Infinity

Consider the rational function $$r(x)= \frac{2x^2+3x-1}{x^2-4}$$.

Saturation of Drug Concentration in Blood

A patient is given a drug with each dose containing 50 mg. However, due to metabolism, only 20% of t

Squeeze Theorem with Oscillatory Behavior

Examine the function $$s(x)=x^2*\sin(1/x)$$ for x ≠ 0.

Trigonometric Limits

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

Chain Rule in Biological Growth Models

A biologist models the growth of a bacterial population by the function $$P(t) = (5*t + 2)^4$$, wher

Composite Function and Chain Rule Application

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

Derivative from First Principles

Let $$f(x)=\sqrt{x}$$. Use the limit definition of the derivative to find $$f'(x)$$.

Derivative of a Composite Function Using the Limit Definition

Consider the function $$h(x)=(2*x+3)^3$$. Use the limit definition of the derivative to answer the f

Derivative via Quotient Rule: Fluid Flow Rate

A function describing the rate of fluid flow is given by $$f(x)= \frac{x^2+2}{3*x-1}$$.

Differentiating a Series Representing a Function

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

Differentiation from First Principles

Let $$h(x)=3*x^2+2*x-1$$. Use the limit definition of the derivative to analyze this function.

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)}}$

Evaluating the Derivative Using the Limit Definition

Consider the function $$f(x) = 3*x^2 - 2*x + 1$$. (a) Use the limit definition of the derivative:

Finding and Interpreting Critical Points and Derivatives

Examine the function $$f(x)=x^3-9*x+6$$. Determine its derivative and analyze its critical points.

Higher-Order Derivatives

Consider the function $$f(x)=x^4 - 2*x^3 + 3*x -1$$. 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 for a Rational Equation

Consider the curve defined by $$\frac{x*y}{x+y} = 3$$.

Instantaneous Rate of Change in Motion

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

Limit Definition of Derivative for a Rational Function

For the function $$f(x)=\frac{1}{x+1}$$, use the limit definition of the derivative to answer the fo

Maclaurin Series and Convergence for 1/(1-x)

An economist is using the function $$f(x)=\frac{1}{1-x}$$ to model economic behavior. Analyze the Ma

Maclaurin Series for e^x Approximation

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

Optimization Using Derivatives

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

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

Pollutant Levels in a Lake

A lake receives pollutants at a rate of $$P_{in}(t)=30-0.5*t$$ concentration units per day and a tre

Population Growth Rates

A city’s population (in thousands) was recorded over several years. Use the data provided to analyze

Product and Quotient Rules in Economic Modeling

A company’s revenue (in thousands of dollars) is modeled by the function $$R(x)= (x+2)(x-1)$$ where

Rate of Change Analysis in a Temperature Model

A temperature model is given by $$T(t)=25+4*t-0.5*t^2$$, where $$t$$ is time in hours. Analyze the t

Related Rates: Changing Shadow Length

A 1.8 m tall man is walking away from a 5 m tall lamp at a constant speed of 1.2 m/s. The lamp casts

Reservoir Management Problem

A reservoir used for irrigation receives water at a rate of $$I(t)=20+2\sin(t)$$ liters per hour and

Satellite Orbit Altitude Modeling

A satellite’s altitude (in kilometers) above Earth is modeled by $$a(t)= 500*\cos\left(\frac{\pi}{6}

Secant and Tangent Approximations from a Graph

A function f(t) has been graphed from t = 0 to 10 seconds. Use the graph to estimate rates of change

Secant Line Approximation in an Experimental Context

A temperature sensor records the following data over a short experiment:

Tangent Line to a Logarithmic Function

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

Widget Production Rate

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

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 Application: Differentiating a Nested Trigonometric Function

Consider the function $$f(x) = \sin(\cos(2*x))$$. Analyze its derivative using the chain rule.

Chain Rule in a Nested Composite Function

Consider the function $$f(x)= \sin\left(\ln((2*x+1)^3)\right)$$. Answer the following parts:

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:

Differentiation of a Log-Exponential Composition with Critical Points

Consider the function $$k(x)=x*\ln(e^{x}+3)$$. Answer the following parts.

Differentiation of an Inverse Trigonometric Form

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

Differentiation of an Inverse Trigonometric Function

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

Implicit Differentiation and Concavity of a Logarithmic Curve

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

Implicit Differentiation 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 in Economic Equilibrium

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

Implicit Differentiation of an Implicit Curve

Consider the curve defined by $$x*y + x^2 - y^2 = 5$$. Answer the following parts.

Implicit Differentiation: Conic Section Analysis

Consider the conic section defined by $$x^2 + 3*x*y + y^2 = 5$$. Answer the following:

Implicit Equation with Logarithmic and Exponential Terms

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

Inverse 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 Derivative in an Exponential Model

Let $$f(x)= e^{2*x} + x$$. Given that $$f$$ is one-to-one and differentiable, answer the following p

Inverse Function Differentiation Basics

Let $$f$$ be a one-to-one differentiable function with $$f(3)=5$$ and $$f'(3)=2$$, and let $$g$$ be

Inverse Function Differentiation with a Logarithmic Function

Let the function $$f(x)=\ln(2+x^2)$$ be differentiable and one-to-one, and let its inverse be $$g(y)

Inverse Function Differentiation with Combined Logarithmic and Exponential Terms

Let $$f(x)=e^{x}+\ln(x)$$ for $$x>1$$ and let g be its inverse function. Answer the following.

Inverse Function Differentiation with Composite Trigonometric Functions

Let $$f(x)= \sin(2*x) + x$$, which is differentiable and one-to-one. It is given that $$f(\pi/6)= 1$

Logarithmic and Exponential Composite Function with Transformation

Let $$g(x)=\ln((3*x+1)^2)-e^{x}$$. Answer the following questions.

Logarithmic Differentiation of a Variable Exponent Function

Consider the function $$y= (x^2+1)^{\sqrt{x}}$$.

Nested Composite Function Differentiation

Consider the function $$ h(x)= \sqrt{\cos(3*x^2+1)} $$.

Rainwater Harvesting System

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

Analyzing Rate of Approach in a Pursuit Problem

Two cars are traveling on perpendicular roads. Car A is moving east at 60 km/h and is 3 km from the

Chemical Concentration Rate Analysis

The concentration of a chemical in a reactor is given by $$C(t)=\frac{5*t}{t+2}$$ M (moles per liter

Cubic Function with Parameter and Its Inverse

Examine the family of functions given by $$f(x)=x^3+k*x$$, where $$k$$ is a constant.

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 \

Drug Concentration Dynamics

The concentration of a drug in the bloodstream is modeled by $$C(t)= 50*e^{-0.2*t} + 10$$ (in mg/L),

Firework Trajectory Analysis

A firework is launched and its height (in meters) is modeled by the function $$h(t)=-4.9t^2+30t+5$$,

Graphical Analysis of an Inverse Function

Consider a continuously differentiable and strictly increasing function $$f(x)$$ as depicted in the

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

L'Hôpital's Analysis

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

Limit Evaluation via L'Hopital's Rule

Evaluate the limit: $$L=\lim_{x\to 0}\frac{e^{2x}-1}{\ln(1+3x)}$$. 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

Maclaurin Series for ln(1+x)

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

Population Decline Modeled by Exponential Decay

A bacteria population is modeled by $$P(t)=200e^{-0.3t}$$, where t is measured in hours. Answer the

Related Rates in a Circular Pool

A circular pool is being filled such that the surface area increases at a constant rate of $$10$$ ft

Revenue Concavity Analysis

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

Sediment Transport in a Riverbank

In a riverbank environment, sediment is deposited at a rate of $$D(t)=20-0.5t$$ (kg/min) while simul

Series Approximation for a Displacement Function

A displacement function is modeled by $$s(t)= \sum_{n=1}^{\infty} \frac{(-1)^{n+1} t^n}{n}$$, which

Series Approximation in Population Dynamics

A population function is given by $$P(t)= 500 \times \sum_{n=0}^{\infty} \frac{(0.03t)^n}{n!}$$. Ans

Vehicle Motion on a Curved Path

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

Water Tank Flow Analysis

A water tank receives water from an inlet at a rate given by $$I(t)=4+\cos(t)$$ (liters per minute)

Amusement Park Ride Braking Distance

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

Analysis of a Motion Function Incorporating a Logarithm

A particle's position is given by $$s(t)= \ln(t+1)+ t$$, where $$t$$ is in seconds. Analyze the moti

Analysis of a Quartic Function as a Perfect Power

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

Application of Rolle's Theorem

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

Concavity & Inflection Points for a Rational Polynomial Function

Examine the function $$f(x)= \frac{x}{x^2+1}$$ to determine its concavity and identify any inflectio

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$

Determining the Meeting Point of Two Functions

Consider the functions $$f(x)= e^x$$ and $$g(x)= 3 + \ln(x)$$ representing two different processes.

Evaluating an Improper Integral using Series Expansion

The function $$I(x)=\sum_{n=0}^\infty (-1)^n * \frac{(2*x)^{n}}{n!}$$ converges to a known function.

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 Theorem: Finding Global Extrema

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

Function Behavior Analysis

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

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

Loan Amortization with Increasing Payments

A loan of $$20000$$ dollars is to be repaid in equal installments over 10 years. However, the repaym

Mean Value Theorem Application

Let $$f(x)=\ln(x)$$ be defined on the interval $$[1, e^2]$$. Answer the following parts using the Me

Mean Value Theorem in Motion

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

Optimization in Particle Routing

A delivery robot’s distance along its predetermined path is described by $$s(t)=\sin(t)+0.5*t$$, whe

Relative Extrema Using the First Derivative Test

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

Series Convergence and Differentiation in Thermodynamics

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

Sign Chart Construction from the Derivative

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

Taylor Series for $$\cos(2*x)$$

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

Taylor Series for $$e^{\sin(x)}$$

Let $$f(x)=e^{\sin(x)}$$. First, obtain the Maclaurin series for $$\sin(x)$$ up to the $$x^3$$ term,

Verifying the Mean Value Theorem

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

Volume of a Solid of Revolution Using the Washer Method

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

Analyzing a Cumulative Distribution Function (CDF)

A chemical reaction has a time-to-completion modeled by the cumulative distribution function $$F(t)=

Antiderivative with an Initial Condition

Given the function $$f(x)=6*x$$, find a function $$F(x)$$ such that $$F'(x)=f(x)$$ and $$F(2)=5$$.

Approximating an Exponential Integral via Riemann Sums

Consider the function $$h(x)=e^{-x}$$ on the interval $$[0,2]$$. A table of values is provided below

Area Estimation with Riemann Sums

A water flow rate function f(x) (in m³/s) is measured at various times. The table below shows the me

Area Under an Even Function Using Symmetry

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

Cost and Inverse Demand in Economics

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

Cyclist's Distance Accumulation Function

A cyclist’s total distance traveled is modeled by $$D(t)= \int_{0}^{t} (5+\sin(u))\, du + 2$$ kilom

Definite Integration of a Polynomial Function

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

Determining Antiderivatives and Initial Value Problems

Suppose that $$F(x)$$ is an antiderivative of the function $$f(x)=5*x^4 - 2*x + 3$$, and that it is

Displacement vs. Total Distance Traveled

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

Distance from Acceleration Data

A car's acceleration is recorded in the table below. Given that the initial velocity is $$v(0)= 10$$

Estimating Area Under a Curve from Tabular Data

A function $$f(t)$$ is sampled at discrete time points as given in the table below. Using these data

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

Fuel Consumption Estimation with Midpoint Riemann Sums

A vehicle’s fuel consumption rate (in liters per hour) over a trip is recorded at various times. The

Graphical Transformations and Inverse Functions

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

Improper Integral and the p-Test

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

Integration by U-Substitution and Evaluation of a Definite Integral

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

Integration Involving Trigonometric Functions

Evaluate the definite integral $$\int_{0}^{\frac{\pi}{2}} \cos(2*t)\,dt.$$

Integration of a Trigonometric Product via U-Substitution

Evaluate the indefinite integral $$\int \sin(2*x)\cos(2*x)\,dx$$.

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 with Variable Acceleration and Displacement Analysis

A particle moves along a straight line with acceleration given by $$a(t)=4-2*t$$ (in m/s²). At time

Population Growth from Birth Rate

In a small town, the birth rate is modeled by $$B(t)= \frac{100}{1+t^2}$$ people per year, where $$t

Rainfall Accumulation Over Time

A storm produces rainfall at a rate modeled by the function $$r(t)=6 * t^(1/2)$$, where $$0 \le t \l

Recovering Accumulated Change

A company’s revenue rate changes according to $$R'(t)=8*t-12$$ (in dollars per day). If the revenue

Reservoir Water Level

A reservoir experiences a net water inflow modeled by $$W(t)=40*\sin\left(\frac{\pi*t}{12}\right)-5$

Rocket Height Determination via U-Substitution

A rocket’s velocity is modeled by the function $$v(t)=t * e^(t^(2))$$ (in m/s) for $$t \ge 0$$. With

Series Representation and Term Operations

Consider the power series for $$\arctan(x)$$ given by $$\sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{2n+

Temperature Change Analysis

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

Total Work Done by a Variable Force

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

Vehicle Motion and Inverse Time Function

A vehicle’s displacement (in meters) is modeled by the function $$f(t)= t^2 + 4$$ for $$t \ge 0$$ se

Volume by Cross-Section: Squares on a Parabolic Base

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

Volume of a Solid: Cross-Sectional Area

A solid has cross-sectional area perpendicular to the x-axis given by $$A(x)= (4-x)^2$$ for $$0 \le

Capacitor Charging with Leakage

A capacitor is being charged by a constant current source of $$5$$ A, but it also leaks charge at a

Chemical Reactor Mixing

In a chemical reactor, the concentration $$C(t)$$ (in M) of a chemical is governed by the equation $

Compound Interest and Investment Growth

An investment account grows according to the differential equation $$\frac{dA}{dt}=r*A$$, where the

Compound Interest with Continuous Payment

An investment account grows with a continuous compound interest rate $$r$$ and also receives continu

Cooling Model Using Newton's Law

Newton's law of cooling states that the temperature T of an object changes at a rate proportional to

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

Estimating Instantaneous Rate from a Table

A function $$f(x)$$ is defined by the following table of values:

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

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 and Decay

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

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 3: Population Growth and Logistic Model

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

FRQ 7: Projectile Motion with Air Resistance

A projectile is launched vertically upward with an initial velocity of 50 m/s. Its vertical motion i

FRQ 9: Epidemiological Model Differential Equation

An epidemic evolves according to the differential equation $$\frac{dI}{dt}=r*I*(M-I)$$, where $$I(t)

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 $$

Logistic Growth Model

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

Mixing Problem in a Tank

A tank initially contains a certain amount of salt dissolved in 100 L of water. Brine with a known s

Modeling Cooling in a Variable Environment

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

Newton's Law of Cooling

An object cools according to Newton's Law of Cooling: $$\frac{dT}{dt} = -k*(T-20)$$, where the ambie

Optimization in Construction: Minimizing Material for a Container

A manufacturer is designing an open-top cylindrical container with fixed volume $$V$$. The material

Particle Motion with Damping

A particle moving along a straight line is subject to damping and its motion is modeled by the secon

Projectile Motion with Air Resistance

A projectile is fired vertically upward with an initial velocity of $$50\,m/s$$. The projectile expe

Radioactive Decay

A radioactive substance decays according to the differential equation $$\frac{dA}{dt}=-kA$$, where $

Reservoir Contaminant Dilution

A reservoir has a constant volume of 10,000 L and contains a pollutant with amount $$Q(t)$$ (in kg)

Salt Tank Mixing Problem

A tank contains $$100$$ L of water with $$10$$ kg of salt. Brine containing $$0.5$$ kg of salt per l

Separable Differential Equation with a Logarithmic Integral

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

Slope Field and Solution Curve Sketching

Consider the differential equation $$\frac{dy}{dx} = x - 1$$. A slope field for this differential eq

Temperature Control in a Chemical Reaction Vessel

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

Area Between a Parabola and a Line

Consider the region bounded by the curves $$y=5*x - x^2$$ and $$y=x$$ where they intersect. Answer t

Area Between Curves in a Business Context

A company’s revenue and cost (in dollars) for producing items (in hundreds) are modeled by the funct

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 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: Parabola and Line

Consider the functions $$f(x)= 5*x - x^2$$ and $$g(x)= x$$. These curves enclose a region in the pla

Area of One Petal of a Polar Curve

The polar curve defined by $$r = \cos(2\theta)$$ forms a rose with four petals. Find the area of one

Average and Instantaneous Analysis in Periodic Motion

A particle moves along a line with its displacement given by $$s(t)= 4*\cos(2*t)$$ (in meters) for $

Average Chemical Concentration Analysis

In a chemical reaction, the concentration of a reactant (in M) is recorded over time as given in the

Average Fuel Consumption and Optimization

A vehicle's fuel consumption rate is modeled by the function $$f(x)=2*x^2-8*x+10$$, where $$x$$ repr

Average Population Density on a Road

A town's population density along a road is modeled by the function $$P(x)=50*e^{-0.1*x}$$ (persons

Average Reaction Concentration in a Chemical Reaction

In a chemical reaction, the concentration of a reactant is modeled by $$C(t)=20*\exp(-0.5*t)$$ (in m

Average Velocity and Displacement from a Polynomial Function

A car's velocity in m/s is given by $$v(t)=t^2-4*t+3$$ for $$t\in[0,5]$$ seconds. Answer the followi

Center of Mass of a Rod with Variable Density

A rod extending along the x-axis from $$x=0$$ to $$x=10$$ meters has a density given by $$\rho(x)=2+

Complex Integral Evaluation with Exponential Function

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

Determining the Arc Length of a Curve

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

Displacement from a Velocity Graph

A runner’s velocity is given by $$v(t)=8-0.5*t$$ (m/s) for $$0\le t\le 12$$ seconds. A graph of this

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[

Implicit Function Differentiation

Consider the implicitly defined function $$\sin(x * y) + x^2 = \ln(y)$$. Answer the following:

Moment of Inertia of a Thin Plate

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

Particle Motion from Acceleration

A particle has an acceleration given by $$a(t)=3*t-6$$ (m/s²). With initial conditions $$v(0)=2$$ m/

Polar Coordinates: Area of a Region

A region in the plane is described in polar coordinates by the equation $$r= 2+ \cos(θ)$$. Determine

Shadow Length Related Rates

A 1.8-meter tall man is walking away from a 5-meter tall lamp post at a constant speed of $$1.5$$ m/

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

Volume by Shell Method: Rotating a Region

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

Volume of a Solid of Revolution Between Curves

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

Volume of a Solid Using the Shell Method

The region in the first quadrant bounded by $$y=x$$, $$y=0$$, and $$x=4$$ is rotated about the y-axi

Volume of a Solid via Shell Method

Consider the region bounded by $$y=x^2$$ and $$y=4$$ in the first quadrant. This region is revolved

Volume of a Solid: ln(x) Region Rotated

Consider the region in the $$xy$$-plane bounded by $$y=\ln(x)$$, $$y=0$$, $$x=1$$, and $$x=e$$. This

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

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

Analyzing a Walker's Path: A Vector-Valued Function

A pedestrian's path is modeled by the vector function $$\vec{r}(t)= \langle t^2 - 4, \sqrt{t+5} \ran

Arc Length of a Parametric Curve

The curve defined by $$x(t)=\ln(t+1)$$ and $$y(t)=\sqrt{t}$$ for $$t \in [0,4]$$ is given.

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

Area Between Polar Curves

In the polar coordinate plane, consider the region bounded by the curves $$r = 2 + \cos(\theta)$$ (t

Area between Two Polar Curves

Given two polar curves: $$r_1 = 1+\cos(\theta)$$ and $$r_2 = 2\cos(\theta)$$, consider the region wh

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 and Differentiation of a Polar Curve

Consider the polar curve defined by $$ r=2+\sin(\theta) $$. Study its conversion to Cartesian coordi

Conversion from Polar to Cartesian Coordinates

The polar equation $$r(\theta)=4*\cos(\theta)$$ represents a curve.

Conversion of Parametric to Polar: Motion Analysis

An object moves along a path given by the parametric equations $$x(t)=t^2$$ and $$y(t)=2*t$$ for $$t

Converting Polar to Cartesian and Computing Slope

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

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:

Differentiability of a Piecewise-Defined Vector Function

Consider the vector-valued function $$\textbf{r}(t)= \begin{cases} \langle t, t^2 \rangle & \text{i

Exponential and Logarithmic Dynamics in a Polar Equation

Consider the polar curve defined by $$r=e^{\theta}$$. Answer the following:

Exponential Growth in Parametric Representation

A model for population growth is given by the parametric equations $$x(t)=t$$ and $$y(t)=e^{0.3t}$$,

Finding the Slope of a Tangent to a Parametric Curve

Consider the parametric equations $$x(t)=e^t$$ and $$y(t)=e^{-t}$$, where $$t \in \mathbb{R}$$.

Implicit Differentiation with Implicitly Defined Function

Consider the equation $$x^2+xy+y^2=7$$, which defines $$y$$ implicitly as a function of $$x$$.

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

Kinematics in Polar Coordinates

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

Modeling Projectile Motion with Parametric Equations

A projectile is launched with an initial speed of \(20\) m/s at an angle of \(45^\circ\) above the h

Numerical Integration Techniques for a Parametric Curve

A curve is defined by the parametric equations $$x(t)=\ln(t+1)$$ and $$y(t)=\sqrt{t}$$ for $$0 \le t

Parameter Elimination in Logarithmic and Quadratic Relationships

Given the parametric equations $$x(t)= \ln(t)$$ and $$y(t)= t^2 - 4*t + 3$$ for $$t > 0$$, eliminate

Parametric Curve with a Loop and Tangent Analysis

Consider the parametric curve defined by $$x(t)=t^3-3*t$$ and $$y(t)=t^2$$, where $$t\in[-2,2]$$. An

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 Spiral Curve Analysis

The curve defined by $$x(t)=t\cos(t)$$ and $$y(t)=t\sin(t)$$ for $$t \in [0,4\pi]$$ represents a spi

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 Motion in Polar Coordinates

A particle moves in polar coordinates with \(r(\theta)=4-\theta\) and the angle is related to time b

Vector-Valued Functions and Curvature

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

Vector-Valued Functions: Position, Velocity, and Acceleration

Let $$\textbf{r}(t)= \langle e^t, \ln(t+1) \rangle$$ represent the position of a particle in the pla

Vector-Valued Motion: Acceleration and Maximum Speed

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

Velocity and Acceleration of a Particle

A particle’s position in three-dimensional space is given by the vector-valued function $$\mathbf{r}

Work Done by a Force along a Path

A force acting on an object is given by the vector function $$\vec{F}(t)= \langle 3t,\; 2,\; t^2 \ra