AP Calculus BC FRQ Room

Analyzing a Function with a Removable Discontinuity

Consider the function $$r(x)=\frac{x^2-9}{x-3}$$ for $$x\neq3$$ and $$r(3)=2.$$ Answer the follow

Complex Rational Function and Continuity Analysis

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

Continuity Analysis Using a Piecewise Defined Function

Let $$f(x)=\begin{cases} x^2, & x \leq 1 \\ 2x-1, & x> 1 \end{cases}$$.

Continuity in a Parametric Function Context

A particle moves such that its coordinates are given by the parametric equations: $$x(t)= t^2-4$$ an

Continuity of a Trigonometric Function Near Zero

Consider the function defined by $$ f(x)= \begin{cases} \frac{\sin(5*x)}{x}, & x \neq 0 \\ L, & x =

Continuity of Log‐Exponential Function

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

End Behavior Analysis of a Rational Function

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

Environmental Pollution Modeling

In a lake, a pollutant is added every year at a constant amount of 5 units. However, due to natural

Epsilon-Delta Style (Conceptual) Analysis

Consider the function $$f(x)=\begin{cases} 3*x+2, & x\neq1, \\ 6, & x=1. \end{cases}$$ Answer the

Establishing Continuity in a Piecewise Function

Consider the piecewise-defined function $$p(x)= \begin{cases} \frac{x^2-4}{x-2} & x \neq 2, \\ k & x

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

Exponential Inflow with a Shift in Outflow Rate

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

Graph Analysis of Discontinuities

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

Graphical Analysis of a Removable Discontinuity

Consider the function $$f(x)=\frac{x^2-1}{x-1}$$ for x \neq 1, with a defined value of f(1) = 3. Ans

Identifying and Removing a Discontinuity

Consider the function $$g(x)=\frac{x^2-4}{x-2}$$ for $$x\neq2$$, which is undefined at $$x=2$$.

Implicitly Defined Curve and Its Tangent Line

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

Indeterminate Limit with Exponential and Log Functions

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

Limit at an Infinite Discontinuity

Consider the function $$g(x)= \frac{1}{(x-2)^2}$$. Analyze its behavior near the point where it is u

Limits and Continuity in Particle Motion

A particle moves along a straight line with velocity given by $$v(t)=\frac{t^2-4}{t-2}$$ for t ≠ 2 s

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 Involving Absolute Value Functions

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

Limits with Composite Logarithmic Functions

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

Logarithmic Function Limits

Consider the function $$f(x)=\frac{\ln(1+3*x)}{x}$$ for $$x \neq 0$$. Answer the following:

Modeling with a Removable Discontinuity

A water reservoir’s inflow rate is modeled by the function $$R(t)=\frac{t^2-16}{t-4}$$ liters per mi

Rational Function with Removable Discontinuity

Consider the function $$f(x)= \frac{x^2-9}{x-3}$$ for $$x \neq 3$$.

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 a Quadratic Function

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

Analyzing Car Speed from a Distance-Time Table

A car's position (in meters) is recorded at various times (in seconds) as shown in the table. Use th

Application of Derivative to Relative Rates in Related Variables

Water is being pumped into a conical tank, and the volume of water is given by $$V=\frac{1}{3}\pi*r^

Chemical Reaction Rate Analysis

The concentration of a reactant in a chemical reaction (in M) is recorded over time (in seconds) as

Composite Function and Chain Rule Application

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

Cooling Model Rate Analysis

The temperature of a cooling object is modeled by $$T(t)=e^{-2*t}+\ln(t+3)$$, where $$t$$ is time in

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

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

Derivative of Inverse Functions

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

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

Error Bound Analysis for $$e^{2x}$$

In a study of reaction rates, the function $$f(x)=e^{2*x}$$ is used. Analyze the error in approximat

Exponential Growth and Its Derivative

A culture of bacteria grows according to the model $$P(t)= 100*e^{0.03*t},$$ where \(P(t)\) is th

Finding the Derivative of a Logarithmic Function

Consider the function $$g(x)=\ln(3*x+1)$$. Answer the following:

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 with Exponential and Trigonometric Functions

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

Oil Spill Containment

Following an oil spill, containment efforts recover oil at a rate of $$O_{in}(t)=40-2*t$$ (accumulat

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 Lines: Analysis of Rate of Change

Consider the function $$f(x)=x^3-6*x^2+9*x+1$$. This function represents a model of a certain proces

Sine Function Analysis

Let $$g(x)=3*\sin(x)+2$$, where $$x$$ is in radians. Analyze its rate of change.

Tangent Line Approximation for a Combined Function

Consider the function $$f(x)= \sin(x) + x^2$$. Use the concept of the tangent line to approximate ne

Tangent Line Approximation for a Parabolic Arch

Engineers design a parabolic arch described by $$y(x)= -0.5*x^2 + 4*x$$.

Temperature Change Rate

The temperature in a chemical reactor is modeled by $$T(t)=\frac{\sin(2*t)}{t}$$ for \(t>0\), where

Analysis of a Composite Chemical Concentration Model

The concentration of a chemical in a reaction is modeled by the composite function $$C(t)= \ln(0.5*t

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

Bacterial Culture: Nutrient Inflow vs Waste Outflow

In a bioreactor, the nutrient inflow rate is given by $$N(t)=\ln(2*t+1)$$ (in units/min). The waste

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 Chain Rule with Exponential and Trigonometric Functions

Consider the function $$f(x) = e^{\cos(x)}$$. Analyze its derivative and explain the role of the cha

Dam Water Release and River Flow

A dam releases water into a river at a rate given by the composite function $$R(t)=c(b(t))$$, where

Differentiation of a Log-Exponential Composition with Critical Points

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

Differentiation of 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\

Enzyme Kinetics in a Biochemical Reaction

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

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 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 Involving Exponential Functions

Consider the relation defined implicitly by $$e^{x*y} + x^2 - y^2 = 7$$.

Implicit Differentiation Involving Product and Logarithm

Consider the curve defined by $$x*y + \ln(y) = x^2$$. Answer the following parts:

Implicit Differentiation: Circle and Tangent Line

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

Inverse Function Derivative in an Exponential Model

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

Inverse Function Differentiation in a Logarithmic Context

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

Inverse Function Differentiation in a Trigonometric Context

Let $$f(x)= \sin(x) + x$$, defined on the interval $$[0, \frac{\pi}{2}]$$, and let $$g$$ be its inve

Logarithmic Differentiation of a Variable Exponent Function

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

Particle Motion with Composite Position Function

A particle moves along a line with its position given by $$s(t)= \sin(t^2)$$, where $$s$$ is in mete

Second Derivative via Chain Rule

Let $$h(x)=(e^{2*x}+1)^4$$. Answer the following parts.

Tangent Line for a Parametric Curve

A curve is given parametrically by $$x(t)= t^2 + 1$$ and $$y(t)= t^3 - t$$.

Trigonometric Composite Inverse Function Analysis

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

Applying L'Hôpital's Rule to an Exponential Function

Consider the function $$F(x)=\frac{e^{2*x}-1}{x}$$, which models the change in a certain system's qu

Applying L'Hospital's Rule to a Transcendental Limit

Evaluate the limit $$\lim_{x\to 0}\frac{e^{2*x}-1}{\sin(3*x)}$$.

Approximating Function Values Using Linearization

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

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

Chemistry: Rate of Change in a Reaction

In a chemical reaction, the concentration of a reactant is modeled by $$C(t)= 50e^{-0.2*t}+5$$, wher

Conical Tank Filling - Related Rates

A conical water tank has its volume given by $$V= \frac{1}{3}\pi*r^2*h$$, where \(r\) is the radius

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

Cycloid Tangent Line

A cycloid is defined by the parametric equations $$x(t)=2*(t-\sin(t))$$ and $$y(t)=2*(1-\cos(t))$$ f

Differentiating a Product: f(x)=x sin(x)

Let \(f(x)=x\sin(x)\). Analyze the behavior of \(f(x)\) near \(x=0\).

Expanding Circular Ripple

A stone is thrown in a pond, creating circular ripples. The area of the circle defined by the ripple

Exponential and Trigonometric Bounded Regions

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

Implicit Differentiation on a Circle

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

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

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

Interpreting Position Graphs: From Position to Velocity

A graph of position (in meters) versus time (in seconds) is provided in the stimulus. The graph show

Interpreting the Derivative in Straight Line Motion

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

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)$

Logarithmic Differentiation and Asymptotic Behavior

Let $$f(x)=\frac{(x^2+1)^3}{e^{2x}(x-1)^2}$$ for x > 1. Answer the following:

Minimum Time to Cross a River

A person must cross a 100-meter-wide river. They can swim at 2 m/s and run along the bank at 5 m/s.

Motion 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

Motion on a Straight Line with a Logarithmic Position Function

A particle moves along a straight line with its position given by $$s(t)=\ln(t+2)+t^2$$ (in meters),

Motion with Non-Uniform Acceleration

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

Parametric Curve Motion

A particle’s trajectory is given by the parametric equations $$x(t)=t^2-1$$ and $$y(t)=2*t+3$$ for $

Parametric Motion in the Plane

A particle moves in the plane according to the parametric equations $$x(t)=t^2-2*t$$ and $$y(t)=3*t-

Parametric Motion with Logarithmic and Radical Components

A particle’s motion is described by the vector function $$\mathbf{r}(t)=\langle \ln(t+1),\sqrt{t} \r

Particle Motion Analysis Using Cubic Position Function

Consider a particle moving along a straight line with its position given by $$x(t)=t^3 - 6*t^2 + 9*t

Particle Motion with Measured Positions

A particle moves along a straight line with its velocity given by $$v(t)=4*t-1$$ for $$t \ge 0$$ (in

Polar Coordinates: Arc Length of a Spiral

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

Related Rates: Expanding Circular Oil Spill

In a coastal region, an oil spill is spreading uniformly and forms a circular region. The area of th

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.

Savings Account and Interest Accrual

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

Analysis of a Function with Oscillatory Behavior

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

Analyzing Extrema for a Rational Function

Let $$f(x)= \frac{x^2+2}{x+1}$$ be defined on the interval $$[0,4]$$. Use calculus methods to analyz

Application in Motion: Approximate Velocity using Taylor Series

A particle’s position is given by $$s(t)=e^{-t}+t^2$$. Using Taylor series approximations near $$t=0

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 the Mean Value Theorem

Let $$f(x)=\frac{x}{x^2+1}$$ be defined on the interval $$[0,2]$$. Answer the following questions us

Area and Volume of Region Bounded by Exponential and Linear Functions

Consider the functions $$f(x)=e^{x}$$ and $$g(x)=x+2$$. The region enclosed by these curves will be

Car Motion: Velocity and Total Distance

A car’s position along a straight road is modeled by $$s(t) = t^3 - 6*t^2 + 9*t + 15$$ (in meters),

Chemical Reaction Rate

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

Concavity Analysis in a Revenue Model

A company’s revenue (in thousands of dollars) is modeled by the function $$R(x) = -0.5*x^3 + 6*x^2 -

Determining the Meeting Point of Two Functions

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

Economic Optimization: Maximizing Profit

The profit function for a product is given by $$P(x) = -2*x^3 + 27*x^2 - 108*x + 150$$, where \(x\)

Extreme Value Theorem in Temperature Variation

A metal rod’s temperature (in °C) along its length is modeled by the function $$T(x) = -2*x^3 + 12*x

Fractal Tree Branch Lengths

A fractal tree is constructed as follows: The trunk has a length of 10 meters. At each generation, e

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

Motion with a Piecewise-Defined Velocity Function

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

Optimization in a Log-Exponential Model

A firm's profit is given by the function $$P(x)= x\,e^{-x} + \ln(1+x)$$, where x (in thousands) repr

Optimizing Material for a Container

An open-top rectangular container with a square base must have a fixed volume of $$32$$ cubic feet.

Parameter-Dependent Concavity Conditions

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

Piecewise Function Discontinuities Analysis

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

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)\

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 Analysis

A projectile is launched at a 45° angle with an initial speed of 20 m/s. Its motion is modeled by th

Second Derivative Test for Critical Points

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

Ski Resort Snow Accumulation and Melting

At a ski resort, snow accumulates naturally at a rate given by $$S(t)=50*\exp(-0.1*t)$$ cm/hour due

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

Stress Analysis in Engineering Structures

A beam experiences stress along its length given by $$S(x)=5*x^3-30*x^2+45*x$$ where x is the distan

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

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.

Water Tank Rate of Change

The volume of water in a tank is modeled by $$V(t)= t^3 - 6*t^2 + 9*t$$ (in cubic meters), where $$t

Accumulated Displacement from Acceleration

A particle moving along a straight line has an acceleration of $$a(t)=6-4*t$$ (in m/s²), with an ini

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 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 Estimation Using Trapezoidal Sums from Tabulated Data

The table below provides values of $$h(t)$$ over time for a process: | Time (t) | 0 | 2 | 5 | 8 | |

Area Estimation with Riemann Sums

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

Area Under a Parametric Curve

A curve is defined parametrically by $$x(t)=t^2$$ and $$y(t)=t^3-3*t$$ for $$t \in [-2,2]$$.

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

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

Comparing Riemann Sums with Definite Integral in Estimating Distance

A vehicle's velocity (in m/s) is recorded at discrete times during a trip. Use these data to estimat

Continuous Antiderivative for a Piecewise Function

A function $$f(x)$$ is defined piecewise as follows: for $$x<2$$, $$f(x)=3*x^2$$, and for $$x\ge2$$,

Convergence of an Improper Integral Representing Accumulation

Consider the improper integral $$\int_{1}^{\infty} \frac{1}{t^2}dt$$, which can be interpreted as th

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 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

Economic Applications: Consumer and Producer Surplus

In a market, the demand function is given by $$D(p)=100-2p$$ and the supply function by $$S(p)=20+3p

Economics: Accumulated Earnings

A company’s instantaneous revenue rate (in dollars per day) is modeled by the function $$R(t)=1000\s

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

Evaluating an Integral Using U-Substitution

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

Fuel Consumption Estimation with Midpoint Riemann Sums

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

Fundamental Theorem of Calculus Application

Let $$F(x)=\int_{2}^{x} (t^{2} - 4*t + 3) dt$$. Answer the following:

Improper Integral and the p-Test

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

Integration of a Rational Function

Consider the function $$f(x)=\frac{1}{x^2+4}$$ on the interval $$[0,2]$$. Evaluate the area under th

Integration Using U-Substitution

Evaluate the indefinite integral $$\int (4*x+2)^5\,dx$$ using u-substitution.

Net Displacement vs. Total Distance Traveled

A particle moving along a straight line has a velocity function given by $$v(t)= t^2 - 4*t + 3$$ (in

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 $$x\ge0$$.

Reservoir Water Level

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

Riemann Sum Estimation from Tabular Data

The following table lists values of a function $$f(x)$$ at selected points, which are used to approx

Solving for Unknowns using Logarithmic Properties in Integration

Consider the definite integral $$\int_(a)^(b) \frac{3}{x} dx$$ which is given to equal 6, where a is

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

Total Work Done by a Variable Force

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

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

Water Accumulation Using Trapezoidal Sum

A reservoir is monitored over time and its water level (in meters) is recorded at various times (in

Water Tank Inflow and Outflow

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

Analysis of a Nonlinear Differential Equation

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

Analysis of a Piecewise Function with Potential Discontinuities

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

Bacteria Growth with Antibiotic Treatment

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

Capacitor Charging with Leakage

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

Chemical Reaction Rate and Series Approximation

A chemical reaction is modeled by the differential equation $$\frac{dC}{dt} = -0.2 * C^2$$ with the

Coffee Cooling: Differential Equation Application

A cup of coffee is cooling according to Newton's Law of Cooling. The following table provides measur

Cooling Coffee Data Analysis

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

Differential Equation with Exponential Growth and Logistic Correction

Consider the modified differential equation $$\frac{dy}{dt} = a*y - b*y^2$$, where $$a$$ and $$b$$ a

Dye Dilution in a Stream

A river has dye added at a constant rate of $$0.5$$ kg/min, and the dye is removed at a rate proport

FRQ 16: Harvesting in a Predator-Prey Model

A prey population $$P(t)$$ in a marine ecosystem is modeled by the differential equation $$\frac{dP}

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 in a Tank

A tank initially contains 50 liters of pure water. A brine solution with a salt concentration of $$3

Modeling the Spread of a Disease Using Differential Equations

Suppose the spread of a disease in a population is modeled by the differential equation $$\frac{dI}{

Motion Under Gravity with Air Resistance

An object falling under gravity experiences air resistance proportional to its velocity. Its motion

Newton's Law of Cooling

A hot liquid is cooling in a room. The temperature $$T(t)$$ (in degrees Celsius) of the liquid at ti

Phase-Plane Analysis of a Nonlinear Differential Equation

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

Population Growth with Harvesting

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

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

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

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 Initial Condition

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

Tank Draining Problem

A tank with a variable cross-sectional area is being emptied. The height \(h(t)\) of the water satis

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

Analyzing Acceleration Data from Discrete Measurements

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

Area Between a Parabola and a Line

Consider the curves given by $$y=5*x-x^2$$ and $$y=x$$. These curves intersect at certain $$x$$-valu

Area Between Curves: Park Design

A park layout is bounded by two curves: $$f(x)=10-x^2$$ and $$g(x)=2*x+2$$. Answer the following par

Area Between Two Curves: Parabola and Line

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

Average Daily Temperature

The temperature during a day is modeled by $$T(t)=10+5*\sin((\pi/12)*t)$$ (in °C), where $$t$$ is th

Average Temperature Calculation

The temperature (in $$^\circ C$$) in City A is recorded at specific times over a 12-hour period. Est

Average Temperature Computation

Consider a scenario in which the temperature (in °C) in a region is modeled by the function $$T(t)=

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 Value and Critical Points of a Trigonometric Function

Consider the function $$f(x)=\sin(2*x)+\cos(2*x)$$ on the interval $$\left[0,\frac{\pi}{2}\right]$$.

Average Velocity and Displacement from a Polynomial Function

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

Center of Mass of a Lamina with Constant Density

A thin lamina occupies the region in the first quadrant bounded by $$y=x^2$$ and $$y=4$$. The densit

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

Comparing Average and Instantaneous Rates of Change

For the quadratic function $$f(x)= 3*x^2 - 4*x + 1$$ on the interval $$[1,3]$$, investigate both its

Designing a Bridge Arch

A bridge arch is modeled by the curve $$y = 10 - 0.25*x^2$$, where $$x$$ is measured in meters and $

Determining the Arc Length of a Curve

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

Drug Concentration Profile Analysis

The functions $$f(t)=5*t+10$$ and $$g(t)=2*t^2+3$$ (where t is in hours and concentration in mg/L) r

Integration in Cost Analysis

In a manufacturing process, the cost per minute is modeled by $$C(t)=t^2 - 4*t + 7$$ (in dollars per

Net Cash Flow Analysis

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

Particle Motion from Acceleration

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

Particle Motion with Variable Acceleration

A particle's acceleration is given by $$a(t)=4*e^{-t} - 2$$ for $$t$$ in seconds over the interval $

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

Shadow Length Related Rates

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

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 via Shell Method

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

Volume of a Solid via the Disc Method

The region bounded by the curve $$y=\sqrt{x}$$ and the x-axis for $$x\in[0,4]$$ is revolved about th

Volume of a Water Tank with Varying Cross-Sectional Area

A water tank has a cross-sectional area given by $$A(x)=3*x^2+2$$ in square meters, where $$x$$ (in

Work Done by a Variable Force

A variable force applied to move an object along a straight line is given by $$F(x)=3*x^2$$ (in newt

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

Work Done by a Variable Force

A variable force acting along a straight line is given by $$F(x) = \frac{10}{x+2}$$ (in Newtons). Fi

Work Done by a Variable Force

A force acting along a straight line is given by $$F(x)=10 - 0.5*x$$ newtons for $$0 \le x \le 12$$

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, &

Work to Pump Water from a Tank

A cylindrical tank of radius 2 ft and height 10 ft is partially filled with water to a depth of 8 ft

Analyzing a Clock's Second Hand with Polar Coordinates

A clock's second hand rotates uniformly, and its tip traces a circle of radius 12 cm. Its position i

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

Arc Length of a Polar Curve

Consider the polar curve given by $$r=2+\cos(\theta)$$ for $$0\le \theta \le \pi$$. Answer the follo

Arc Length of a Polar Curve

Consider the polar curve given by $$r = 2 + 2*\sin(\theta)$$ for $$0 \le \theta \le \pi$$.

Comparing Arc Lengths in Parametric and Polar Systems

Consider the curve given in parametric form by $$x(t)=\cos(2*t)$$ and $$y(t)=\sin(2*t)$$ for $$0\le

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 Differentiation of a Polar Curve

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

Conversion and Tangents in Polar Coordinates

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

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 $$-2 \le t \le 3$$.

Cycloid and Its Arc Length

Consider the cycloid defined by the parametric equations $$x(t)=t-\sin(t)$$ and $$y(t)=1-\cos(t)$$ f

Determining Curvature from a Vector-Valued Function

Consider the curve defined by $$\mathbf{r}(t)=\langle t, t^2, t^3 \rangle$$ for $$t \ge 0$$. Analyze

Differentiation and Integration of a Vector-Valued Function

Let $$\mathbf{r}(t)=\langle e^{-t}, \sin(t), \cos(t) \rangle$$ for $$t \in [0,\pi]$$.

Exponential Growth in Parametric Representation

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

Implicit Differentiation with Implicitly Defined Function

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

Intersection of Parametric Curves

Consider the parametric curves $$C_1$$ given by $$x(t)= t^2,\; y(t)= 2t$$ and $$C_2$$ given by $$x(s

Kinematics in the Plane: Parametric Motion

A particle moves in the plane with its position given by the parametric equations $$ x(t)=t^2-2*t $$

Modeling Projectile Motion with Parametric Equations

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

Motion Along a Helix

A particle moves along a helix described by the vector-valued function $$\vec{r}(t)=<\cos(t),\, \sin

Parametric Curves and Concavity

Consider the parametric equations $$x(t)= \sin(t)$$ and $$y(t)= \cos(2*t)$$ for $$t \in [0, 2\pi]$$.

Particle Motion in the Plane

A particle moves in the plane with its position described by the parametric equations $$x(t)=3*\cos(

Particle Motion on an Elliptical Arc

A particle moves along a curve described by the parametric equations $$x(t)= 2*cos(t)$$ and $$y(t)=

Polar Coordinates: Area Between Curves

Consider two polar curves: the outer curve given by $$R(\theta)=4$$ and the inner curve by $$r(\thet

Spiral Intersection on the X-Axis

Consider the spiral defined by the parametric equations $$x(t) = e^{-t}\cos(2*t)$$ and $$y(t)= e^{-t

Spiral Motion with a Damped Vector Function

An object moves according to the spiral vector function $$\mathbf{r}(t)=\langle e^{-t}\cos(t),\; e^{

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 Integrals in Motion

A particle's acceleration is given by the vector function $$\vec{a}(t)=<\ln(t),\, t^{-1},\, e^{t}>$$

Wind Vector Analysis in Navigation

A boat on a river is propelled by its engine giving a constant velocity of \(\langle 3, 4 \rangle\)

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

Work Done by a Force along a Vector Path

A force field is given by $$\mathbf{F}(t)=\langle2*t,\;3\sin(t)\rangle$$ and an object moves along a