AP Calculus AB FRQ Room

Absolute Value Function Limits

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

Algebraic Simplification and Limit Evaluation of a Log-Exponential Function

Consider the function $$z(x)=\ln\left(\frac{e^{3*x}+e^{2*x}}{e^{3*x}-e^{2*x}}\right)$$ for $$x \neq

Analyzing a Piecewise Function for Continuity

Consider the piecewise function $$ f(x)=\begin{cases} 2x+1, & x<2 \\ x^2-1, & x\geq2 \end{cases}$$.

Analyzing Process Data for Continuity

A manufacturing process produces items whose lengths (in mm) are recorded as a function f(x) of time

Area and Volume Setup with Bounded Regions

Consider the region R bounded by the curves $$y = x^2$$, $$y = 4$$, and $$x = 0$$. Though integratio

Continuity Analysis with a Piecewise-defined Function

A particle’s displacement is described by the piecewise function $$s(t)= \begin{cases} t^2+1, & t <

Continuity and Asymptotic Behavior of a Rational Exponential Function

Consider the function $$q(x)= \frac{e^{2*x} - 4}{e^{x} - 2}$$. Notice that the function is not defin

Continuity of a Radical Function

Consider the function $$f(x)=\sqrt{x-1}$$. Answer the following parts.

Evaluating Limits Near Vertical Asymptotes

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

Exponential Limit Parameter Determination

Consider the function $$f(x)=\frac{e^{3*x} - e^{k*x}}{x}$$ for $$x \neq 0$$, and define $$f(0)=L$$,

Graph Reading: Left and Right Limits

A graph of a function f is provided below which shows a discontinuity at x = 2. Use the graph to det

Graph Transformations and Continuity

Let $$f(x)=\sqrt{x}$$ and consider the function $$g(x)= f(x-2)+3= \sqrt{x-2}+3$$.

Graphical Estimation of a Limit

The following graph shows the function $$f(x)$$. Use the graph to answer the subsequent questions re

Implicit Differentiation Involving Logarithms

Consider the curve defined implicitly by $$\ln(x) + \ln(y) = \ln(5)$$. Answer the following:

Intermediate Value Theorem Application

Let $$p(x)=x^3-4x-5$$. Use the Intermediate Value Theorem (IVT) to show that the equation \(p(x)=0\)

Intermediate Value Theorem in Equation Solving

A continuous function defined on [0, 10] is given by $$f(x)= \frac{x}{10} - \sin(x)$$.

Limits Involving a Removable Discontinuity

Consider the function $$g(x)= \frac{(x+3)(x-2)}{x-2}$$ defined for $$x \neq 2$$. Answer the followin

Optimization and Continuity in a Manufacturing Process

A company designs a cylindrical can (without a top) for which the cost function in dollars is given

Parameter Determination from a Logarithmic-Exponential Limit

Let $$v(x)=\frac{\ln(e^{2*x}+1)-2*x}{x}$$ for $$x\neq0$$. Find the value of the limit $$\lim_{x \to

Piecewise Function Continuity Analysis

The function f is defined by $$ f(x) = \begin{cases} \frac{x^2 - 1}{x - 1}, & x \neq 1 \\ k, & x

Real-World Analysis of Vehicle Deceleration Using Data

A study measures the speed of a car (in m/s) as it approaches a stop sign. The recorded speeds at di

Removable Discontinuity and Direct Limit Evaluation

Consider the function $$f(x)=\frac{(x-2)(x+3)}{x-2}$$ defined for $$x\neq2$$. The function is not de

Removable Discontinuity in a Rational Function

Consider the function $$f(x)=\begin{cases} \frac{x^2-16}{x-4} & x\neq4 \\ 3*x+1 & x=4 \end{cases}$$.

Robotic Arm and Limit Behavior

A robotic arm moving along a linear axis has a velocity function given by $$v(t)= \frac{t^3-8}{t-2}$

Squeeze Theorem with a Trigonometric Function

Consider the function $$f(x) = x^2 \cdot \sin\left(\frac{1}{x}\right)$$ for $$x \neq 0$$, and define

Squeeze Theorem with an Oscillatory Function

Consider the function $$f(x) = x \cdot \sin\left(\frac{1}{x}\right)$$ for $$x \neq 0$$ and define $$

Trigonometric Limit Computation

Consider the function $$f(x)= \frac{\sin(5*(x-\pi/4))}{x-\pi/4}$$.

Water Tank Inflow-Outflow Analysis

Consider a water tank operating over the time interval $$0 \le t \le 12$$ minutes. The water inflow

Analyzing Function Behavior Using Its Derivative

Consider the function $$f(x)=x^4 - 8*x^2$$.

Approximating the Tangent Slope

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

Average vs. Instantaneous Rate of Change

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

Bacterial Culture Growth with Washout

In a bioreactor, bacteria grow at a rate of $$f(t)=50*e^{0.05*t}$$ (cells/min) while simultaneous wa

Chemical Reactor Flow Rates

A chemical reactor is operated so that reactants are added at a rate of $$f(t)=12-t$$ liters/min (fo

Concavity and the Second Derivative

Consider the function $$f(x)=x^4-4*x^3+6*x^2$$. Answer the following:

Finding Derivatives of Composite Functions

Let $$f(x)= (3*x+1)^4$$.

Graph Interpretation of the Derivative

Consider the function $$f(x)= x^3 - 6*x^2 + 9*x + 1$$. A graph of this function is provided below.

Graphical Interpretation of Rate of Change

Consider the graph of a function provided in the stimulus which shows a vehicle's displacement over

Higher-Order Derivatives in Motion

A particle moves along a line with its position given by $$s(t)= t^3 - 6*t^2 + 9*t + 5$$, where $$t$

Implicit Differentiation in Motion

A particle’s motion is given by the implicit equation $$y^2 + x*y = 10$$, where x represents time (i

Instantaneous Velocity from a Position Function

A ball is thrown upward, and its height in feet is modeled by $$s(t)= -16*t^2 + 64*t + 5$$, where $$

Interpreting Derivative Graphs and Tangent Lines

A graph of the function $$f(x)=x^2 - 2*x + 1$$ along with its tangent line at $$x=2$$ is provided. A

Inverse Function Analysis: Cubic Function

Consider the function $$f(x)=x^3+2*x+1$$ defined for all real numbers.

Inverse Function Analysis: Exponential-Linear Function

Consider the function $$f(x)=e^x+x$$ defined for all real numbers.

Marginal Cost Analysis

A company's total cost function is given by $$C(x)=5*x^2+20*x+100$$, where $$x$$ represents the numb

Marginal Profit Calculation

A company’s profit (in thousands of dollars) is given by $$P(x)= -2*x^2 + 50*x - 100$$, where $$x$$

Polynomial Rate of Change Analysis

Consider the function $$f(x)= x^3 - 2*x^2 + x$$, which models a physical process. Analyze the rates

Profit Function Analysis

A company's profit function is given by $$P(x)=-2x^2+12x-5$$, where x represents the production leve

Secant and Tangent Lines to a Curve

Consider the function $$f(x)=x^2 - 4*x + 5$$. Answer the following questions:

Secant Approximation Convergence and the Derivative

Consider the natural logarithm function $$f(x)= \ln(x)$$. Investigate its rate of change using the d

Secant Line Slope Approximations in a Laboratory Experiment

In a chemistry lab, the concentration of a solution is modeled by $$C(t)=10*\ln(t+1)$$, where $$t$$

Secant Slope from Tabulated Data

A table below gives values of a function $$f(x)$$ representing the concentration of a solution at di

Secant vs. Tangent Rate Comparison

For the function $$f(x)=x^2$$, we analyze the relationship between the secant and tangent approximat

Tangent and Normal Lines in Road Construction

A road is modeled by the quadratic function $$f(x)= \frac{1}{2}*x^2 + 3*x + 10$$.

Tangent Line Approximation

Suppose a continuous function $$f(x)$$ is differentiable with $$f(2)=8$$ and $$f'(2)=5$$. Use this i

Tangent to an Implicit Curve

Consider the curve defined implicitly by \(x^2 + y^2 = 25\). Answer the following parts.

Using the Quotient Rule for a Rational Function

Let $$f(x) = \frac{3*x+5}{x-2}$$. Differentiate $$f(x)$$ using the quotient rule.

Advanced Implicit Differentiation: Second Derivative Analysis

Consider the curve defined implicitly by the equation $$x^2*y+\sin(y)= x^3$$.

Analyzing Motion in the Plane using Implicit Differentiation

A particle moves in the xy-plane along a path defined implicitly by $$x^2+x*y+y^2=7$$. Determine the

Chain Rule in a Light Intensity Model

The intensity of light is modeled by $$I(r) = \frac{1}{r^2}$$, where r is the distance (in meters) f

Chain Rule in Angular Motion

An object rotates such that its angular position is given by $$\theta(t)= \arctan(3*t^2)$$, where $$

Chain Rule with Exponential and Polynomial Functions

Let $$h(x)= e^{3*x^2+2*x}$$ represent a function combining exponential and polynomial elements.

Combining Composite and Implicit Differentiation

Consider the equation $$e^{x*y}+x^2-y^2=7$$.

Comparing the Rates between a Function and its Inverse

Let $$f(x)=x^5+2*x$$. Answer the following:

Composite and Product Rule Combination

The function $$F(x)= (3*x^2+2)^{4} * \cos(x^3)$$ arises in modeling a complex system. Answer the fol

Composite Function in Finance

An account balance is modeled by the function $$B(t)=(2*t+1)^{3/2}$$ dollars, where $$t$$ represents

Composite Function Involving Exponential and Cosine

Consider the function $$f(x)= e^(\cos(x^2))$$. Address the following:

Composite Function with Multiple Layers

Suppose an economic model is described by the function $$F(x)=\sqrt{\ln(3*x^2+2)}$$, where $$x$$ rep

Differentiating an Inverse Trigonometric Function

Let $$y = \arcsin\left(\frac{2*x}{1+x^2}\right)$$.

Differentiation of a Composite Rational Function

Let $$f(x)=\frac{(2*x+1)^3}{\sqrt{5*x-2}}$$. Use the chain rule and the quotient (or product) rule t

Differentiation of Inverse Trigonometric Composite Function

Given the function $$y = \arctan(\sqrt{x})$$, answer the following parts.

Differentiation of Inverse Trigonometric Functions in Physics

In an optics experiment, the angle of refraction \(\theta\) is given by $$\theta= \arcsin\left(\frac

Estimating Derivatives Using a Table

An experiment measures a one-to-one function $$f$$ and its inverse $$g$$, yielding the following dat

Implicit Differentiation in an Elliptical Orbit

Consider the ellipse defined by $$4*x^2 + 9*y^2 = 36$$, which can model the orbit of a satellite.

Implicit Differentiation Involving Exponential Functions

Let the equation $$x*e^{y}+y*e^{x}=10$$ define $$y$$ implicitly as a function of $$x$$. Use implicit

Implicit Differentiation with Exponential Terms

Consider the equation $$e^{x} + y = x + e^{y}$$ which relates $$x$$ and $$y$$ via exponential functi

Implicit Differentiation with Exponential-Trigonometric Functions

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

Implicit Differentiation with Product Rule

Consider the implicit equation $$x*y+y^2=6$$ which defines $$y$$ as a function of $$x$$. Use implici

Implicit Differentiation with Product Rule

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

Inverse Function Derivative and Recovery

Let $$f(x)=x^3+x$$, which is one-to-one on a suitable interval. Answer the following parts.

Inverse Function Derivative for a Log-Linear Function

Let $$f(x)= x+ \ln(x)$$ for $$x > 0$$ and let g be the inverse of f. Solve the following parts:

Inverse Function Differentiation

Let $$f(x)=x^3+x$$ which is one-to-one on its domain. Its inverse function is denoted by $$g(x)$$.

Inverse Function Differentiation in a Biological Growth Model

In a bacterial growth experiment, the population $$P$$ (in colony-forming units) at time $$t$$ (in h

Inverse Function Differentiation in a Piecewise Scenario

Consider the piecewise function $$f(x)=\begin{cases} x^2+1, & x \geq 0 \\ -x+1, & x<0 \end{cases}$$

Inverse Function Differentiation with Exponentials and Trigonometry

Let $$f(x)=\sin(e^{x})+e^{x}$$ and assume that it is invertible. Answer the following:

Inverse Trigonometric Differentiation

Let $$y = \arcsin\left(\frac{2*x}{1+ x^2}\right)$$. Answer the following parts.

Logarithmic Differentiation of a Composite Function

For the function $$y= (x^2+1)^(\tan(x))$$, use logarithmic differentiation to address the following

Population Dynamics via Composite Functions

A biological population is modeled by $$P(t)= \ln\left(20*e^(0.1*t^2)+ 5\right)$$, where t is measur

Related Rates via Chain Rule

A spherical balloon is being inflated so that its volume increases at a rate of $$\frac{dV}{dt}=150\

Water Tank Flow Analysis using Composite Functions

A water tank is equipped with an inflow system and an outflow system. At time $$t$$ (in minutes), wa

Water Tank Optimization Using Composite Functions

A water tank has an inflow rate given by $$I(t)= 3+2\sin(0.1*t)$$ and an outflow rate given by $$O(t

Analysis of Experimental Data

The graph below shows the displacement of an object moving in a straight line. Analyze the object's

Analysis of Particle Motion

A particle moves along a horizontal line with velocity function $$v(t)=4t-t^2$$ (m/s) for $$t \geq 0

Balloon Altitude and Temperature

A hot air balloon rises such that its altitude is given by $$h(t)=3*t^{2/3}$$ meters, where t is in

Biochemical Reaction Rate Analysis

A biochemical reaction proceeds with a rate modeled by $$R(t)=50t(1-t)^2$$ for $$0\le t\le1$$ (where

Cost Efficiency in Production

A firm's cost function for producing $$x$$ items is given by $$C(x)=0.1*x^2 - 5*x + 200$$. Analyze t

Cost Estimation using Linearization

The cost (in dollars) to manufacture $$x$$ items is given by $$C(x) = 0.005x^3 - 0.2x^2 + 50x + 200$

Deceleration with Air Resistance

A car’s velocity is modeled by $$v(t) = 30e^{-0.2t} + 5$$ (in m/s) for $$t$$ in seconds.

Demand Function Inversion and Analysis

The product demand is modeled by $$p(q)=\frac{100}{q+1}+20$$, where p is the price (in dollars) and

Economic Cost Function Linearization

A company's production cost is modeled by $$C(x)= 0.02*x^3 - 1.5*x^2 + 40*x + 200$$ dollars, where $

Economic Inflation Rate

The cost of a commodity is modeled by $$C(t)=100e^{0.03*t}$$ dollars, where t is in years.

Expanding Balloon: Related Rates with a Sphere

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

Expanding Oil Spill

The area of an oil spill is modeled by $$A(t)=\pi (2+t)^2$$ square kilometers, where $$t$$ is in hou

Exponential Decay in Radioactive Material

A radioactive substance decays according to $$M(t)=M_0e^{-0.07t}$$, where $$M(t)$$ is the mass remai

Falling Object Analysis

An object is dropped from a height and its position is modeled by $$s(t)=100-4.9t^2$$ (in meters), w

Falling Object's Velocity Analysis

A rock is thrown upward from the top of a building with a velocity function $$v(t)= 20 - 9.8*t$$ (in

FRQ 1: Vessel Cross‐Section Analysis

A designer is analyzing the cross‐section of a vessel whose shape is given by the ellipse $$\frac{x^

FRQ 3: Ladder Sliding Problem

A 13­m ladder leans against a vertical wall. Its position satisfies the equation $$x^2 + y^2 = 169$$

FRQ 6: Particle Motion Analysis on a Straight Line

A particle moving along a straight line has its velocity described by $$v(t) = 3*t^2 - 4*t + 2$$, wh

Implicit Differentiation in Related Rates

A 5-foot ladder leans against a wall such that its bottom slides away from the wall. The relationshi

Inverse Trigonometric Analysis for Navigation

A navigation system relates the angle of elevation $$\theta$$ to a mountain with the horizontal dist

Linear Approximation for Function Values

Consider the function $$f(x) = x^3$$. Use linearization at $$x = 4$$ to approximate the value of $$f

Local Linearization Approximation

Let $$f(x)=x^3.$$ We want to approximate $$f(4.02)$$ using linearization near $$x=4$$.

Logarithmic Profit Optimization

A company’s profit is modeled by $$P(x) = 50x \ln(x) - 100x$$, where $$x$$ (in thousands) is the num

Marginal Profit Analysis

A company's profit in thousands of dollars is given by $$P(x)= -0.5*x^2+20*x-50$$, where $$x$$ (in h

Medicine Dosage: Instantaneous Rate of Change

The concentration of a medicine in the bloodstream is given by $$C(t) = 25e^{-0.2t}+5$$, where $$t$$

Multi‐Phase Motion Analysis

A car's motion is described by a piecewise velocity function. For $$0 \le t < 2$$ seconds, the veloc

Optimization: Minimizing Material for a Box

A company wants to design an open-top box with a square base that holds 32 cubic meters. Let the bas

Particle Motion Analysis

A particle moves along a straight line with displacement given by $$s(t)=t^3-6t^2+9t+2$$ for $$0\le

Particle Motion Analysis

A particle moves along a straight line with an acceleration given by $$a(t) = 6 - 4*t$$, where $$t$$

Particle Motion and Changing Acceleration

A particle moves along a line with its position given by $$s(t)= \ln(t+1) - t$$, where t (in seconds

Projectile Motion with Velocity Components

A projectile is launched from the ground with a constant horizontal velocity of 15 m/s and a vertica

Rate of Change in Pool Volume

The volume $$V(t)$$ (in gallons) of water in a swimming pool is given by $$V(t)=10t^2-40t+20$$, wher

Related Rates in a Conical Tank

Water is being poured into a conical tank at a rate of $$\frac{dV}{dt}=10$$ cubic meters per minute.

Related Rates: Inflating Spherical Balloon

A spherical balloon is being inflated such that its volume increases at a constant rate of $$20$$ cu

Revenue Function and Marginal Revenue Analysis

A company's revenue is modeled by $$R(x)= -0.5*x^3 + 20*x^2 + 15*x$$, where $$x$$ represents the num

Route Optimization for a Rescue Boat

A rescue boat must travel from a point on the shore to an accident site located 2 km along the shore

Shadow Length: Related Rates

A 15-foot tall lamp post casts light on a 6-foot tall man walking away from the lamp at 4 ft/sec. Le

Tangent Line and Linearization Approximation

Let $$f(x)=\sqrt{x}$$. Use linearization at $$x=16$$ to approximate $$\sqrt{15.7}$$. Answer the foll

Temperature Change in a Cooling Process

A cup of coffee cools according to the function $$T(t)=70+50e^{-0.1*t}$$, where $$T$$ is in degrees

Temperature Change in a Cooling Process

A cup of coffee cools according to the function $$T(t)= 80 + 20e^{-0.3t}$$, where t is measured in m

Temperature Change in Cooling Coffee

A cup of coffee cools according to the model $$T(t) = 90e^{-0.05t} + 20$$, where $$t$$ is the time i

Using L'Hospital's Rule to Evaluate a Limit

Consider the limit $$L=\lim_{x\to\infty}\frac{5x^3-4x^2+1}{7x^3+2x-6}$$. Answer the following:

Water Tank Volume Change

A water tank is being filled and its volume is given by $$V(t)= 4*t^3 - 9*t^2 + 5*t + 100$$ (in gall

Airport Runway Deicing Fluid Analysis

An airport runway is being de-iced. The fluid is applied at a rate $$F(t)=12*\sin(\frac{\pi*t}{4})+1

Area Bounded by $$\sin(x)$$ and $$\cos(x)$$

Consider the functions $$f(x)= \sin(x)$$ and $$g(x)= \cos(x)$$ on the interval $$[0, \frac{\pi}{2}]$

Area Growth of an Expanding Square

A square has a side length given by $$s(t)= t + 2$$ (in seconds), so its area is $$A(t)= (t+2)^2$$.

Chemical Reaction Rate and Exponential Decay

In a chemical reaction, the concentration of a reactant declines according to $$C(t)= C_0* e^{-k*t}$

Concavity and Inflection Points in a Quartic Function

Analyze the concavity and determine any points of inflection for the function $$f(x)= x^4 - 4*x^3$$.

Designing an Enclosure along a River

A farmer wants to build a rectangular enclosure adjacent to a river, using the river as one side of

Finding Local Extrema Using the First Derivative Test

Consider the function $$f(x) = x^3 - 6*x^2 + 9*x + 2$$. Answer the following:

Increasing/Decreasing Behavior in a Financial Model

A financial analyst models the performance of an investment with the function $$f(x)= \ln(x) - \frac

Inverse Analysis of a Function with an Absolute Value Term

Consider the function $$f(x)=x+|x-2|$$ with the domain restricted to $$x\ge 2$$. Analyze the inverse

Inverse Analysis of a Function with Parameter

Consider the function $$f(x)=x^3+a*x$$ where a is a real parameter. Analyze the invertibility of f a

Inverse Analysis of a Logarithm-Exponential Hybrid Function

Consider the function $$f(x)=\ln(x+2)+e^(x)$$ defined for $$x>-2$$. Address the following regarding

Inverse Analysis of a Logarithmic Function

Consider the function $$f(x)=\ln(x-1)$$ defined for $$x>1$$. Answer the following questions about it

Inverse Analysis of a Quadratic Function (Restricted Domain)

Consider the function $$f(x)=x^2-4*x+7$$ defined on the restricted domain $$[2, \infty)$$. Analyze t

Jump Discontinuity in a Piecewise Linear Function

Consider the piecewise function $$ f(x) = \begin{cases} 2x + 1, & x < 3, \\ 2x - 4, & x \ge 3. \end

Logistic Growth Model and Derivative Interpretation

Consider the logistic growth model given by $$f(t)= \frac{5}{1+ e^{-t}}$$, where $$t$$ represents ti

Mean Value Theorem for a Logarithmic Function

Consider the function $$f(x)= \ln(x)$$ defined on the interval $$[1, e^2]$$. Use the Mean Value Theo

Motion Analysis with Acceleration Function

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

Pharmacokinetics: Drug Concentration Decay

A drug in the bloodstream decays according to $$D(t)= D_0*e^{-k*t}$$, where $$t$$ is in hours. Answe

Polynomial Rational Discontinuity Investigation

Consider the function $$ g(x) = \begin{cases} \frac{x^3 - 8}{x - 2}, & x \neq 2, \\ 5, & x = 2. \en

Reservoir Evaporation and Rainfall

A reservoir gains water through rainfall and loses water by evaporation. Rainfall occurs at a rate g

Reservoir Sediment Accumulation

A reservoir experiences sediment deposition from rivers and sediment removal via dredging. The sedim

Temperature Analysis Over a Day

The temperature $$f(x)$$ (in $$^\circ C$$) at time $$x$$ (in hours) during the day is modeled by $$f

Accumulated Rainfall Estimation

A meteorological station recorded the rainfall rate (in mm/hr) at various times during a rainstorm.

Antiderivative of a Transcendental Function

Consider the function $$f(x)=\frac{2}{x}$$. Answer the following parts:

Application of the Fundamental Theorem of Calculus

A particle moves along a straight line with an instantaneous velocity given by $$v(t)=3*t^2+2*t$$ (i

Area Under a Curve with a Discontinuous Function

Consider the function $$h(x)= \begin{cases} x+2 & \text{if } 0 \le x < 3,\\ 7 & \text{if } x = 3,\\

Area Under a Parabola

Consider the quadratic function $$f(x)=x^2 - 4*x + 3$$. Analyze the function on the interval $$[1,4]

Average Value of a Function

The temperature over the first 8 hours of a day is modeled by $$T(t)=-0.5*t^2 + 4*t + 10$$, where t

Average Value of a Log Function

Let $$f(x)=\ln(1+x)$$ for $$x \ge 0$$. Find the average value of $$f(x)$$ on the interval [0,3].

Cell Growth and Accumulation

A biologist models the rate of cell production in a culture by the function $$R(t)=0.1*t^{2} + 2$$ (

Chemical Production via Integration

The production rate of a chemical in a reactor is given by $$r(t)=5*(t-2)^3$$ (in kg/hr) for $$t\ge2

Definite Integral Approximation Using Riemann Sums

Consider the function $$f(x)= x^2 + 3$$ defined on the interval $$[2,6]$$. A table of sample values

Economic Revenue Analysis from Marginal Revenue Data

A company's marginal revenue (in thousands of dollars per hour) is recorded over a 4-hour production

Elevation Profile Analysis on a Hike

A hiker records the elevation (in meters) along a trail at various distances. Use this data to analy

Estimating an Integral Using the Midpoint Rule

For the function $$f(x)=\ln(x)$$ defined on the interval [1, e], answer the following:

Evaluating the Definite Integral of a Power Function

Let $$f(t)= 3*t^{1/3}$$. Evaluate the definite integral $$\int_{27}^{64} 3*t^{1/3}\,dt$$. Answer th

Finding the Area of a Parabolic Arch

An architect designs an arch described by the parabola $$y = 10 - \frac{x^{2}}{5}$$. The arch spans

FRQ1: Analysis of an Accumulation Function and its Inverse

Consider the function $$ F(x)=\int_{1}^{x} (2*t+3)\,dt $$ for $$ x \ge 1 $$. Answer the following pa

FRQ12: Inverse Analysis of a Temperature Accumulation Function

The cumulative temperature above freezing over the morning is modeled by $$ T(t)=\int_{0}^{t} (0.8*t

FRQ19: Inverse Analysis with a Fractional Integrand

Let $$ M(x)=\int_{2}^{x} \frac{t}{t+2}\,dt $$. Answer the following parts.

Logistically Modeled Accumulation in Biology

A biologist is studying the growth of a bacterial culture. The rate at which new bacteria accumulate

Oxygen Levels in a Bioreactor

In a bioreactor, oxygen is introduced at a rate $$O_{in}(t)= 7 - 0.5t$$ mg/min and is consumed at a

Particle Motion with Variable Acceleration

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

Particle Trajectory in the Plane

A particle moves in the plane with its velocity components given by $$v_x(t)=\cos(t)$$ and $$v_y(t)=

Vehicle Distance Estimation from Velocity Data

A car's velocity (in m/s) is recorded at several time points during a trip. Use the table below for

Volume of a Solid by Washer Method

A region is bounded by the curves $$y=\sqrt{x}$$ and $$y=\frac{x}{2}$$. This region, between the cur

Volume of a Solid with Square Cross Sections

A solid has a base in the xy-plane bounded by the curves $$y=x$$ and $$y=x^2$$. Cross sections perpe

Bernoulli Differential Equation

Solve the Bernoulli differential equation $$\frac{dy}{dx}-\frac{1}{x}y=-x*y^2$$ for $$x>0$$ with the

CO2 Absorption in a Lake

A lake absorbs CO2 from the atmosphere. The concentration $$C(t)$$ of dissolved CO2 (in mol/m³) in t

Cooling of Electronic Components

After shutdown, the temperature $$T$$ (in °C) of an electronic component is recorded over time (in s

Disease Spread Modeling

The spread of an infection in a closed population is modeled by the differential equation $$\frac{dI

Epidemic Model: Logistic Growth of Infected Individuals

In a closed population, the spread of an infection is modeled by the logistic differential equation

Epidemic Spread Modeling

An epidemic in a closed population of 1000 individuals is modeled by the logistic equation $$\frac{d

Epidemic Spread with Limited Capacity

In a closed community, the number of infected individuals $$I(t)$$ (in people) is modeled by the log

Evaporation of a Liquid

A liquid evaporates from an open container and its volume $$V$$ (in liters) changes over time (in ho

Heating a Liquid in a Tank

A liquid in a tank is being heated by mixing with an incoming fluid whose temperature oscillates ove

Implicit Differential Equation and Asymptotic Analysis

Consider the differential equation $$\frac{dy}{dx}= \frac{y(1-y)}{x}$$ for $$x > 0$$ with the initia

Implicit IVP with Substitution

Solve the initial value problem $$\frac{dy}{dx}=\frac{y}{x}+\frac{x}{y}$$ with $$y(1)=2$$. (Hint: Us

Integrating Factor Method

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

Logistic Equation with Harvesting

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

Logistic Growth Model Analysis

A population $$y(t)$$ grows according to the logistic differential equation $$\frac{dy}{dt} = k * y

Logistic Growth Model for Population Dynamics

A population $$P$$ is modeled by the logistic differential equation $$\frac{dP}{dt}=0.5*P\left(1-\fr

Logistic Population Growth

A population $$P(t)$$ grows according to the logistic model $$\frac{dP}{dt}=0.3P\Bigl(1-\frac{P}{100

Mixing of a Pollutant in a Lake

A lake with a constant volume of $$10^6$$ m\(^3\) receives polluted water from a river at a rate of

Mixing Tank Problem

A tank initially contains $$100$$ liters of pure water. A salt solution with a concentration of $$0.

Motion Along a Curve with Implicit Differentiation

A particle moves along the curve defined by $$x^2+ y^2- 2*x*y= 1$$. At a certain instant, its horizo

Newton's Law of Cooling

A hot object is placed in a room with constant temperature $$20^\circ C$$. Its temperature $$T$$ sat

Nonlinear Cooling of a Metal Rod

A thin metal rod cools in an environment at $$15^\circ C$$ according to the differential equation $$

Population Dynamics with Harvesting

A fish population is governed by the differential equation $$\frac{dP}{dt} = 0.4*P\left(1-\frac{P}{1

Population with Constant Harvesting

A fish population in a lake grows according to the differential equation $$\frac{dy}{dt} = r*y - H$$

Radioactive Decay

A radioactive substance decays according to $$\frac{dy}{dt} = -0.05\,y$$ with an initial mass of $$y

Radioactive Decay with Production

A radioactive substance decays while being produced at a constant rate, and its mass $$M(t)$$ (in kg

Radioactive Isotope in Medicine

A radioactive isotope used in medical imaging decays according to $$\frac{dA}{dt}=-kA$$, where $$A$$

Radioactive Material with Constant Influx

A laboratory receives radioactive waste material at a constant rate of $$3$$ g/day. Simultaneously,

Radioactive Material with Continuous Input

A radioactive substance decays at a rate proportional to its amount while being produced continuousl

RC Circuit Discharge

In an RC circuit, the voltage across a capacitor decays according to $$\frac{dV}{dt}=-\frac{1}{RC}V$

Reaction Rate Model: Second-Order Decay

The concentration $$C$$ of a reactant in a chemical reaction obeys the differential equation $$\frac

Related Rates: Shadow Length

A 2 m tall lamp post casts a shadow of a 1.8 m tall person who is walking away from the lamp post at

Separable Differential Equation involving $$y^{1/3}$$

Consider the differential equation $$\frac{dy}{dx} = y^{1/3}$$ with the initial condition $$y(8)=27$

Separable Differential Equation: Growth Model

Consider the separable differential equation $$\frac{dy}{dx} = 3*x*y$$ with the initial condition $$

Separable Differential Equation: y and x

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

Slope Field Sketching for $$\sin(x)$$ Model

Given the slope field for the differential equation $$\frac{dy}{dx} = \sin(x)$$, sketch a solution c

Tank Mixing and Salt Concentration

A tank initially contains 100 L of solution with 5 kg of dissolved salt. A salt solution with concen

Temperature Regulation in a Greenhouse

The temperature $$T$$ (in °F) inside a greenhouse is recorded over time (in hours) as shown. The war

Arch of a Bridge

An arch of a bridge is modeled by the function $$y=10-0.5*(x-5)^2$$, where $$x$$ is in meters and th

Area Between a Function and Its Tangent

A function $$f(x)$$ and its tangent line at $$x=a$$, given by $$L(x)=m*x+b$$, are considered on the

Area Between an Exponential Function and a Linear Function

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

Area Between Curves in an Ecological Study

In an ecological study, the population densities of two species are modeled by the functions $$P_1(x

Area Between Curves: River Cross-Section

A river's cross-sectional profile is modeled by two curves. The bank is represented by $$y = 10 - 0.

Average Density of a Rod

A rod of length $$10$$ cm has a linear density given by $$\rho(x)= 4 + x$$ (in g/cm) for $$0 \le x \

Average Force and Work Done on a Spring

A spring is compressed according to Hooke's Law, where the force required to compress the spring is

Average Temperature Analysis

A researcher models the temperature during a day using the function $$T(t)=10+15*\sin\left(\frac{\pi

Average Temperature Analysis

A local weather station recorded the temperature throughout a day using the model $$T(t)=-0.5*t+35$$

Average Value Calculation for a Polynomial Function

Consider the function $$f(x)=2*x^2-3*x+1$$ defined on the interval $$[0,5]$$. Compute the average va

Cooling Process Analysis

A cup of coffee cools in a room, and its temperature (in °C) is modeled by $$T(t)=30*e^{-0.1*t}+5$$

Determining Velocity and Position from Acceleration

A particle moves along a line with acceleration given by $$a(t)=4-2*t$$ (in $$m/s^2$$). At time $$t=

Discontinuities in a Piecewise Function

Consider the function $$f(x)=\begin{cases} \frac{\sin(x)}{x} & \text{if } x \neq 0 \\ 2 & \text{if }

Filling a Container: Volume and Rate of Change

Water is being poured into a container such that the height of the water is given by $$h(t)=2*\sqrt{

Finding the Area Between Two Curves

Let the functions $$f(x)=5*x-x^2$$ and $$g(x)=x$$ be given. Find the area of the region bounded by t

Free Workout Class Attendance

The attendance at a free workout class increases by a fixed number of people each session. The first

Hollow Rotated Solid

Consider the region in the first quadrant bounded by the curves $$y=x$$ and $$y=x^2$$. This region i

Implicit Differentiation in an Electrical Circuit

In an electrical circuit, the voltage $$V$$ and current $$I$$ are related by the equation $$V^2 + (3

Manufacturing Output Increase

A factory produces goods with weekly output that increases by a constant number of units each week.

Motion along a Straight Path

A particle moving along the x-axis has its acceleration given by $$a(t)=6-4*t$$ (in m/s²) for $$t \g

Particle Motion with Exponential Acceleration

A particle moves along a straight line with acceleration given by $$a(t)=2*e^{-t} - 1$$ (in m/s²) fo

Pipeline Installation Cost Analysis

The cost to install a pipeline along a route is given by $$C(x)=100+5*\sin(x)$$ (in dollars per mete

Population Growth with Variable Growth Rate

A city's population changes with time according to a non-constant growth rate given in thousands per

Population Model Using Exponential Function

A bacteria population is modeled by $$P(t)=100*e^{0.03*t} - 20$$, where $$t$$ is measured in hours.

River Current Analysis

The velocity of a river is given by $$v(x)=2+x-0.1*x^2$$ (in m/s) for 0 ≤ x ≤ 10, where x measures t

Total Distance Traveled from a Velocity Profile

A particle’s velocity over the interval $$[0, 6]$$ seconds is given in the table below. Note that th

Voltage and Energy Dissipation Analysis

The voltage across an electrical component is modeled by $$V(t)=12*e^{-0.1*t}*\ln(t+2)$$ (in volts)

Volume by the Cylindrical Shells Method

A region bounded by $$y=\ln(x)$$, $$y=0$$, and the vertical line $$x=e$$ is rotated about the y-axis

Volume of a Solid of Revolution using Shells

Consider the region under the curve $$f(x)=e^{-x}$$ for $$x \in [0,1]$$. This region is revolved abo

Volume of a Solid of Revolution Using the Disk Method

Consider the region bounded by the graph of $$f(x)=\sqrt{x}$$, the x-axis, and the vertical line $$x

Volume of a Solid with Square Cross-Sections

A solid has a base in the xy-plane bounded by $$y=x$$ and $$y=x^2$$ for $$0 \le x \le 1$$. Every cro

Water Flow into a Reservoir

Water flows into a reservoir at a rate given by $$R(t)= 20 - 2*t$$ cubic meters per hour, where $$t$

Work to Pump Water from a Cylindrical Tank

A cylindrical tank with a radius of 3 m and a height of 10 m is completely filled with water (densit