AP Calculus BC FRQ Room

Algebraic Manipulation in Limit Computations

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

Analyzing a Composite Function Involving a Limit

Let $$f(x)=\sin(x)$$ and define the function $$g(x)=\frac{f(x)}{x}$$ for $$x\neq0$$, with the conven

Analyzing Continuity on a Closed Interval

Suppose a function $$f(x)$$ is continuous on the closed interval $$[0,5]$$ and differentiable on the

Approximating Limits Using Tabulated Values

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

Continuity Analysis in Road Ramp Modeling

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

Continuity and the Intermediate Value Theorem in Temperature Modeling

A temperature function $$T(t)$$, continuous for $$t\in[0,24]$$ (in hours), satisfies $$T(6)=10^\circ

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 of an Integral-Defined Function

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

Evaluating Limits Involving Exponential and Rational Functions

Consider the limits involving exponential and polynomial functions. (a) Evaluate $$\lim_{x\to\infty}

Identifying and Removing a Discontinuity

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

Inverse Function Analysis and Continuity

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

Limits and Asymptotic Behavior of Rational Functions

Let $$k(x)=\frac{5*x^2-2*x+7}{x^2+4}.$$ Answer the following:

Limits and Removable Discontinuity in Rational Functions

Consider the rational function $$g(x) = \frac{(x-2)(x+3)}{x-2}.$$ Use this expression to answer the

Limits Involving Radical Functions

Examine the function $$m(x)=\frac{\sqrt{x}-2}{x-4}$$.

Limits of Composite Trigonometric Functions

Let $$p(x)= \frac{\sin(3x)}{\sin(5x)}$$.

Mixed Function Inflow Limit Analysis

Consider the water inflow function defined by $$R(t)=10+\frac{\sqrt{t+4}-2}{t}$$ for \(t\neq0\). Det

Parameter Determination for Continuity

Let $$h(x)= \begin{cases} \frac{e^{2x} - 1 - a\,\ln(1+bx)}{x} & x \neq 0 \\ c & x = 0 \end{cases}.$$

Pendulum Oscillations and Trigonometric Limits

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

Piecewise Function Critical Analysis

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

Radioactive Material Decay with Intermittent Additions

A laboratory maintains a radioactive material by adding 10 units of the substance at the beginning o

Removable Discontinuity in a Rational Function

Consider the function given by $$f(x)= \frac{(x+3)*(x-1)}{(x-1)}$$ for $$x \neq 1$$. Answer the foll

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

Trigonometric Limits Analysis

Evaluate the following limits involving trigonometric functions.

Analysis of Derivatives: Tangents and Normals

Consider the curve defined by $$y = x^3 - 6*x^2 + 9*x + 2.$$ (a) Compute the derivative $$y'$$ an

Analyzing a Function with an Oscillatory Component

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

Average vs Instantaneous Rates

Consider the function $$f(x)=\frac{\sin(x)}{x}$$ for \(x\neq0\), with $$f(0)=1$$. Answer the followi

Chemical Mixing Tank

In an industrial process, a mixing tank receives a chemical solution at a rate of $$C_{in}(t)=25+5*t

Differentiability of an Absolute Value Function

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

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

Estimating Instantaneous Acceleration from Velocity Data

An object's velocity (in m/s) is recorded over time as shown in the table below. Use the data to ana

Evaluation of Derivative at a Point Using the Limit Definition

Let $$f(x)=3*x^2-7$$. Use the limit definition of the derivative to evaluate $$f'(2)$$.

Exploration of the Definition of the Derivative as a Limit

Consider the function $$f(x)=\frac{1}{x}$$ for $$x\neq0$$. Answer the following:

Fuel Storage Tank

A fuel storage tank receives oil at a rate of $$F_{in}(t)=40\sqrt{t+1}$$ liters per hour and loses o

Graphical Derivative Analysis

A series of experiments produced the following data for a function $$f(x)$$:

Growth Rate of a Bacterial Colony

The radius of a bacterial colony is modeled by $$r(t)= \sqrt{4*t+1}$$, where t (in hours) represents

Manufacturing Cost Function and Instantaneous Rate

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

Motion Along a Line

An object moves along a line with its position given by $$s(t)=4*t^3 - 12*t^2 + 9*t$$, where $$s$$ i

Population Growth Rates

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

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*

Revenue Change Analysis via the Product Rule

A company’s revenue (in thousands of dollars) is modeled by $$R(x) = (2*x + 3)*(x^2 - x + 4)$$, wher

Satellite Orbit Altitude Modeling

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

Tangent Line Estimation from Experimental Graph Data

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

Testing Differentiability at a Junction Point

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

Traffic Flow and Instantaneous Rate

The number of cars passing through an intersection per minute is modeled by $$F(t)= 3t^2 + 2t + 10$$

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 Reservoir Depth Analysis

The depth of water (in meters) in a reservoir is modeled by $$d(t)=10+3*t-0.5*t^2$$, where $$t$$ is

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

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 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

Composite Differentiation in Polynomial Functions

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

Continuity and Differentiability of a Piecewise Function

Consider the function defined by $$ f(x)= \begin{cases} x^2, & x < 1, \\ 2*x + c, & x \ge 1. \end{ca

Financial Flow Analysis: Investment Rates

An investment fund experiences deposits at a rate modeled by the composite function $$D(t)=g(h(t))$$

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 and Inverse Functions in a Trigonometric Equation

Consider the equation $$x*y + \sin(x+y)= 1$$ which defines y implicitly as a function of x.

Implicit Differentiation in a Nonlinear Equation

Consider the equation $$x*y + y^3 = 10$$, which defines y implicitly as a function of x.

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 with Logarithms and Products

Consider the equation $$ \ln(x+y) + x*y = \ln(4)+4 $$.

Inverse Function Differentiation for Cubic Functions

Let $$f(x)= x^3 + 2*x$$, and let $$g(x)$$ be its inverse function. Answer the following:

Inverse Function Differentiation in a Science Experiment

In an experiment, the relationship between an input value $$x$$ and the output is given by $$f(x)= \

Inverse Function Differentiation in Economics

A product’s demand is modeled by a one-to-one differentiable function $$Q = f(P)$$, where $$P$$ is t

Inverse Function Differentiation with a Cubic Function

Let $$f(x)= x^3+ x + 1$$ be a one-to-one function, and let $$g$$ be its inverse function. Answer the

Inverse Trigonometric Differentiation in Navigation

A ship's course angle is given by $$ \theta= \arcsin\left(\frac{3*x}{5}\right) $$, where x is the ho

Lake Water Level Dynamics: Seasonal Variation

A lake's water inflow is modeled by the composite function $$I(t)=p(q(t))$$, where $$q(t)=0.5*t-1$$

Polar and Composite Differentiation: Arc Slope for a Polar Curve

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

Population Dynamics in a Fishery

A lake is being stocked with fish as part of a conservation program. The number of fish added per da

Projectile Motion and Composite Exponential Functions

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

Rate of Change in a Biochemical Process Modeled by Composite Functions

The concentration of a biochemical in a cell is modeled by the function $$C(t) = \sin(0.2*t) + 1$$,

Analysis of a Piecewise Function with Discontinuities

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

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 $$s$$

Analyzing Runner's Motion

A runner's displacement is modeled by the function $$s(t)=-t^3+9t^2+1$$, where s(t) is in meters and

Applying L'Hospital's Rule to a Transcendental Limit

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

Chemical Reaction Rate Model

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

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

Complex Limit Analysis with L'Hôpital's Rule

Evaluate the limit $$\lim_{x \to 0} \frac{e^{2*x} - 1 - 2*x}{x^2}$$ using L'Hôpital's Rule. Answer t

Cooling Coffee: Temperature Change

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

Cooling Temperature Model

The temperature of a heated object cooling in a room is modeled by $$T(t)= 80 + 120*e^{-0.25*t}$$, w

Cost Function Analysis in Production

A company's cost for producing $$x$$ items is given by $$C(x)=0.5*x^3-4*x^2+10*x+500$$ dollars.

Cubic Function with Parameter and Its Inverse

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

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

Economic Model: Revenue and Cost Rates

A company's revenue (in thousands of dollars) is modeled by $$R(x)=120-4*x^2+0.5*x^3$$, where $$x$$

Financial Model Inversion

Consider the function $$f(x)=\ln(x+2)+x$$ which models a certain financial indicator. Although an ex

Fuel Consumption Rate Analysis

The fuel consumption of a car (in gallons per 100 miles) is modeled by $$C(v)=0.05*v^2+1$$, where $$

Graphical Interpretation of Slope and Instantaneous Rate

A graph (provided below) displays a linear function representing a physical quantity over time. Use

Implicit Differentiation on an Ellipse

An ellipse representing a racetrack is given by $$\frac{x^2}{25}+\frac{y^2}{9}=1$$. A runner's x-coo

Inflating Spherical Balloon

A spherical balloon is being inflated so that the volume increases at a constant rate of $$dV/dt = 1

Inflating Spherical Balloon

A spherical balloon is being inflated so that its volume increases at a constant rate of $$\frac{dV}

Instantaneous vs. Average Speed in a Race

An athlete’s displacement during a 100 m race is modeled by $$s(t)=2*t^3-t^2+1$$, where $$s(t)$$ is

Linearization and Differentials: Approximating Function Values

Consider the function $$f(x)= x^4$$. Use linearization to estimate the value of the function for a s

Linearization Approximation

Let $$f(x)=x^4$$. Linearization can be used to approximate small changes in a function's values. Ans

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 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

Logarithmic Transformation and Derivative Limits

Consider the function $$f(x)=\ln\left(\frac{e^{3x}+1}{1+e^{-x}}\right)$$. Answer the following:

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

Optimizing Factory Production with Log-Exponential Model

A factory's production is modeled by $$P(x)=200x^{0.3}e^{-0.02x}$$, where x represents the number of

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 with Changing Acceleration

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

Particle on Implicit Curve

A particle moves so that its coordinates $$(x(t), y(t))$$ always satisfy the equation $$x^2 + x*y +

Population Decline Modeled by Exponential Decay

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

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

Rational Function Particle Motion Analysis

A particle moves along a straight line with its position given by $$s(t)=\frac{t^2+1}{t-1}$$, where

Reactant Flow in a Chemical Reactor

In a chemical reactor, a reactant is introduced at a rate $$I(t)=15\sin(\frac{t}{2})$$ (grams per mi

Related Rates in Conical Tank Draining

Water is draining from a conical tank. The volume of water is given by $$V=\frac{1}{3}\pi*r^2*h$$, a

Related Rates: Pool Water Level

Water is being drained from a circular pool. The surface area of the pool is given by $$A=\pi*r^2$$.

Series Analysis in Profit Optimization

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

Solids of Revolution: Washer vs Shell Methods

Consider the region enclosed by $$y = \sin(x)$$ and $$y = \cos(x)$$ for $$0 \le x \le \frac{\pi}{4}$

Trigonometric Implicit Relation

Consider the implicit equation $$\sin(x*y) + x - y = 0$$.

Analysis of an Absolute Value Function

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

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.

Application of the Extreme Value Theorem in Economics

A company's revenue is modeled by $$R(x)= -2*x^2+40*x+100$$, where $$x$$ is the number of units sold

Chemical Reactor Rate Analysis

In a chemical reactor, a reactant is added at a rate given by $$A(t)=8*\sqrt{t}$$ grams/min and is s

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 -

Concavity and Inflection Points in Particle Motion

Consider the position function of a particle $$s(x)=x^3-6*x^2+9*x+2$$.

Critical Point Analysis for Increasing/Decreasing Intervals

Consider the function $$f(x)= x^3 - 9*x^2 + 24*x + 5$$. Analyze the intervals where the function is

Determining Absolute Extrema for a Trigonometric-Polynomial Function

Consider the function $$f(x)= x+\cos(x)$$ defined on the closed interval $$[0, 2\pi]$$. Determine th

Dynamic Analysis Under Time-Varying Acceleration in Two Dimensions

A particle moves in the plane with acceleration given by $$\vec{a}(t)=\langle3\cos(t),-2\sin(t)\rang

Extremum Analysis Using the Extreme Value Theorem

Given the function $$f(x)= \cos(x)-x$$ defined on the interval $$\left[0, \frac{\pi}{2}\right]$$, an

Implicit Differentiation and Tangent to an Ellipse

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

Investigating a Composite Function Involving Logarithms and Exponentials

Let $$f(x)= \ln(e^x + x^2)$$. Analyze the function by addressing the following parts:

Loan Amortization with Increasing Payments

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

Maximizing the Area of an Inscribed Rectangle

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

Mean Value Theorem Application

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

Modeling Real World with the Mean Value Theorem

A car travels along a straight road with its position at time $$t$$ (in seconds) given by $$ s(t)=0.

Motion Analysis: Particle’s Position Function

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

Optimization: Maximum Area with Fixed Perimeter

A rectangle has a fixed perimeter of $$100$$ meters. Determine the dimensions of the rectangle that

Optimizing Fencing for a Rectangular Garden

A homeowner plans to build a rectangular garden adjacent to a river (so the side along the river nee

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)\

Rate of Change in a Logarithmic Temperature Model

A cooling process is modeled by the temperature function $$T(t)= 100 - 20\,\ln(t+1)$$, where t is me

Reservoir Sediment Accumulation

A reservoir accumulates sediment at a rate of $$S_{in}(t)=3*t$$ tonnes/day but also loses sediment v

Road Trip Analysis

A car's speed (in mph) during a road trip is recorded at various times. Use the table provided to an

Rolle's Theorem: Modeling a Car's Journey

An object moves along a straight line and its position is given by $$s(t)= t^3-6*t^2+9*t$$ for $$t$$

Volume of a Solid of Revolution Using the Washer Method

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

Wastewater Treatment Reservoir

At a wastewater treatment reservoir, wastewater enters at a rate of $$W_{in}(t)=12+2*t$$ m³/min and

Accumulated Displacement from a Piecewise Velocity Function

A particle moves along a line with velocity given by $$ v(t)= \begin{cases} t^2-1, & 0 \le t < 2, \\

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

Area Between the Curves: Linear and Quadratic Functions

Consider the curves $$y = 2*t$$ and $$y = t^2$$. Answer the following parts to find the area of th

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

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

Chemical Reaction: Rate of Concentration Change

A chemical reaction features a rate of change of concentration given by $$R(t)= 5*e^{-0.5*t}$$ (in m

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 via Integration

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

Displacement and Distance from a Velocity Function

A particle moves along a straight line with its velocity given by $$v(t)=3\sin(t)$$ (in m/s) for $$t

Distance from Acceleration Data

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

Evaluating a Trigonometric Integral

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

Integration of a Complex Trigonometric Function

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

Integration Using U-Substitution

Evaluate the integral $$\int (3*x+2)^5\,dx$$ using u-substitution.

Investment Growth Analysis with Exponentials

An investment grows according to the function $$f(t)= 100*e^{0.05*t}$$ for $$t \ge 0$$ (in years). A

Marginal Cost and Production

A factory's marginal cost function is given by $$MC(x)= 4 - 0.1*x$$ dollars per unit, where $$x$$ is

Modeling a Car's Journey with a Time-Dependent Velocity

A car's velocity is modeled by $$ v(t)= \begin{cases} 4t, & 0 \le t < 3, \\ 12, & 3 \le t \le 5, \en

Optimizing the Inflow Rate Strategy

A municipality is redesigning its water distribution system. The water inflow rate is modeled by $$F

Probability Density Function and Expected Value

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

Rainfall Accumulation and Runoff

During a storm, rainfall intensity is modeled by $$R(t)=3*t$$ inches per hour for $$0 \le t \le 4$$

Riemann and Trapezoidal Sums with Inverse Functions

Consider the function $$f(x)= 3*\sin(x) + 4$$ defined on the interval \( x \in [0, \frac{\pi}{2}] \)

Riemann Sum Approximation with Irregular Intervals

A set of experimental data provides the values of a function $$f(x)$$ at irregularly spaced points a

Riemann Sum from a Table: Plant Growth Data

A function $$f(t)$$ describes the height (in cm) of a plant over time (in days). The measurements ar

River Flow and Inverse Rate Functions

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

Taylor/Maclaurin Series Approximation and Error Analysis

Consider the function $$f(x)=\ln(1+x)$$. This function is infinitely differentiable at x = 0 and has

Total Rainfall Accumulation from a Discontinuous Rate Function

Rain falls at a rate (in mm/hr) given by $$ R(t)= \begin{cases} 3t, & 0 \le t < 2, \\ 5, & t = 2, \\

Vehicle Motion and Inverse Time Function

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

Water Accumulation Using Trapezoidal Sum

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

Chemical Reaction Kinetics

A first-order chemical reaction has its concentration $$C(t)$$ (in mol/L) governed by the differenti

Compound Interest and Investment Growth

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

Cooling of an Electronic Component

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

Direction Fields and Phase Line Analysis

Consider the autonomous differential equation $$\frac{dy}{dt}=(y-2)(3-y)$$. Answer the following par

Euler's Method and Differential Equations

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

FRQ 5: Mixing Problem in a Tank

A tank initially contains 100 liters of water with 10 kg of dissolved salt. Brine with a salt concen

Implicit Differentiation and Homogeneous Equation

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

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 $$x$$

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 Population Growth Model

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

Mixing Problem in a Tank

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

Mixing Problem with Differential Equations

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

Modeling 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 cools according to Newton's Law of Cooling, which is modeled by the differential equation

Population Dynamics with Harvesting

A species population is modeled by the differential equation $$\frac{dP}{dt} = 0.2*P\left(1-\frac{P}

Population Growth with Harvesting

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

Radioactive Decay with Constant Production

A radioactive substance decays at a rate proportional to its current amount but is also produced at

Saltwater Mixing Problem

A tank initially contains 1000 L of a salt solution with a concentration of 0.2 kg/L (thus 200 kg of

Separable DE with Exponential Function

Solve the differential equation $$\frac{dy}{dx}=y\cdot\ln(y)$$ for y > 0 given the initial condition

Separable Differential Equation with Initial Condition

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

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:

Tank Mixing Problem

A tank contains 1000 L of a well‐mixed salt solution. Brine containing 0.5 kg/L of salt flows into t

Water Tank Inflow-Outflow Model

A water tank is subject to an inflow and outflow. The inflow rate is given by $$I(t)=3*t+2$$ liters

Area Between a Rational Function and Its Asymptote

Consider the function $$f(x)=\frac{2*x+3}{x+1}$$ and its horizontal asymptote $$y=2$$ over the inter

Area Between Curves from Experimental Data

In an experiment, researchers recorded measurements for two functions, $$f(t)$$ and $$g(t)$$, repres

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 Economic Curves

In an economic model, two cost functions are given by $$C_1(x)=100-2*x$$ and $$C_2(x)=60-x$$, where

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 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 Temperature in a City

An urban planner recorded the temperature variation over a 24‐hour period modeled by $$T(t)=10+5*\si

Average Temperature of a Day

In a certain city, the temperature during the day is modeled by a continuous function $$T(t)$$ 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\

Average Value of a Population Growth Rate

The instantaneous growth rate of a bacterial population is modeled by the function $$r(t)=0.5*\cos(0

Car Braking and Stopping Distance

A car decelerates with an acceleration given by $$a(t)=-2*t$$ (in m/s²) and has an initial velocity

Center of Mass of a Non-uniform Rod

A thin rod of length 10 m has a linear density given by $$\lambda(x)= 3 + 0.5*x$$ (in kg/m) for $$0

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

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]

Cyclist's Journey: Displacement versus Total Distance

A cyclist's velocity is given by $$v(t)=\sin(t)$$ (in m/s) for $$t\in[0,2\pi]$$. Answer the followin

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 $$[0,2]$$.

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 Charge Distribution Along a Rod

A rod of length 10 m has a linear charge density given by $$\lambda(x) = 3e^{-0.5*x}$$ coulombs per

Electric Current and Charge

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

Implicit Differentiation with Trigonometric Function

Consider the equation $$\cos(x * y) + x = y$$. Answer the following:

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

Motion Analysis on a Particle with Variable Acceleration

A particle moves along a straight line with acceleration given by $$a(t)=6-4*t$$ (in m/s²). The init

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 Position and Distance Traveled

A particle moves along a line with velocity $$v(t)=t^3-6*t^2+9*t$$ (m/s) for $$t\in[0,5]$$. Given th

Polar Coordinates: Area of a Region

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

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

Projectile Motion Analysis

A projectile is launched vertically upward with an initial velocity of $$20$$ m/s. The only accelera

Series and Integration Combined: Error Bound in Integration

Consider the integral $$\int_{0}^{0.5} \frac{1}{1+x^2} dx$$. Use the Taylor series expansion of the

Total Change in Temperature Over Time (Improper Integral)

An object cools according to the function $$\Delta T(t) = e^{-0.5*t}$$, where $$t\ge 0$$ is time in

Volume of a Solid Using the Shell Method

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

Volume of a Solid with Variable Cross Sections

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

Volume of a Wavy Dome

An auditorium roof has a varying cross-sectional area given by $$A(x)=\pi*(1 + 0.1*\sin(x))^2$$ (in

Volume with Equilateral Triangle Cross Sections

The region bounded by $$y=4-x^2$$ and $$y=0$$ for $$x\in[-2,2]$$ serves as the base of a solid. Cros

Work Done by a Variable Force

A variable force is applied along a straight line and is given by $$F(x)=3*\ln(x+1)$$ (in Newtons),

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 force acting on an object is given by $$F(x)=5+3*x$$ (in newtons), where $$x$$ is the displacement

Analyzing a Looping Parametric Curve

The curve is defined by the equations $$x(t)=t^3-3t$$ and $$y(t)=t^2$$ for \(-2\le t\le 2\). Due to

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 equations $$x(t)=\sin(t)$$ and $$y(t)=\cos(t)$$ for $$0\leq t\leq \frac{\pi}

Arc Length of a Parametric Curve

Consider the curve defined by $$x(t)= 3*\sin(t)$$ and $$y(t)= 3*\cos(t)$$ for $$0 \le t \le \frac{\p

Area Between Polar Curves

Consider the polar curves $$ r_1=2+\cos(\theta) $$ and $$ r_2=1+\cos(\theta) $$. Although the curves

Area Between Polar Curves: Annulus with a Hole

Two polar curves are given by \(R(\theta)=3\) and \(r(\theta)=2+\cos(\theta)\) for \(0\le\theta\le2\

Area Between Two Polar Curves

Consider the polar curves $$ r_1=2*\sin(\theta) $$ and $$ r_2=\sin(\theta) $$. Determine the area of

Area Enclosed by a Polar Curve

Consider the polar curve given by $$r = 2*\sin(\theta)$$.

Area Enclosed by a Polar Curve

Let the polar curve be defined by $$r=3\sin(\theta)$$ with $$0\le \theta \le \pi$$. Answer the follo

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

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

Finding the Slope of a Tangent to a Parametric Curve

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

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

Intersection of Polar and Parametric Curves

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

Motion in a Damped Force Field

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

Optimization on a Parametric Curve

A curve is described by the parametric equations $$x(t)= e^{t}$$ and $$y(t)= t - e^{t}$$.

Parametric Curve Intersection

Two curves are defined parametrically as follows: For curve C1, $$x(t) = t^2$$ and $$y(t) = 2*t + 1$

Parametric Curves and Concavity

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

Parametric Tangent Line and Curve Analysis

For the curve defined by the parametric equations $$x(t)=t^{2}$$ and $$y(t)=t^{3}-3t$$, answer the f

Particle Motion in the Plane

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

Polar Coordinates: Area Between Curves

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

Polar Curve Sketching and Area Estimation

A polar curve is described by sample data given in the table below.

Polar Equations and Slope Analysis

Given the polar curve $$r = 4\sin(\theta)$$, analyze its Cartesian form and tangent properties.

Polar to Cartesian Conversion

Consider the polar curve defined by $$r = 4*\cos(\theta)$$.

Projectile Motion with Parametric Equations

An object is launched with projectile motion as described by the parametric equations $$x(t)=50*t$$

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 Fields and Particle Trajectories

A particle moves in the plane with velocity given by $$\vec{v}(t)=\langle \frac{e^{t}}{t+1}, \ln(t+2