AP Calculus BC FRQ Room

Algebraic Manipulation in Limit Computations

Let $$s(x)=\frac{x^3-8}{x-2}.$$ Answer the following:

Analysis of Rational Function Asymptotes and Removable Discontinuities

Consider the rational function $$h(x)= \frac{3*x^2+5*x-2}{x^2-4}$$. Answer the following questions r

Continuity in Composition of Functions

Let $$g(x)=\frac{x^2-4}{x-2}$$ for x ≠ 2 and undefined at x = 2, and let f(x) be a continuous functi

Continuity of an Integral-Defined Function

Consider the function defined by the integral $$F(x)= \int_{0}^{x} \frac{t}{t^2+1} \; dt$$.

Horizontal Asymptote of a Rational Function

Consider the rational function $$f(x)= \frac{2*x^3+5*x^2-3}{x^3-4*x+1}$$. Answer the questions regar

Horizontal Asymptote of a Rational Function

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

Indeterminate Limit with Exponential and Log Functions

Examine the limit $$\lim_{x \to 0} \frac{e^{2x} - \cos(x) - 1}{\ln(1+x^2)}.$$

Intermediate Value Theorem Application with a Cubic Function

A function f(x) is continuous on the interval [-2, 2] and its values at certain points are given in

Internet Data Packet Transmission and Error Rates

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

Investment Portfolio Rebalancing

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

Limits at Infinity and Horizontal Asymptotes

Consider the rational function $$g(x)= \frac{4*x^3-x+2}{2*x^3+3*x^2-5}$$.

Limits with Composite Logarithmic Functions

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

Oscillatory Behavior and Squeeze Theorem

Consider the function $$h(x)= x^2 \cos(1/x)$$ for $$x \neq 0$$ with $$h(0)=0$$.

Rational Function Analysis with Removable Discontinuities

Consider the function $$f(x)=\frac{(x+3) * (x-1)}{(x-1)}$$ for $$x \neq 1$$. This function exhibits

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:

Temperature Change Analysis

The function $$T(t)$$ represents the temperature (in $$^\circ C$$) in a chemical reactor as a functi

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

Water Treatment Plant Discontinuity Analysis

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

Analyzing a Function with an Oscillatory Component

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

Average vs Instantaneous Rate of Change in Temperature Data

The table below shows the temperature (in °C) recorded at several times during an experiment. Use t

Bacterial Culture Growth: Discrete to Continuous Analysis

In a controlled laboratory, a bacterial culture doubles every hour. The discrete model after n hours

Composite Exponential-Log Function Analysis

Consider the function $$f(x)=e^{x}*\ln(x+3)$$, which arises in the study of compound reactions in ch

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

Determining Rates of Change with Secant and Tangent Lines

A function is graphed by the equation $$f(t)=t^3-3*t$$ and its graph is provided. Use the graph to a

Electricity Consumption: Series and Differentiation

A household's monthly electricity consumption increases geometrically due to seasonal variations. Th

Higher-Order Derivatives

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

Implicit Differentiation: Elliptic Curve

Consider the curve defined by $$2*x^2 + 3*x*y + y^2 = 20$$.

Implicit Differentiation: Inverse Trigonometric Equation

Consider the function defined implicitly by $$\arctan(y) + y = x$$.

Instantaneous Velocity from a Displacement Function

A particle moves along a straight line with its position at time $$t$$ (in seconds) given by $$s(t)

Logarithmic Differentiation

Let $$T(x)= (x^2+1)^{3*x}$$ model a quantity with variable growth characteristics. Use logarithmic d

Logarithmic Differentiation Simplification

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

Maclaurin Polynomial for √(1+x)

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

Parametric Analysis of a Curve

A particle moves along a path given by the parametric equations $$x(t)=t^2+1$$ and $$y(t)=t^3-3*t$$,

Population Growth Approximation using Taylor Series

A biologist models population growth with the exponential function $$P(t)=e^{0.05*t}$$. To estimate

Product Rule Application in Kinematics

A particle’s distance along a path is given by $$s(t)= t*e^(2*t)$$, where $$t$$ is in seconds. Answe

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*

Rate Function Involving Logarithms

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

Rate of Change in a Logarithmic Function

Consider the function $$f(x)=\frac{\ln(x)}{x}$$ defined for \(x>0\). Answer the following:

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

River Flow and Differentiation

The rate of water flow in a river is modeled by $$Q(t)= 5t^2 + 8t + 3$$ in cubic meters per second,

Secants and Tangents in Profit Function

A firm’s profit is modeled by the quadratic function $$f(x)=-x^2+6*x-8$$, where $$x$$ (in thousands)

Second Derivative of a Composite Function

Consider the function $$f(x)=\cos(3*x^2)$$. Answer the following:

Tangent Line Estimation from Experimental Graph Data

A function $$f(x)$$ is represented by the following graph of experimental data approximating $$f(x)=

Tangent Lines and Related Approximations

For the function $$f(x)= \sin(x) + x^2,$$ (a) Compute \(f'(0)\). (b) Write the equation of the t

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

Using the Limit Definition for a Non-Polynomial Function

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

Water Tank: Inflow-Outflow Dynamics

A water tank initially contains $$1000$$ liters of water. Water enters the tank at a rate of $$R_{in

Analyzing a Composite Function from a Changing Systems Model

The displacement of an object is given by $$s(t) = \sqrt{t^2 + 4*t + 5}$$ (in meters), where $$t$$ i

Chain Rule and Inverse Trigonometric Differentiation

Consider the function $$f(x)= 3*\arccos\left(\frac{x}{4}\right) + \sqrt{1-\frac{x^2}{16}}$$. Answer

Chain Rule and Quotient Rule for a Rational Composite Function

Let $$f(x)= \frac{(3*x^2 + 2)^4}{(1+x)^{1/2}}$$. Answer the following:

Composite Function Analysis

Consider the function $$f(x)= \sqrt{3*x^2+2*x+1}$$ which arises in an experimental study of motion.

Composite Function Rates in a Chemical Reaction

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

Differentiation of a Product Involving Inverse Trigonometric Components

Let $$m(x)=\arctan(x)+x*\arcsin\left(\frac{x}{2}\right)$$, where the domain of arcsin requires $$-2\

Differentiation of Composite Exponential and Trigonometric Functions

Let $$f(x) = e^{\sin(x^2)}$$ be a composite function. Differentiate $$f(x)$$ and simplify your answe

Differentiation of Inverse Trigonometric Functions

Consider the function $$f(x)= \sin(x)$$ for $$x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$$ and

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 Involving a Mixed Function

Consider the equation $$x*e^{y}+y*\ln(x)=10$$, where x > 0 and y is defined implicitly as a function

Implicit Differentiation of a Circle

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

Implicit Differentiation of a Product Equation

Consider the equation $$ x*y + x + y = 10 $$.

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

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

Implicit Differentiation: Circle and Tangent Line

The equation $$x^2 + y^2 = 25$$ represents a circle. Use implicit differentiation to find the deriva

Inverse Analysis of a Radical Function

Consider the function $$f(x)=\sqrt{2*x+3}$$ defined for $$x \ge -\frac{3}{2}$$. Analyze its invertib

Inverse Function Derivative for the Natural Logarithm

Consider the function $$f(x) = \ln(x+1)$$ for $$x > -1$$ and let $$g$$ be its inverse function. Anal

Inverse Function Differentiation for a Quadratic Function

Let $$ f(x)= (x+1)^2 $$ with the domain $$ x\ge -1 $$. Consider its inverse function $$ f^{-1}(y) $$

Inverse Function Differentiation in Navigation

A vehicle’s distance traveled is modeled by $$f(t)= t^3 + t$$, where $$t$$ represents time in hours.

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.

Nested Composite Function Differentiation

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

Population Growth Analysis Using Composite Functions

A population model is defined by $$P(t)= f(g(t))$$ where $$g(t)= e^{-t} + 3$$ and $$f(u)= 2*u^2$$. H

Analysis of Particle Motion

A particle’s velocity is given by $$v(t)= 4t^3 - 3t^2 + 2$$. Analyze the particle’s motion by invest

Analyzing Motion on an Inclined Plane

A sled moves along an inclined plane with displacement given by $$s(t)=5*t^2-0.5*t^3$$, where $$s$$

Approximating Changes with Differentials

Given that the surface area of a sphere is $$A = 4\pi r^2$$, use differentials to approximate the ch

Bacterial Growth and Linearization

A bacterial population is modeled by $$P(t)=100e^{0.3*t}$$, where $$t$$ is in hours. Answer the foll

Bacterial Population Growth Analysis

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

Biological Growth Rate

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

Biology: Logistic Population Growth Analysis

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

Car Motion with Changing Acceleration

A car's velocity is given by $$v(t) = 3*t^2 - 4*t + 2$$, where $$t$$ is in seconds. Answer the follo

Chemical Reaction Rate Model

A chemical reaction has its reactant concentration modeled by $$C(t)= 0.5*t^2 - 3*t + 4$$, where \(C

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

Cooling Coffee Temperature

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

Cooling Coffee Temperature Change

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

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 and Concavity for a Non-Motion Problem: Water Filling a Tank

The volume of water in a tank is given by $$V(t)=4*t^3-12*t^2+9*t+15$$, where $$V$$ is in gallons an

Graphical Analysis of Derivatives

A function $$f(x)$$ is plotted on the graph provided below. Using this graph, answer the following:

Instantaneous vs. Average Rate of Change in Temperature

A rod's temperature along its length is modeled by $$T(x)=20\ln(x+1)+e^{-x}$$, where x (in meters) i

Integration Region: Exponential and Polynomial Functions

Let the region be bounded by the curves $$y = x^2$$ and $$y = e^x$$. Analyze the area of the region

L’Hôpital’s Rule for an Exponential Ratio

Analyze the limit of the function $$f(t)=\frac{e^{2*t}-1}{t}$$ as $$t\to 0$$. Answer the following:

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

Logistic Population Model Inversion

Consider the logistic population model given by $$f(t)=\frac{50}{1+e^{-0.3*(t-5)}}$$. This function

Marginal Analysis in Economics

The cost function for producing $$x$$ items is given by $$C(x)= 0.1*x^3 - 2*x^2 + 20*x + 100$$ dolla

Mixing a Saline Solution: Related Rates

A tank contains a saline solution with a constant volume of 50 liters. Salt is added at a rate of 2

Motion along a Curved Path

A particle moves along the curve defined by $$y=\sqrt{x}$$. At the moment when $$x=9$$ and the x-coo

Optimization in Design: Maximizing Inscribed Rectangle Area

A rectangle is inscribed in a semicircle of radius $$R$$ (with the rectangle's base along the diamet

Polar Coordinates: Arc Length of a Spiral

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

Popcorn Sales Growth Analysis

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

Series Analysis in Profit Optimization

A company's profit function near a break-even point is approximated by $$\pi(x)= 1000 + \sum_{n=1}^{

Series Approximation for Investment Growth

An investment accumulation function is modeled by $$A(t)= 1 + \sum_{n=1}^{\infty} \frac{(0.07t)^n}{n

Series Approximation of a Temperature Function

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

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

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

Series Differentiation and Approximation of Arctan

Consider the function $$F(x)= \sum_{n=0}^{\infty} \frac{(-1)^n x^{2*n+1}}{2*n+1}$$, which represents

Sliding Ladder

A 10 m long ladder rests against a vertical wall. Let $$x$$ be the distance from the foot of the lad

Vehicle Motion on a Curved Path

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

Absolute Extrema and the Candidate’s Test

Let $$f(x)=x^3-3x^2-9x+5$$ be defined on the closed interval $$[-2,5]$$. Answer the following parts:

Aircraft Climb Analysis

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

Analysis of a Function with Oscillatory Behavior

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

Analysis of a Rational Function and the Mean Value Theorem

Consider the function $$g(x)=\frac{x^2-4}{x-2}$$. Answer the following parts.

Application of Rolle's Theorem

Consider the function $$g(x)=x^3-3x$$ on the interval $$[-\sqrt{3},\sqrt{3}]$$. Answer the following

Application of Rolle's Theorem

Let f be a function that is continuous on $$[2,5]$$ and differentiable on $$(2,5)$$, with $$f(2) = f

Asymptotic Behavior and Limits of a Logarithmic Model

Examine the function $$f(x)= \ln(1+e^{-x})$$ and its long-term behavior.

Bank Account Growth and Instantaneous Rate

A bank account balance is modeled by the function $$B(t) = 1000*e^{0.05*t}$$, where t (in years) rep

Candidate’s Test for Absolute Extrema in Projectile Motion

A projectile is launched such that its height at time $$t$$ is given by $$h(t)= -16*t^2+32*t+5$$ (in

Concavity and Inflection Points

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

Construction Payment Milestones

A construction project is structured around milestone payments. The first payment is $$10000$$ dolla

Curve Sketching Using Derivatives

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

Derivative Analysis of a Rational Function

Consider the function $$s(x)=\frac{x}{x^2+1}$$. Answer the following parts:

Economic Production Optimization

A company’s cost function is given by $$C(x) = 0.5*x^3 - 3*x^2 + 4*x + 200$$, where x represents the

Energy Consumption Rate Model

A household's energy consumption rate (in kW) is modeled by $$E(t) = 2*t^2 - 8*t + 10$$, where t is

Epidemic Infection Model

In a community experiencing an epidemic, the rate of new infections is modeled by $$I(t)=\frac{200}{

Extreme Value Analysis

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

Extreme Value Analysis of a Logarithmic-Exponential-Polynomial Function

Examine the function $$f(x)= x^2\,e^{-x} + \ln(x)$$ defined for $$x > 0$$. Apply methods to find its

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

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

Logarithmic Function Derivative Analysis

Consider the function $$f(x)= \ln(x^2+1)$$. Answer the following questions about its behavior.

Manufacturing Optimization in Production

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

Mean Value Theorem 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 Temperature Analysis

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

Minimizing Production Cost

A company’s production cost is modeled by the function $$C(x)=0.5*x^2 - 20*x + 300$$, where $$x$$ re

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

Motion with a Piecewise-Defined Velocity Function

A particle travels along a line with a piecewise velocity function defined by $$ v(t)=\begin{cases}

Parameter-Dependent Concavity Conditions

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

Pharmaceutical Dosage and Metabolism

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

Radius of Convergence and Series Manipulation in Substitution

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

Retirement Savings with Diminishing Deposits

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

Rolle's Theorem on a Cubic Function

Consider the cubic function $$f(x)= x^3-3*x^2+2*x$$ defined on the interval $$[0,2]$$. Verify that t

Skier's Speed Analysis

A skier's speed (in m/s) on a slope was recorded at various time intervals. Use the data provided to

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

Applying the Fundamental Theorem of Calculus

Consider the function $$f(x)=2*x$$. Use the Fundamental Theorem of Calculus to evaluate the definite

Area Estimation Using Riemann Sums for $$f(x)=x^2$$

Consider the function $$f(x)=x^2$$ on the interval $$[1,4]$$. A table of computed values for the lef

Area Under a Piecewise-Defined Curve with a Jump Discontinuity

Consider the function $$ g(x)= \begin{cases} 2x+1 & \text{if } 0 \le x < 2, \\ 3x-2 & \text{if } 2 \

Bacteria Population Accumulation

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

Car Acceleration, Velocity, and Distance

In a physics experiment, the acceleration of a car is modeled by the function $$a(t)=4*t-1$$ (in m/s

Chemical Reaction Rates

A chemical reaction in a vessel occurs at a rate given by $$R(t)= 8*e^{-t/2}$$ mmol/min. Determine t

Comparing Integration Approximations: Simpson's Rule and Trapezoidal Rule

A student approximates the definite integral $$\int_{0}^{4} (x^2+1)\,dx$$ using both the trapezoidal

Cost Accumulation via Integration

A manufacturing process has a marginal cost function given by $$MC(x)= 4 + 3*x$$ dollars per item, w

Cost Function Accumulation

A manufacturer’s marginal cost function is given by $$C'(x)= 0.1*x + 5$$ dollars per unit, where x

Estimating Chemical Production via Riemann Sums

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

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 a Trigonometric Integral

Evaluate the integral $$\int_{0}^{\pi/2} \cos(3*x)\,dx$$.

Integration via U-Substitution for a Composite Function

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

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

Population Growth with Logistic Differential Equation

Suppose a population $$P(t)$$ grows according to the logistic differential equation $$\frac{dP}{dt}

Population Growth: Rate to Accumulation

A population's growth rate (in thousands of individuals per year) is modeled by $$P'(t)=2*t - 1$$ fo

Probability Density Function and Expected Value

Let the probability density function (pdf) be defined by $$f(x)=k*x*e^{-x}$$ for $$x\ge0$$.

Riemann Sums and Inverse Analysis from Tabular Data

Let $$f(x)= 2*x + 1$$ be defined on the interval $$[0,5]$$. Answer the following questions about $$f

River Flow and Inverse Rate Functions

The rate of water flow in a river is modeled by \( f(t)= 2*t + \sin(t) \) for \( t \in [0, \pi] \) (

Transportation Model: Distance and Inversion

A transportation system is modeled by $$f(t)= (t-1)^2+3$$ for $$t \ge 1$$, where \(t\) is time in ho

Trapezoidal and Riemann Sums from Tabular Data

A scientist collects data on the concentration of a chemical over time as given in the table below.

Volume Accumulation in a Reservoir

A reservoir is being filled at a rate given by $$R(t)= e^{2*t}$$ liters per minute. Determine the t

Volume of a Solid with Known Cross-sectional Area

A solid extends from $$x=0$$ to $$x=5$$, and its cross-sectional area perpendicular to the x-axis is

Water Volume Accumulation with a Faulty Sensor Reading

Water flows into a container at a rate given by $$ r(t)= \begin{cases} 4t, & 0 \le t < 3, \\ 10, & t

Work Done by an Exponential Force

A variable force acting along the x-axis is given by $$F(x)=5 * e^(0.5 * x)$$ (in Newtons) for 0 \(\

Analysis of a Nonlinear Differential Equation

Consider the nonlinear differential equation $$\frac{dy}{dx} = y^3-3*y$$.

Bacteria Culture with Regular Removal

A bacterial culture has a population $$B(t)$$ that grows at a continuous rate of $$12\%$$ per hour,

Bacterial Growth with Time-Dependent Growth Rate

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

Bank Account Growth with Continuous Compounding

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

Drug Concentration in the Bloodstream

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

Existence and Uniqueness in an Implicit Differential Equation

Consider the implicit initial value problem given by $$y\,e^{y}+x=0$$ with the initial condition $$y

FRQ 4: Newton's Law of Cooling

A cup of coffee cools according to Newton's Law of Cooling, where the temperature $$T(t)$$ satisfies

FRQ 17: Slope Field Analysis and Particular Solution

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

FRQ 18: Enzyme Reaction Rates

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

Homogeneous Differential Equation

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

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 Differentiation in a Differential Equation Context

Suppose the function $$y(x)$$ satisfies the implicit equation $$x\,e^{y}+y^2=7$$. Differentiate both

Interpreting Slope Fields for a Differential Equation

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

Logistic Model in Population Dynamics

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

Logistic Model in Product Adoption

A company models the adoption rate of a new product using the logistic equation $$\frac{dP}{dt} = 0.

Logistic Population Growth Model

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

Modeling Ambient Temperature Change

The ambient temperature $$T(t)$$ of a city changes according to the differential equation $$\frac{dT

Modeling Cooling and Heating: Temperature Differential Equation

Suppose the temperature of an object changes according to the differential equation $$\frac{dT}{dt}

Modeling Medication Concentration in the Bloodstream

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

Modeling Temperature in a Biological Specimen

A biological specimen initially at $$37^\circ C$$ is cooling in an environment where the ideal ambie

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

Nonlinear Differential Equation with Implicit Solution

Consider the nonlinear differential equation $$\frac{dy}{dx} = \frac{x^2+y^2}{2xy}$$. An implicit so

Population Dynamics with Harvesting

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

Population Growth with Logistic Differential Equation

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

Radioactive Decay Data Analysis

A radioactive substance is decaying over time. The following table shows the measured mass (in grams

Rainfall in a Basin: Differential Equation Model

During a rainstorm, the depth of water $$h(t)$$ (in centimeters) in a basin is modeled by the differ

Separable and Implicit Solution for $$\frac{dy}{dx}= \frac{x}{1+y^2}$$

Consider the differential equation $$\frac{dy}{dx}= \frac{x}{1+y^2}$$, which is defined for all real

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 Equilibrium Analysis for $$\frac{dy}{dx}= y*(1-y)$$

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

Temperature Control in a Chemical Reaction Vessel

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

Traffic Flow on a Highway

A highway segment experiences an inflow of cars at a rate of $$200+10*t$$ cars per minute and an out

Accumulated Rainfall

The rate of rainfall over a 12-hour storm is modeled by $$r(t)=4*\sin\left(\frac{\pi}{12}*t\right) +

Analyzing the Inverse of an Exponential Function

Let $$f(x)=\ln(2*x+1)$$, defined for $$x\ge0$$.

Approximating Functions using Taylor Series

Consider the function $$f(x)= \ln(1+2*x)$$. Use Taylor series methods to approximate and analyze thi

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: Enclosed Region

The curves $$f(x)=x^2$$ and $$g(x)=x+2$$ enclose a region. Answer the following:

Area Between Curves: Parabolic & Linear Regions

Consider the curves $$f(x)=5*x-x^2$$ and $$g(x)=x$$. Answer the following questions regarding the re

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

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 Parabola and Tangent

Consider the parabola defined by $$y^2 = 4 * x$$. Let $$P = (1, 2)$$ be a point on the parabola. Ans

Area Under a Curve with a Discontinuity

Consider the function $$f(x)=\frac{1}{x+2}$$ defined on $$[0,3]$$.

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 Temperature Over a Day

The temperature in a city over a 24-hour period is modeled by $$T(t)=10+5\sin\left(\frac{\pi}{12}*t\

Center of Mass of a Lamina

A thin lamina occupies the region under the curve $$y=\sqrt{x}$$ on the interval $$[0,4]$$ and has a

Cyclist Average Speed Calculation

A cyclist’s velocity is given by $$v(t) = t^2 - 4*t + 6$$ (in m/s) for $$t$$ in the interval $$[0,4]

Determining Average Value of a Velocity Function

A runner’s velocity is given by $$v(t)=2*t+3$$ (in m/s) for $$t\in[0,5]$$ seconds.

Drone Motion Analysis

A drone’s vertical acceleration is modeled by $$a(t) = 6 - 2*t$$ (in m/s²) for time $$t$$ in seconds

Economic Analysis: Consumer and Producer Surplus

In a market analysis, the demand and supply for a product are modeled by the equations: Demand: $$D(

Electric Current and Charge

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

Particle Motion with Velocity Reversal

A particle moves along a straight line with an acceleration given by $$a(t)=12-6*t$$ (in m/s²) for $

Population Change via Rate Integration

A small town’s population changes at a rate given by $$P'(t)=100*e^{-0.3*t}$$ (persons per year) wit

Pumping Water from a Conical Tank

An inverted right circular conical tank has a height of $$10$$ meters and a top radius of $$4$$ mete

Radioactive Decay Accumulation

The rate of decay of a radioactive substance is given by $$R(t)=100*e^{-0.3*t}$$ decays per day. Ans

Surface Area of a Solid of Revolution

Consider the curve $$y=\sqrt{x}$$ on the interval $$[0,9]$$. When this curve is rotated about the x-

Volume about a Vertical Line using Two Methods

A region in the first quadrant is bounded by $$y=x$$, $$y=0$$, and $$x=2$$. This region is rotated a

Volume by Cross‐Sectional Area in a Variable Tank

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

Volume of a 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 of Revolution Using the Disc Method

Let R be the region bounded by $$y=\sqrt{x}$$, $$y=0$$, and $$x=4$$. This region is rotated about th

Volume of a Solid with Equilateral Triangle Cross Sections

Consider the region bounded by $$y= \sqrt{x}$$ and $$y=0$$ for $$x \in [0,1]$$. A solid is formed by

Work Done in Lifting a Cable

A cable of length 10 m with a uniform mass density of 5 kg/m hangs vertically from a winch. The winc

Work Done with a Discontinuous Force Function

A force acting on an object is defined piecewise by $$F(x)=\begin{cases} x^2, & 0 \le x < 4,\\ 16, &

Analyzing Concavity for a Polar Function

Consider the polar function given by \(r=5-2\sin(\theta)\). Answer the following:

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 and Tangent for a Polar Curve

The polar curve is defined by $$r = 2+\cos(\theta)$$.

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

Circular Motion Analysis

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

Conversion between Polar and Cartesian Coordinates

The polar equation $$r = 2 + 2\cos(\theta)$$ describes a limaçon. Analyze this curve by converting i

Conversion to Cartesian and Analysis of a Parametric Curve

Consider the parametric equations $$x(t)= 2*t + 1$$ and $$y(t)= (t - 1)^2$$ for $$-2 \le t \le 3$$.

Converting and Analyzing a Polar Equation

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

Curvature and Oscillation in Vector-Valued Functions

Given the vector function $$\vec{r}(t)=\langle \ln(t+1), \cos(t) \rangle$$ for $$t \ge 0$$, answer t

Curvature of a Parametric Curve

Consider the curve defined by the parametric equations $$x(t)=\ln(t)$$ and $$y(t)=t^2$$ for \(t>0\).

Integration of Vector-Valued Acceleration

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

Intersections in Polar Coordinates

Two polar curves are given by $$r = 3 - 2*\sin(\theta)$$ and $$r = 1 + \cos(\theta)$$.

Logarithmic Exponential Transformations in Polar Graphs

Consider the polar equation $$r=2\ln(3+\cos(\theta))$$. Answer the following:

Optimization in Parametric Projectile Motion

A projectile is launched from the ground with an initial speed of $$20\,m/s$$ at an angle $$\alpha$$

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 Motion and Change of Direction

A particle moves along a path defined by the parametric equations $$x(t)=t^3-3t$$ and $$y(t)=2t^2$$

Parametric Particle Motion

A particle moves along a path described by the parametric equations: $$x(t)=t^2-2*t$$ and $$y(t)=3*t

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 with Air Resistance: Parametric Analysis

A projectile is launched with air resistance, and its motion is modeled by the parametric equations:

Time of Nearest Approach on a Parametric Path

An object travels along a path defined by $$x(t)=5*t-t^2$$ and $$y(t)=t^3-6*t$$ for $$t\ge0$$. Answe

Vector-Valued Function Integration

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

Vector-Valued Functions: Tangent and Normal Components

A car’s motion on a plane is described by the vector-valued function $$\mathbf{r}(t)=\langle t^2, 4*

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

Work Done Along a Path in a Force Field

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