AP Calculus BC FRQ Room

Algebraic Manipulation in Limit Evaluation

Consider the function $$f(x)=\frac{x^3-8}{x-2}$$ defined for $$x \neq 2$$. Answer the following:

Algebraic Manipulation with Radical Functions

Let $$f(x)= \frac{\sqrt{x+5}-3}{x-4}$$, defined for $$x\neq4$$. Answer the following:

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 One-Sided Limits and Discontinuities in a Piecewise Function

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

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

Continuity Analysis of a Rational-Piecewise Function

Consider the function $$r(x)=\begin{cases} \frac{x^2-1}{x-1} & x<0, \\ 2*x+c & x\ge0. \end{cases}$$

Continuity Analysis of an Integral Function

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

Evaluating a Complex Limit for Continuous Extension

Consider the function $$ f(x)= \begin{cases} \frac{\ln(1+x+e^x) - (x+e^x-1)}{x^2}, & x \neq 0 \\ C,

Evaluating a Logarithmic Limit

Given the limit $$\lim_{x \to 2} \frac{\ln(x-1)}{x^2-4} = k$$, find the value of $$k$$ using algebra

Evaluating a Rational Function Limit Using Algebraic Manipulation

Consider the function $$f(x)=\frac{x^2-9}{x-3}$$. Analyze the limit as $$x \to 3$$.

Exploring Infinite and Vertical Asymptotes in Rational Functions

Consider the function $$q(x)= \frac{2x^3-x}{x^2-1}$$.

Fuel Efficiency and Speed Graph Analysis

A car manufacturer records data on fuel efficiency (in mpg) as a function of speed (in mph). A graph

Graph Analysis of Discontinuities

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

Horizontal Asymptote of a Rational Function

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

Inflow Function with a Vertical Asymptote

A water reservoir is fed by an inflow given by $$R_{in}(t)=\frac{50\,t}{t-5}$$ liters per minute, de

Intermediate Value Theorem Application

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

Investment Portfolio Rebalancing

An investment portfolio is rebalanced periodically, yielding profits that form a geometric sequence.

Limit Evaluation Involving Radicals and Rationalization

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

Limits with Composite Logarithmic Functions

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

Manufacturing Process Tolerances

A manufacturing company produces components whose dimensional errors are found to decrease as each c

Rational Function and Removable Discontinuity

Consider the function $$g(x) = \frac{(x+3)(x-2)}{(x-2)}$$, defined for $$x \neq 2$$, and suppose tha

Squeeze Theorem with an Oscillatory Factor

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

Trigonometric Limits

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

Vertical Asymptote Analysis in a Rational Function

Consider the function $$g(x)=\frac{x+1}{x-3}$$, which is undefined at $$x=3$$. 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

Analysis of Increasing and Decreasing Intervals

Let $$f(x)=x^4 - 8*x^2$$. Answer the following parts.

Applying the Quotient Rule

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

Composite Function and Chain Rule Application

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

Composite Function Behavior

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

Continuous Compound Interest Analysis

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

Cooling Tank System

A laboratory cooling tank has heat entering at a rate of $$H_{in}(t)=200-10*t$$ Joules per minute an

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 of Inverse Functions

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

Differentiability of an Absolute Value Function

Consider the function $$f(x) = |x|$$.

Epidemiological Rate Change Analysis

In epidemiology, the spread of a disease is sometimes modeled by a combination of logarithmic and ex

Higher Order Derivatives: Concavity and Inflection Points

Consider the function $$f(x)= x^4 - 4*x^3+6*x^2.$$ (a) Find the first derivative \(f'(x)\) and th

Implicit Differentiation on an Ellipse

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

Instantaneous Rate of Change and Series Approximation for √(1+x)

A company models its cost using the function $$C(x)=\sqrt{1+x}$$. To understand small changes in cos

Irrigation Reservoir Analysis

An irrigation reservoir has an inflow rate modeled by $$I(t)=\frac{100}{1+e^{-0.5*(t-10)}}$$ liters

Manufacturing Cost Function and Instantaneous Rate

The total cost (in dollars) to produce x units of a product is given by $$C(x)= 0.2x^3 - 3x^2 + 50x

Marginal Cost Analysis Using Composite Functions and the Chain Rule

A company's cost function is given by $$C(x)= e^{2*x} + \sqrt{x+5}$$, where x (in hundreds) represen

Optimization in Engineering Design

A manufacturer designs a cylindrical can with a fixed volume of $$1000\,cm^3$$. The surface area of

Piecewise Function and Discontinuity Analysis

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

Related Rates in a Conical Tank

Water is draining from a conical tank. The tank has a total height of 10 m and its radius is always

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 and Tangent Slope Analysis

Consider the function $$f(x)=\frac{1}{x}$$ for $$x \neq 0$$. Answer the following:

Tangent Line Approximation

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

Tangent Line Estimation in Transportation Modeling

A vehicle's displacement along a highway is modeled by $$s(t)=\ln(3*t+1)*e^{t}$$, where $$t$$ denote

Tangent Line to a Logarithmic Function

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

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

Taylor Series Expansion of ln(x) About x = 2

For a financial model, the function $$f(x)=\ln(x)$$ is expanded about $$x=2$$. Use this expansion to

Taylor Series for Cos(x) in Temperature Modeling

An engineer uses the cosine function to model periodic temperature variations. Approximate $$\cos(x)

Temperature Change: Secant vs. Tangent Analysis

A scientist recorded the temperature $$T$$ (in °C) at various times $$t$$ (in seconds) as shown in t

Using the Limit Definition for a Non-Polynomial Function

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

Analysis of a Piecewise Function with Discontinuities

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

Chain Rule for a Multi-layered Composite Function

Let $$f(x)= \sqrt{\ln((3*x+2)^5)}$$. Answer the following:

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

Chain Rule in the Context of Light Intensity Decay

The light intensity as a function of distance from the source is given by $$I(x) = 500 * e^{-0.2*\sq

Combined Differentiation: Inverse and Composite Function

Let $$f(x)= \ln(2*x+1)$$ and let $$g$$ be the inverse function of $$f$$. Answer the following parts:

Composite Differentiation in Polynomial Functions

Consider the function $$f(x)= (2*x^3 - x + 1)^4$$. Use the chain rule to differentiate f(x).

Composite Exponential Logarithmic Function Analysis

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

Differentiation Involving Absolute Values and Composite Functions

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

Differentiation of the Inverse Function in a Mechanics Experiment

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

Ice Cream Storage Dynamics

An ice cream storage facility receives ice cream at a rate given by the composite function $$I(t)=d(

Implicit Differentiation in a Chemical Reaction

In a chemical process, the concentrations of two reactants, $$x$$ and $$y$$, satisfy the relation $$

Implicit Differentiation in a Circle

Consider the circle defined by $$ x^2+y^2=49 $$.

Implicit Differentiation in a Conic Section

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

Implicit Differentiation in Economic Equilibrium

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

Inverse Function Differentiation for a Cubic Function

Let $$f(x)= x^3 + x$$ be an invertible function with inverse $$g(x)$$. Use the inverse function deri

Inverse Function Differentiation in a Logarithmic Context

Let $$f(x)= \ln(x+2) - x$$, and let $$g$$ be its inverse function. Answer the following:

Inverse of a Composite Function

Let $$f(x)=\sqrt{3*x+1}$$ and $$g(x)=x^2-1$$, and define $$h(x)=f(g(x))$$. Analyze the invertibility

Investigating the Inverse of a Rational Function

Consider the function $$f(x)=\frac{2*x-1}{x+3}$$ with $$x \neq -3$$. Analyze its inverse.

Optimization in Manufacturing Material

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

Projectile Motion and Composite Exponential Functions

A projectile’s height at time $$t$$ (in seconds) is modeled by the function $$h(t)= e^{- (t-2)^2}$$.

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 $$C(t) = \sin(0.2*t) + 1$$,

Second Derivative of an Implicit Function

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

Trigonometric Composite Inverse Function Analysis

Consider the function $$f(x)=\sin(x)+x$$ defined on the interval $$\left[-\frac{\pi}{2}, \frac{\pi}{

Water Tank Composite Rate Analysis

A water tank receives water from an inflow pipe where the inflow rate is given by the composite func

Air Conditioning Refrigerant Balance

An air conditioning system is charged with refrigerant at a rate given by $$I(t)=12-0.5t$$ (kg/min)

Analyzing Experimental Temperature Data

A laboratory experiment records the temperature of a chemical reaction over time. The temperature (i

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

The population of a bacterial culture is modeled by $$P(t)=1000e^{0.3*t}$$, where $$P(t)$$ is the nu

Compound Interest Rate Change

An investment grows according to $$A(t)=5000e^{0.07t}$$, where t is measured in years. Answer the fo

Curvature Analysis in the Design of a Bridge

A bridge's vertical profile is modeled by $$y(x)=100-0.5*x^2+0.05*x^3$$, where $$y$$ is in meters an

Differentiation of a Product Involving Exponentials and Logarithms

Consider the function $$f(t)=e^{-t}\ln(t+2)$$, defined for t > -2. Answer the following:

Draining Conical Tank

Water is draining from a conical tank at a rate of $$5$$ m³/min. The tank has a height of $$10$$ m a

Economic Optimization: Profit Maximization

A company's profit (in thousands of dollars) is modeled by $$P(x) = -2x^2 + 40x - 150$$, where $$x$$

Estimation Error with Differentials

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

Filling a Conical Tank

A conical water tank has its radius related to its height by $$r=\frac{h}{2}$$, and its volume is gi

Implicit Differentiation: Tangent to a Circle

Consider the circle given by $$x^2 + y^2 = 25$$.

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

Infrared Sensor Distance Analysis

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

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 Revenue in a Business Model

A company’s revenue as a function of price is given by $$R(p)= p*(100 - 2*p)$$, where $$p$$ is the p

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

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

Ozone Layer Recovery Simulation

In a simulation of ozone layer dynamics, ozone is produced at a rate of $$I(t)=\frac{25}{t+1}$$ (Dob

Pool Water Volume Change

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

Population Growth Rate Analysis

A population grows exponentially according to $$P(t)=1200e^{0.15t}$$, where t is measured in months.

Related Rates: Expanding Circular Ripple

A circular ripple in a pond expands such that its area, given by $$A=\pi r^2$$, is increasing at a c

Related Rates: Inflating Spherical Balloon with Exponential Volume Rate

A spherical balloon is being inflated so that its volume changes at a rate of $$\frac{dV}{dt}=8e^{0.

Revenue and Marginal Analysis

A company’s revenue function is given by $$R(p)= p*(1000 - 5*p)$$, where $$p$$ is the price per unit

Surface Area of a Solid of Revolution

Consider the curve $$y = \ln(x)$$ for $$x \in [1, e]$$. Find the surface area of the solid formed by

Urban Traffic Flow Analysis

An urban highway ramp experiences an inflow of cars at a rate of $$I(t)=40+2t$$ (cars per minute) an

Analysis of a Piecewise Function's Differentiability and Extrema

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

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

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

Concavity and Inflection Points in a Trigonometric Function

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

Concavity and Inflection Points of an Exponential Log Function

Consider the function $$f(x)= x\,e^{-x} + \ln(x)$$ for $$x > 0$$. Analyze the concavity of f.

Echoes in an Auditorium

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

Implicit Differentiation and Tangent to an Ellipse

Consider the ellipse defined by the equation $$4*x^2 + 9*y^2 = 36$$. Answer the following parts:

Instantaneous vs. Average Rates in a Real-World Model

A company’s monthly revenue is modeled by $$ R(t)=0.5t^3-4t^2+12t+100, \quad 0 \le t \le 6,$$ where

Inverse Analysis of a Cubic Polynomial

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

Inverse Function and Critical Points in a Business Context

A company models its profit (in thousands of dollars) by $$f(x)= \ln(4*x+7)$$ for $$x \ge 0$$, where

Loan Amortization with Increasing Payments

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

Logarithmic-Quadratic Combination Inverse Analysis

Consider the function $$f(x)= \ln(x^2+1)$$ for $$x \ge 0$$. Answer the following parts.

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 $$

Maximizing the Area of an Inscribed Rectangle

A rectangle is inscribed in a circle of radius $$5$$. Determine the dimensions of the rectangle that

Mean Value Theorem Application

Let \( f(x) = x^3 - 3*x^2 + 2 \) be defined on the closed interval \([0,3]\). Answer the following p

Mean Value Theorem in a Temperature Model

The temperature over a day (in °C) is modeled by $$T(t)=10+8*\sin\left(\frac{\pi*t}{12}\right)$$ for

Mean Value Theorem in Motion

A car travels along a straight road and its position is modeled by $$s(x) = x^2$$ (in kilometers), w

Motion Analysis: Particle’s Position Function

A particle moves along a straight line and its position is given by $$s(t)=t^4 - 8*t^2 + 16$$ (in me

Optimization with a Combined Logarithmic and Exponential Function

A company's revenue is modeled by $$R(x)= x\,e^{-0.05x} + 100\,\ln(x)$$, where x (in hundreds) repre

Pharmaceutical Dosage and Metabolism

A patient is administered a medication with an initial dose of 50 mg. Due to metabolism, the amount

Planar Particle Motion with Time-Dependent Accelerations

A particle moves in the plane with its position given by $$\vec{s}(t)=\langle t^2-4*t+4,\; \ln(t+1)\

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

Profit Maximization in Business

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

Projectile Motion and Maximum Height

A projectile is launched with its height (in meters) given by the function $$h(t) = -5*t^2 + 20*t +

Rate of Change in a Chemical Reaction

The concentration of a reactant in a chemical reaction is modeled by $$C(t)=10*e^{-0.5*t} + 2$$ (in

Region Area and Volume: Polynomial and Linear Function

A region in the x-y plane is bounded by the curves $$f(x)=x^2$$ and $$g(x)=2 - x$$. Answer the follo

Relative Motion in Two Dimensions

A point moves in the plane along the path defined by $$x(t)=t^2-4*t+3$$ and $$y(t)=\ln(t+1)$$ for $$

Second Derivative Test for Critical Points

Consider the function $$f(x)=x^3-9*x^2+24*x-16$$.

Series Representation in a Biological Growth Model

A biological process is approximated by the series $$B(t)=\sum_{n=0}^\infty (-1)^n * \frac{(0.3*t)^n

Volume by Cross Sections Using Squares

A region in the xy-plane is bounded by $$y=x$$, $$y=0$$, and $$x=3$$. Perpendicular to the x-axis, c

Accumulated Change via U-Substitution

Evaluate the definite integral representing the accumulated amount of a substance given by $$\int_{1

Accumulation Function Analysis

A function $$A(x) = \int_{0}^{x} (e^{-t} + 2)\,dt$$ represents the accumulated amount of a substance

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

Application of the Fundamental Theorem

Consider the function $$f(x)=x^2+2*x$$ defined on the interval $$[1,4]$$. Evaluate the definite inte

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 \

Area Under the Curve for a Quadratic Function

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

Average Value of a Function on an Interval

Let $$f(x)=\ln(x)$$ be defined on the interval $$[1,e]$$. Determine the average value of $$f(x)$$ on

Bacteria Growth with Nutrient Supply

A bacterial culture in a laboratory is provided with nutrients at a rate of $$N(t)=6*\ln(t+1)$$ mg/m

Biomedical Modeling: Drug Concentration Dynamics

A drug concentration in the bloodstream is modeled by $$f(t)= 5\left(1 - e^{-0.3*t}\right)$$ for $$t

Convergence of an Improper Integral

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

Cost Accumulation from Marginal Cost Function

A company’s marginal cost function $$MC(q)$$ (in dollars per unit) for producing $$q$$ units is give

Definite Integral using U-Substitution

Evaluate the integral $$\int_{1}^{5} (2*x - 3)^4\,dx$$ using the method of u-substitution.

Definite Integral via U-Substitution

Evaluate the definite integral $$\int_{1}^{3} (2*x-1)^6\,dx$$ using u-substitution.

Distance vs. Displacement from a Velocity Function

A runner's velocity is modeled by $$v(t)=5-0.5*t$$ (in m/s) for $$0\le t\le10$$. The runner may chan

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 Area Under a Curve via Riemann Sums

The following table shows values of a function f(t): | t | 0 | 2 | 4 | 6 | 8 | |---|---|---|---|---

Evaluating a Piecewise Function with a Removable Discontinuity

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

Evaluating an Integral Using U-Substitution

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

Evaluation of an Improper Integral

Consider the integral $$\int_{1}^{\infty} \frac{1}{x^{2}} dx$$. Answer the following:

Finding Area Between Two Curves

Consider the curves $$f(x)=x^2$$ and $$g(x)=4*x$$.

Finding the Area Between Curves

Find the area of the region bounded by the curves $$y=4-x^2$$ and $$y=x$$.

Inverse Functions in Economic Models

Consider the function $$f(x) = 3*x^2 + 2$$ defined for $$x \ge 0$$, representing a demand model. Ans

Modeling Water Inflow Using Integration

Water flows into a tank at a rate given by $$R(t)=4-0.5*t$$ (in liters per minute) for $$t\in[0,8]$$

Power Series Analysis and Applications

Consider the function with the power series representation $$f(x)=\sum_{n=0}^\infty \frac{(-1)^n x^{

Revenue Estimation Using the Trapezoidal Rule

A company recorded its daily revenue (in thousands of dollars) over four days. Use the data in the t

Riemann Sum Approximation of Area

Given the following table of values for the function $$f(x)$$ on the interval $$[0,4]$$, use Riemann

Volume of a Solid with Square Cross-Sections

Consider the region bounded by the curve $$y=x^{2}$$ and the line $$y=4$$. Cross-sections taken perp

Volume of Water Flow in a River

The water flow rate through a river, given in cubic meters per second, is measured at different time

Work on a Nonlinear Spring

A nonlinear spring exerts a force given by $$F(x)=8 * e^(0.3 * x)$$ (in Newtons) as a function of di

Autocatalytic Reaction Dynamics

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

Bacterial Growth with Predation

A bacterial culture grows according to the differential equation $$\frac{dB}{dt}= r*B - P$$, where $

Chemical Reaction Rate

In a chemical reaction, the concentration $$C(t)$$ of a reactant decreases according to the first-or

Cooling Coffee Data Analysis

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

Euler's Method and Differential Equations

Consider the initial value problem $$\frac{dy}{dx} = x*y$$ with the initial condition $$y(0)=1$$. Eu

Exact Differential Equation

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

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 12: Bacterial Growth with Limiting Resources

A bacterial culture grows according to the differential equation $$\frac{dP}{dt}=r*P-c*P^2$$, where

Homogeneous Differential Equation

Solve the homogeneous differential equation $$\frac{dy}{dx}= \frac{x^2+y^2}{x*y}$$ using the substit

Investment Account Growth with Fees

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

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

Mixing Problem with Differential Equations

A tank initially contains $$S(0)=S_0$$ grams of salt dissolved in a volume $$V$$ liters of water. Br

Modeling Currency Exchange Rates with Differential Equations

Suppose the exchange rate $$E(t)$$ (domestic currency per foreign unit) evolves according to the dif

Modeling Medication Concentration in the Bloodstream

A patient receives an intravenous drug at a constant rate $$R$$ (mg/min) and the drug is eliminated

Motion along a Line with a Separable Differential Equation

A particle moves along a straight line according to the differential equation $$\frac{dy}{dx} = \fra

Newton's Law of Cooling

Newton's Law of Cooling is given by the differential equation $$\frac{dT}{dt} = -k*(T-T_a)$$, where

Newton's Law of Cooling

An object cooling in a room follows Newton's law of cooling described by $$\frac{dT}{dt} = -k*(T-A)$

Optimization in Construction: Minimizing Material for a Container

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

Parametric Equations and Differential Equations

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

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

Population Dynamics with Harvesting

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

Population Dynamics with Harvesting

A fish population in a lake is modeled by the logistic equation with harvesting: $$\frac{dP}{dt}=r\,

Projectile Motion with Drag

A projectile is launched horizontally with an initial velocity $$v_0$$. Due to air resistance, the h

RC Circuit Differential Equation

In an RC circuit, the capacitor charges according to the differential equation $$\frac{dQ}{dt}=\frac

Relative Motion with Acceleration

A car starts from rest and its velocity $$v(t)$$ (in m/s) satisfies the differential equation $$\fra

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

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

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

Slope Field Analysis for $$\frac{dy}{dx}=x$$

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

Slope Field and Sketching a Solution Curve

The differential equation $$\frac{dy}{dx}=x-y$$ has been represented by a slope field. Answer the fo

Solving a Linear Differential Equation using an Integrating Factor

Consider the linear differential equation $$\frac{dy}{dx} + \frac{2}{x} * y = \frac{\sin(x)}{x}$$ wi

Solving a Separable Differential Equation

Solve the differential equation $$\frac{dy}{dx} = \frac{4*x}{y}$$ with the initial condition $$y(0)=

Temperature Control in a Chemical Reaction Vessel

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

Temperature Regulation in Biological Systems

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

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

Arc Length of a Logarithmic Curve

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

Area Between 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 in a Physical Context

The heights of two particles moving along parallel tracks are given by $$h_1(t)=t^2$$ and $$h_2(t)=4

Area Between Curves: Parabolic and Linear Functions

Consider the functions $$f(x)=5*x-x^2$$ and $$g(x)=x$$. Determine the area enclosed between these cu

Average and Instantaneous Acceleration

For a particle, the acceleration is given by $$a(t)=4*\sin(t)-t$$ (in m/s²) for $$t\in[0,\pi]$$. Giv

Average Chemical Concentration Analysis

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

Average Population Density

In an urban study, the population density (in thousands per km²) of a city is modeled by the functio

Average Power Consumption

A household's power consumption is modeled by the function $$P(t)=3+2*\sin\left(\frac{\pi}{12}*t\rig

Average Velocity of a Runner

A runner's velocity is modeled by $$v(t)=5+3\cos(0.5*t)$$ (m/s) for $$0\le t\le10$$ seconds. Answer

Bonus Payout: Geometric Series vs. Integral Approximation

A company issues monthly bonuses that decrease by 20% each month. The bonus in the first month is $5

Determining the Arc Length of a Curve

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

Error Analysis in Taylor Polynomial Approximations

Let $$h(x)= \cos(3*x)$$. Analyze the error involved when approximating $$h(x)$$ by its third-degree

Inflow vs Outflow: Water Reservoir Capacity

A reservoir receives water with an inflow rate given by $$I(t)=20+5\sin(t)$$ (liters/min) and discha

Optimization of Material Usage in a Container

A container's volume is given by $$V(h)=\int_0^h \pi*(3-0.5*\ln(1+x))^2dx$$, where $$h$$ is the heig

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 Motion: Position, Velocity, and Acceleration

A particle moves along a straight line with acceleration $$a(t)=4-2*t$$ (in m/s²), initial velocity

Projectile Motion under Gravity

An object is projected vertically upward with an initial velocity of $$20$$ m/s and from an initial

Rainfall Accumulation Analysis

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

Rainfall Accumulation Analysis

The rainfall rate (in cm/hour) at a location is modeled by $$r(t)=0.5+0.1*\sin(t)$$ for $$0 \le t \l

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 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 Hollow Cylinder Using the Washer Method

A manufacturer designs a hollow cylindrical container. The outer surface is modeled by $$y=10-\sqrt{

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 Revolution between sin(x) and cos(x)

Consider the region bounded by $$y = \sin(x)$$ and $$y = \cos(x)$$ over the interval where they inte

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 $$ r=3*\cos(3\theta) $$.

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 Polar Curve

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

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

Circular Motion Analysis

A particle moves in a circle according to the vector-valued function $$\vec{r}(t)=<3\cos(t),\, 3\sin

Comparing Parametric, Polar, and Cartesian Representations

An object moves along a curve described by the parametric equations $$x(t)= \frac{t}{1+t^2}$$ and $$

Concavity and Inflection Points of a Parametric Curve

For the curve defined by $$x(t)=e^{t}-t$$ and $$y(t)=\ln(1+t^2)$$ for $$t \ge 0$$, answer the follow

Conversion and Tangents in Polar Coordinates

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

Converting and Analyzing a Polar Equation

Examine the polar equation $$r=2+3\cos(\theta)$$.

Differentiability of a Piecewise-Defined Vector Function

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

Drone Altitude Measurement from Experimental Data

A drone’s altitude (in meters) is recorded at various times (in seconds) as shown in the table below

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\

Helical Particle Motion

A particle travels along a helical path described by $$\vec{r}(t)= \langle \cos(t),\; \sin(t),\; t \

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 in a Damped Force Field

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

Motion in the Plane: Logarithmic and Radical Components

A particle’s position in the plane is given by the vector-valued function $$\mathbf{r}(t)=\langle \l

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 $$t > 1$$

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

Projectile Motion Modeled by Vector-Valued Functions

A projectile is launched with an initial velocity vector $$\vec{v}_0=\langle 10, 20 \rangle$$ (in m/

Projectile Motion using Parametric Equations

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

Projectile Motion with Parametric Equations

A ball is launched from ground level with an initial speed of $$20 \text{ m/s}$$ at an angle of $$\f

Vector-Valued Function and Particle Motion

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