AP Calculus BC FRQ Room

Bacterial Growth Experiment

A laboratory experiment involves a bacterial culture whose population at hour $$n$$ is modeled by a

Caffeine Metabolism in the Human Body

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

Complex Rational Function and Continuity Analysis

Let $$R(x)= \frac{x^3-8}{x-2}$$ for $$x \neq 2$$.

Composite Function and Continuity

Consider the piecewise function $$ g(x)=\begin{cases} x^2 & \text{if } x<2, \\ 3x-2 & \text{if } x\

Continuity Analysis of a Piecewise Function

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

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 Involving a Radical Expression

Examine the function $$f(x)= \begin{cases} \frac{\sqrt{x+4}-2}{x} & x \neq 0 \\ k & x=0 \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 $$f(0)=L$$.

Economic Growth and Continuity

The function $$E(t)$$ represents an economy's output index over time (in years). A table provides th

End Behavior of an Exponential‐Log Function

Consider the function $$f(x)= e^{-x} \ln(1+x)$$. Analyze its behavior by investigating the limit as

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,

Exploring Removable and Nonremovable Discontinuities

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

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 Curve and Its Tangent Line

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

Intermediate Value Theorem Application

Let $$f(x)=x^3-4*x+1$$, which is continuous on the real numbers. Answer the following:

Intermediate Value Theorem in Water Tank Levels

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

Limit Behavior in a Container Optimization Problem

A manufacturer designs a closed cylindrical container with a fixed volume $$V$$ (in cubic units). Th

Limits and Continuity of Radical Functions

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

Limits at a Point: Removable Discontinuity Analysis

Consider the function $$f(x)=\frac{(x+3)*(x-2)}{(x+3)*(x+5)}$$ which is not defined at $$x=-3$$. Ans

Limits Involving Radicals

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

Limits Involving Trigonometric Functions

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

Limits with a Parameter in a Trigonometric Function

Consider the function $$f(x)= \begin{cases} \frac{\sin(a*x)}{x} & x \neq 0 \\ b & x=0 \end{cases}$$,

Non-Existence of a Limit due to Oscillation

Consider the function $$h(x)= \sin(\frac{1}{x})$$. Answer the following regarding its limit as x app

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} $

Pendulum Oscillations and Trigonometric Limits

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

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

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

Reciprocal Function Behavior and Asymptotes

Examine the function $$f(x)= \frac{1}{x-1}$$.

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:

Trigonometric Function and the Squeeze Theorem

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

Using the Squeeze Theorem for Trigonometric Limits

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

Water Flow Measurement Analysis

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

Analyzing a Function with an Oscillatory Component

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

Calculating Velocity and Acceleration from a Position Function

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

Chemical Reaction Rate

In a chemical reaction, the concentration of a reactant is modeled by $$C(t)=10 - 2*\ln(t+1)$$, wher

Compound Exponential Rate Analysis

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

Comprehensive Analysis of $$e^{-x^2}$$

The function $$f(x)=e^{-x^2}$$ is used to model temperature distribution in a material. Provide a co

Derivative via the Limit Definition: A Rational Function

Consider the function $$f(x)=\frac{1}{x+2}$$. Use the limit definition of the derivative to find $$f

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

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:

Exponential Population Growth in Ecology

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

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

Instantaneous Rate of Change of a Trigonometric Function

Consider the function $$h(t)=\sin(2*t) + \cos(t)$$ which models the displacement (in centimeters) of

Interpreting Graphical Slope Data

A laboratory experiment measures the velocity (in m/s) of a moving object over time. A graph of the

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

Projectile Motion Analysis

A projectile is launched and its height in feet at time $$t$$ seconds is given by $$h(t)=-16*t^2+32*

Radioactive Decay and Derivative

A radioactive substance decays according to $$M(t)= 50e^{-0.03t}$$, where t is in years and M(t) is

Related Rates: Two Moving Vehicles

A car is traveling east at 60 km/h and a truck is traveling north at 80 km/h. Let $$x$$ and $$y$$ be

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:

Second Derivative and Concavity Analysis

Consider the function $$f(x)=x^3-6*x^2+12*x-5$$. Answer the following:

Temperature Change: Secant vs. Tangent Analysis

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

Testing Differentiability at a Junction Point

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

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

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

Widget Production Rate

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

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

Bacterial Growth and Nutrient Concentration

A bacterial culture grows such that the number of bacteria at the end of each hour is given by the g

Chain Rule and Higher-Order Derivatives

Given the function $$f(x)= \ln(\sqrt{1 + e^{3*x}})$$, answer the following parts:

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 Combined with Inverse Trigonometric Differentiation

Let $$h(x)= \arccos((2*x-1)^2)$$. Answer the following:

Chain Rule in a Trigonometric Light Intensity Model

A light sensor records the intensity of light according to the function $$I(x) = \cos(\sqrt{3*x + 2}

Chain Rule in Oscillatory Motion

A mass-spring system has its displacement modeled by $$ s(t)= e^{-0.5*t}\cos(3*t) $$.

Chain, Product, and Implicit: A Motion Problem

A particle moves along a curve defined by the parametric equations $$x(t)=e^{-t}\cos(t)$$ and $$y(t)

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:

Complex Composite and Implicit Function Analysis

Consider the equation $$e^{x*y}+\ln(x+y)=2$$, where y is defined implicitly as a function of x. Answ

Composite and Implicit Differentiation with Trigonometric Functions

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

Composite Function with Hyperbolic Sine

A cable's displacement over time is modeled by $$s(t)= \sinh(\ln(t+1))$$, where $$t$$ is in seconds.

Composite Functions in a Biological Model

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

Composite Implicit Differentiation Involving Trigonometric and Polynomial Terms

Consider the relation $$\sin(x*y) + y^3 = x$$.

Differentiation in a Logistic Population Model

The population of a species is modeled by the logistic function $$P(t)= \frac{1000}{1+e^{-0.3*(t-5)}

Fuel Tank Dynamics

A fuel storage tank is being filled by a pump at a rate given by the composite function $$P(t)=(4*t+

Graphical Analysis of a Composite Function

Let $$f(x)=\ln(4*e^{x}+1)$$. A graph of f is provided. Answer the following parts.

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 for a Spiral Equation

Consider the curve given by the equation $$x^2 + y^2 = 4*x*y$$. Analyze its derivative using implici

Implicit Differentiation in Trigonometric Equations

For the equation $$\cos(x*y) + x^2 - y^2 = 0$$, y is defined implicitly as a function of x.

Inverse Analysis of a Log-Polynomial Function

Consider the function $$f(x)=\ln(x^2+1)$$. Analyze its one-to-one property on the interval $$[0,\inf

Inverse Function Derivative with Logarithms

Let $$f(x)= \ln(x+2) + x$$ with inverse function $$g(x)$$. Find the derivative $$g'(y)$$ in terms of

Inverse Function Differentiation in a Radical Context

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

Inverse Function in Logistic Population Growth

A population model is given by $$P(t)=\frac{100}{1+4e^{-0.5*t}}$$ for t \ge 0. Analyze the inverse f

Inverse Trigonometric Functions in Navigation

A ship navigates such that its angular position relative to a fixed reference is given by $$\theta =

Logarithmic and Exponential Composite Function with Transformation

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

Maximizing the Garden Area

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

Rainwater Harvesting System

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

Related Rates: Temperature Change in a Moving Object

An object moves along a path where its temperature is given by $$T(x)= \ln(3*x + 2)$$ and its positi

Second Derivative of an Implicit Function

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

Arc Length Calculation

Consider the curve $$y = \sqrt{x}$$ for $$x \in [1, 4]$$. Determine the arc length of the curve.

Conical Tank Water Flow

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

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

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

Inverse Trigonometric Composition

Consider the function $$f(x)=2*\sin(x)-1$$ defined on $$\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$$.

L'Hôpital's Rule in Context

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

Linearization in Engineering Load Estimation

In an engineering project, the load on a beam is modeled by $$L(x)=100*x^2+50*x+200$$, where $$L(x)$

Linearization in Inverse Function Approximation

Let $$f(x)=x^5+2*x+1$$ be a one-to-one function. Although its inverse cannot be found explicitly, li

Linearization to Estimate Change in Electrical Resistance

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

Maximizing a Rectangular Enclosure Area

A farmer has 100 m of fencing to enclose a rectangular area. Answer the following:

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

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

Particle Motion Along a Line with Polynomial Velocity

A particle moves along the x-axis with velocity $$v(t)=4*t^3-9*t^2+6*t-1$$ (m/s). Given that $$s(0)=

Particle Motion Analysis

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

Particle Motion with Changing Acceleration

A particle moves along a straight line with an acceleration given by $$a(t)=8-3*t$$ (in m/s²). Given

Popcorn Sales Growth Analysis

A movie theater observes that the number of popcorn servings sold increases by 15% each week. Let $$

Projectile Motion with Exponential Term

A projectile's height is given by $$h(t)=50t-5t^2+e^{-0.5t}$$, where h is measured in meters and t i

Radical Function Inversion

Let $$f(x)=\sqrt{2*x+5}$$ represent a measurement function. Analyze its inverse.

Savings Account and Interest Accrual

A student starts with an initial savings account balance of $$B_0=1000$$ dollars and makes monthly d

Varying Acceleration and Particle Motion

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

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)

Analysis of a Piecewise Function's Differentiability and Extrema

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

Analysis of an Exponential-Linear Function

Consider the function $$p(x)=e^x-4*x$$. Answer the following parts:

Analysis of Relative Extrema and Increasing/Decreasing Intervals

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

Analyzing Convergence of a Modified Alternating Series

Consider the series $$S(x)=\sum_{n=1}^\infty (-1)^n * \frac{(x+2)^n}{n}$$. Answer the following.

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

Chemical Reaction Rate

During a chemical reaction, the concentration of a reagent (in M) is measured over time (in minutes)

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

Exploring Inverses of a Trigonometric Transformation

Consider the function $$f(x)= 2*\tan(x) + x$$ defined on the interval $$(-\pi/4, \pi/4)$$. Answer th

Finding Local Extrema for an Exponential-Logarithmic Function

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

Inverse Function Analysis in an Optimization Scenario

Consider the cost function $$f(x)= x^4 + 2*x^2 + 1$$ defined for $$x \ge 0$$, where f(x) represents

Mean Value Theorem in Temperature Analysis

A city’s temperature is modeled by the function $$T(t)= t^3 - 6*t^2 + 9*t + 5$$ (in °C), where $$t$$

Optimal Timing via the Mean Value Theorem

A particle’s position is given by $$s(t)=t^2e^{-t}+3$$ for $$t\in[0,3]$$.

Optimization in a Geometric Setting: Garden Design

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

Optimization: Maximizing Rectangular Area with a Fixed Perimeter

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

Parameter-Dependent Concavity Conditions

Consider the function $$ f(x)=x^3+a*x^2+2x,$$ where $$a$$ is a real parameter. Answer the following

Particle Motion on a Curve

A particle moves along a straight-line path with its position given by \( s(t)=t^3 - 6*t^2 + 9*t + 1

Projectile Motion Analysis

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

Rate of Change and Inverse Functions

Let $$f(x)=x^3 + 3*x + 1$$, which is one-to-one. Investigate the rate of change of \(f(x)\) and its

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

Retirement Savings with Diminishing Deposits

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

Square Root Function Inverse Analysis

Consider the function $$f(x)= \sqrt{3*x+4}$$ defined for $$x \ge -\frac{4}{3}$$. Answer the followin

Taylor Series for $$\sqrt{1+x}$$

Consider the function $$f(x)=\sqrt{1+x}$$. In this problem, compute its 3rd degree Maclaurin polynom

Taylor Series in Economics: Cost Function

An economic cost function is modeled by $$C(x)=1000\,e^{-0.05*x}+50\,x$$, where x represents the pro

Vector Analysis of Particle Motion

A particle moves in the plane with its position given by the vector function $$\mathbf{r}(t) = \lang

Volume Using Cylindrical Shells

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

Accumulated Population Change from a Growth Rate Function

A population changes at a rate given by $$P'(t)= 0.2*t^2 - 1$$ (in thousands per year) for t between

Accumulation Function and the Fundamental Theorem of Calculus

Let $$F(x) = \int_{2}^{x} \sqrt{1 + t^3}\, dt$$. Answer the following parts regarding this accumulat

Accumulation Function from a Rate Function

The rate at which water flows into a tank is given by $$r(t)=3\sqrt{t}$$ (in liters per minute) for

Advanced Inflow/Outflow Dynamics

A reservoir receives water from a river at a rate given by $$f(t)=50*(1+0.1*t)$$ cubic meters per ho

Analyzing an Invertible Cubic Function

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

Analyzing and Integrating a Function with a Removable Discontinuity

Consider the function $$ f(x)= \begin{cases} \frac{x^2-9}{x-3} & \text{if } x \neq 3,\\ 4 & \text{if

Antiderivatives and the Constant of Integration

Consider the rate function $$ r(t)= 2*t + 3 $$ where t represents time in seconds.

Antiderivatives and the Fundamental Theorem of Calculus

Given the function $$f(x)= 2*x+3$$, use the Fundamental Theorem of Calculus to evaluate the definite

Arc Length Calculation

Find the arc length of the curve $$y=\frac{1}{3}x^{3/2}$$ from $$x=0$$ to $$x=9$$.

Area Between a Curve and Its Tangent

For the function $$f(x)=x^3-3*x^2+2*x$$, analyze the area between the curve and its tangent line at

Average Value of an Exponential Function

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

Comparing Riemann Sum Approximations for an Increasing Function

A function f(x) is given in the table below: | x | 0 | 2 | 4 | 6 | |---|---|---|---|---| | f(x) | 3

Cost Accumulation from Marginal Cost Function

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

Cross-Sectional Area of a River Using Trapezoidal Rule

The depth $$h(x)$$ (in meters) of a river’s cross-section is measured at various points along a hori

Determining Constant in a Height Function

A ball is thrown upward with a constant acceleration of $$a(t)= -9.8$$ m/s² and an initial velocity

Distance Traveled by a Particle

A particle has a velocity function given by $$v(t)=t^2-4*t+3$$ (in m/s) for $$t\in [0,5]$$ seconds.

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

Error Estimation in Riemann Sum Approximations

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

Fundamental Theorem and Total Accumulated Growth

A bacteria culture grows according to the logistic model $$\frac{dN}{dt}=N\left(1-\frac{N}{10000}\r

Heat Energy Accumulation

The rate of heat transfer into a container is given by $$H(t)= 5\sin(t)$$ kJ/min for $$t \in [0,\pi]

Improper Integral Convergence

Examine the convergence of the improper integral $$\int_1^\infty \frac{1}{x^p}\,dx$$.

Integration of a Complex Trigonometric Function

Evaluate the integral $$\int_{0}^{\pi/2} 4*\cos^3(t)*\sin(t) dt$$.

Integration of a Trigonometric Product via U-Substitution

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

Investigating Partition Sizes

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

Midpoint Riemann Sum for $$f(x)=\frac{1}{1+x^2}$$

Consider the function $$f(x)=\frac{1}{1+x^2}$$ on the interval $$[-1,1]$$. Use the midpoint Riemann

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 $$t\in[0,8]$$

Temperature Change in a Material

A laser heats a material such that its temperature changes at a rate given by $$\frac{dT}{dt} = 2*\s

Temperature Function Analysis with Inverses

A temperature profile over time is given by $$f(t)= \ln(2*t + 3)$$ for $$t \ge 0$$ (with temperature

Volume by Disk Method of a Rotated Region

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

Bacteria Growth with Antibiotic Treatment

A bacterial culture has a population $$N(t)$$ that grows at a rate proportional to its size, given b

Chemical Reaction Rate Modeling

In a chemical reaction, the concentration $$C(t)$$ (in moles per liter) of a reactant decreases acco

Compound Interest and Investment Growth

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

Cooling Coffee Data Analysis

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

Cooling of a Smartphone Battery

A smartphone battery cools according to Newton’s law: $$\frac{dT}{dt} = -k*(T-T_{room})$$. Initially

Cooling of an Electronic Component

An electronic component overheats and its temperature $$T(t)$$ (in $$^\circ C$$) follows Newton's La

Disease Spread Model

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

Drug Concentration in the Bloodstream

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

Environmental Modeling Using Differential Equations

The concentration $$C(t)$$ of a pollutant in a lake is modeled by the differential equation $$\frac{

Free-Fall with Air Resistance

An object falling under gravity experiences air resistance proportional to the square of its velocit

FRQ 18: Enzyme Reaction Rates

A chemical concentration $$C(t)$$ in a reaction decreases according to the differential equation $$\

Implicit Differentiation from an Implicitly Defined Relation

Consider the implicit equation $$x^3 + y^3 - 3*x*y = 0$$ which defines $$y$$ as a function of $$x$$

Implicit Solution from Separated Variables

Consider the differential equation $$\frac{dy}{dx} = \frac{x*y}{1+x^2}$$ with the initial condition

Infectious Disease Spread Model

In a closed population of N individuals, the number of infected individuals $$I(t)$$ is modeled by t

Logistic Model in Population Dynamics

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

Logistic Population Growth Model

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

Logistic Population Model

A fish population is modeled by the logistic differential equation $$\frac{dP}{dt}= r*P\left(1-\frac

Mixing Problem in a Tank

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

Mixing Problem with Constant Flow Rate

A tank holds 500 L of water and initially contains 10 kg of dissolved salt. Brine with a salt concen

Modeling Disease Spread with Differential Equations

In a simple model for disease spread, the number of infected individuals, $$I(t)$$, evolves accordin

Newton's Law of Cooling

An object with an initial temperature of $$80^\circ C$$ is placed in a room at a constant temperatur

Optimization in Construction: Minimizing Material for a Container

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

Parameter Identification in a Cooling Process

The temperature of an object cooling in an environment at $$20^\circ C$$ is modeled by Newton's Law

Phase-Plane Analysis of a Nonlinear Differential Equation

Consider the logistic differential equation $$\frac{dy}{dt} = y(1-y)$$, which models a normalized po

Picard Iteration for Approximate Solutions

Consider the initial value problem $$\frac{dy}{dt}=y+t, \quad y(0)=1$$. Use one iteration of the Pic

Population Growth with Harvesting

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

Projectile Motion with Drag

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

RC Circuit: Voltage Decay

In an RC circuit, the voltage across a capacitor satisfies $$\frac{dV}{dt} = -\frac{1}{R*C} * V$$. G

Separation of Variables with Trigonometric Functions

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

Series Solution for a Differential Equation

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

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

Solving a Separable Differential Equation

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

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

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 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 Two Curves

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

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 Speed from a Variable Acceleration Scenario

A particle moves along the x-axis under an acceleration given by $$a(t)= 3*t - 2$$ (in m/s²) and has

Average Temperature Over a Day

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

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

Balloon Inflation Related Rates

A spherical balloon is being inflated such that its radius $$r(t)$$ (in centimeters) increases at a

Car Motion Analysis

A car's acceleration is given by $$a(t) = 4 - 2 * t$$ (in m/s²) for $$0 \le t \le 4$$ seconds. The c

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+

Center of Mass of a Thin Rod

A thin rod extends from $$x=0$$ to $$x=4$$ m and has a density function $$\lambda(x)=1+\frac{\ln(x+2

Environmental Contaminant Spread Analysis

A contaminant enters a lake at a rate given by $$r(t)=4e^{-0.5*t}$$ kilograms per day, where $$t$$ i

Implicit Differentiation with Exponential Terms

Consider the equation $$e^{x * y} + x^2 * y = y^3$$. Answer the following:

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/

Pollution Concentration in a Lake

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

Profit-Cost Area Analysis

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

Projectile Motion under Gravity

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

Salt Concentration in a Mixing Tank

A tank initially contains 50 L of water with 5 g of salt. A salt solution with a concentration of 0.

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 Region via Washer Method

The region in the first quadrant bounded by the curves $$y=x$$ and $$y=x^2$$ is rotated about the x-

Volume of a Solid by Disc Method

The region bounded by $$y=\sqrt{x}$$ and $$y=x$$ for $$x\in[0,1]$$ is rotated about the $$x$$-axis t

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 of an Irregular Tank

A water tank has a varying cross-sectional profile described by $$y(x)=\sqrt{25 - (x-5)^2}$$, for $$

Volume with Square Cross-Sections

Consider the region under the curve $$y = \sqrt{x}$$ between $$x = 0$$ and $$x = 4$$. Squares are co

Analyzing the Concavity of a Parametric Curve

A curve is defined by $$x(t)=t-\sin(t)$$ and $$y(t)=1-\cos(t)$$.

Arc Length in Polar Coordinates

A polar curve is defined by $$r(\theta)=1+\cos(\theta)$$ for $$0 \leq \theta \leq \pi$$.

Arc Length of a Cycloid

A cycloid is generated by a circle of radius \(r=1\) rolling along a straight line. The cycloid is g

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

Arc Length of a Parametric Curve

Consider the curve defined by $$x(t)=t^3-3*t$$ and $$y(t)=t^2+2$$ for $$t \in [0,2]$$.

Arc Length of a Vector-Valued Function

Let $$ r(t)=\langle 3*t,\; 4*\sin(t) \rangle $$ for $$ 0 \le t \le \pi $$. Determine properties rela

Area Between Polar Curves

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

Area between Two Polar Curves

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

Catching a Thief: A Parametric Pursuit Problem

A police car and a thief are moving along a straight road. Initially, both are on the same road with

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

Concavity and Inflection in Parametric Curves

A curve is defined by the parametric functions $$x(t)=t^3-3*t$$ and $$y(t)=t^2$$ for \(-2\le t\le2\)

Conversion of Polar to Parametric Form

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

Curvature of a Vector-Valued Function

Let $$\vec{r}(t)=\langle t, t^2, \ln(t) \rangle$$ for \(t>0\). The curvature \(\kappa(t)\) is given

Displacement from a Vector-Valued Velocity Function

A particle's velocity is given by $$\vec{v}(t)=\langle \cos(t), \sin(t), t \rangle$$ for $$t \in [0,

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

Intersection of Polar and Parametric Curves

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

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$

Modeling Circular Motion with Vector-Valued Functions

An object moves along a circle of radius $$3$$ with its position given by $$\mathbf{r}(t)=\langle 3\

Motion Along a Parametric Curve

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

Parametric Egg Curve Analysis

An egg-shaped curve is modeled by the parametric equations $$x(t)= \cos(t)+0.5\cos(2t)$$ and $$y(t)=

Parametric Equations and Tangent Lines

A curve is defined parametrically by $$x(t)=t^3-3t$$ and $$y(t)=t^2+2$$, where $$t$$ is a real numbe

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 Representation of an Ellipse

An ellipse is represented by the parametric equations $$x(t)=4\cos(t)$$ and $$y(t)=3\sin(t)$$ for $$

Polar Plots and Intersection Points in Design

A designer creates a pattern using the polar equations $$r=5\cos(θ)$$ and $$r=5\sin(θ)$$. Analyze th

Satellite Orbit: Vector-Valued Functions

A satellite’s orbit is modeled by the vector function $$\mathbf{r}(t)=\langle \cos(t)+0.1*\cos(6*t),

Self-Intersection in a Parametric Curve

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

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 Polar Curve

Consider the polar curve $$r=5-2\cos(\theta)$$. Answer the following parts.

Taylor/Maclaurin Series: Approximation and Error Analysis

Let $$f(x)=\ln(1+x)$$. Without using a calculator, generate the third-degree Maclaurin polynomial fo

Vector-Valued Function Integration

A particle moves along a straight line with constant acceleration given by $$ a(t)=\langle 6,\;-4 \r