AP Calculus AB FRQ Room

Algebraic Manipulation in Limit Evaluation

Evaluate the limit $$\lim_{x \to 1} \frac{x^3 - 1}{x - 1}$$.

Analysis of One-Sided Limits and Jump Discontinuity

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

Analyzing Limits from Experimental Data (Table)

The table below shows measured values of a function $$f(x)$$ near $$x = 1$$. | x | f(x) | |-----

Analyzing Process Data for Continuity

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

Asymptotic Analysis of a Rational Function

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

Combined Limit Analysis of a Piecewise Function

Consider the function $$f(x)= \begin{cases} \frac{x^2-1}{x-1} & \text{if } x \neq 1, \\ c & \text{if

Composite Function and Continuity Analysis

Define \(f(x)=\sqrt{1-\ln(x)}\) for \(x>0\) and consider the composite function \(g(x)=f(e^x)\). Ans

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 and Composition of Functions

Consider two functions: $$ f(x)=\frac{x^2-1}{x-1} $$ for $$x\ne1$$, and the piecewise function $$ g(

Continuity in a Piecewise Function with Square Root and Rational Expression

Consider the function $$f(x)=\begin{cases} \sqrt{x+6}-2 & x<-2 \\ \frac{(x+2)^2}{x+2} & x>-2 \\ 0 &

Direct Evaluation of Polynomial Limits

Let $$ f(x)=x^3-5*x+2 $$.

Estimating Derivatives Using Limit Definitions from Data

The position of an object (in meters) is recorded at various times (in seconds) in the table below.

Evaluating Sequential Limits in Particle Motion

A particle’s velocity is given by the function $$v(t)= \frac{(t-2)(t+4)}{t-2}$$ for $$t \neq 2$$, an

Factorization and Limit Evaluation

Consider the function $$f(x) = \frac{x^3 - 8}{x - 2}$$. (a) Factor the numerator and simplify the e

Factorization and Removable Discontinuity

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

Graph Analysis: Identify Limits and Discontinuities

A graph of a function f(x) is provided in the stimulus. The graph shows a removable discontinuity at

Graph Transformations and Continuity

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

Intermediate Value Theorem in Equation Solving

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

Investigating Discontinuities in a Rational Function

Consider the function $$ h(x)=\frac{x^2-4}{x-2} $$ for $$x\ne2$$.

Jump Discontinuity in a Piecewise Function

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

Limit Analysis in Population Modeling

A population is modeled by the function $$P(t)= \frac{1000*t}{t+5}$$ where $$t \geq 0$$ (in years).

Limit Involving a Square Root and Removable Discontinuity

Consider the function $$h(x)=\frac{\sqrt{x+4}-2}{x}$$ for $$x\neq0$$ and $$h(0)=1$$. Answer the foll

Limit Involving Radical Expressions

For the function $$f(x)=\frac{\sqrt{x+9}-3}{x}$$, evaluate the limit as x approaches 0.

Limits Involving Absolute Value Expressions

Evaluate the limit $$\lim_{x \to 0} \frac{|x|}{x}$$.

Limits Involving Absolute Value Functions

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

Limits Involving Composition and Square Roots

Consider the function $$ f(x)=\sqrt{x+4}-2 $$.

Limits Involving Radical Functions

Evaluate the limit $$\lim_{x \to 4} \frac{\sqrt{x} - 2}{x - 4}$$.

Long-Term Behavior of Particle Motion: Horizontal Asymptotes

For a particle, the velocity function is given by $$v(t)= \frac{4*t^2-t+1}{t^2+2*t+3}$$. Answer the

One-Sided Limits and Absolute Value Functions

Let $$f(x) = \frac{|x - 2|}{x - 2}$$. Analyze its behavior as x approaches 2.

One-Sided Limits and Vertical Asymptotes

Consider the function $$ f(x)= \frac{1}{x-4} $$.

Piecewise Function Continuity and IVT

Let $$f(x)=\begin{cases} x^2, & x \leq 1 \\ a*x+b, & x > 1 \end{cases}$$. Determine constants a and

Rational Function with Two Critical Points

Consider the function $$f(x)=\begin{cases} \frac{x^2+x-6}{x^2-9} & x\neq -3,3 \\ \frac{5}{6} & x=-3

Removable Discontinuity and Limit

Consider the function $$ f(x)=\frac{x^2-9}{x-3} $$ for $$ x\ne3 $$, which is not defined at $$ x=3 $

Removable Discontinuity and Limit Evaluation

Consider the function $$f(x) = \frac{(x + 3) * (x - 2)}{x + 3}$$ for $$x \neq -3$$. Answer the follo

Removable Discontinuity and Redefinition

Consider the function $$f(x)=\frac{x^2-4}{x-2}$$ for $$x\neq2$$. Note that f is undefined at $$x=2$$

Removable Discontinuity in a Cubic Function

Consider the function $$f(x)=\begin{cases} \frac{x^3-27}{x-3} & x\neq3 \\ 10 & x=3 \end{cases}$$. An

Removing Discontinuities

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

Squeeze Theorem Application

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

Water Tank Inflow-Outflow Analysis

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

Acceleration Through Successive Differentiation

A particle’s position is given by $$s(t)=t^3-6*t^2+9*t+4$$ (with s in meters and t in seconds). Answ

Analysis of Motion in the Plane

A particle moves in the plane with its position given by $$\mathbf{s}(t)=\langle t^2 - 4*t,\, 3*t +

Analyzing a Function's Derivative from its Graph

A graph of a smooth function is provided. Answer the following questions:

Analyzing Concavity and Inflection Points Using Derivatives

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

Application of the Quotient Rule: Velocity on a Curve

A car's velocity is modeled by $$v(t)= \frac{2*t+3}{t+1}$$, where $$t$$ is measured in seconds. Anal

Approximating the Instantaneous Rate of Change Using Secant Lines

A function $$f(t)$$ models the position of an object. The following table shows selected values of $

Approximating the Tangent Slope

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

Car Fuel Consumption vs. Refuel

A car is being refueled at a constant rate of $$4$$ liters/min while it is being driven. Simultaneou

Derivative from First Principles

Derive the derivative of the polynomial function $$f(x)=x^3+2*x$$ using the limit definition of the

Derivative of an Exponential Decay Function

Consider the function $$f(t)=e^{-0.5*t}$$, which may represent the decay of a substance over time. A

Derivative using the Limit Definition for a Linear Function

For the linear function $$f(x)= 5*x - 3$$, perform an analysis of its derivative using the limit def

Derivatives in Economics: Cost Functions

A company's production cost is modeled by $$C(q)=500+20*q-0.5*q^2$$, where $$q$$ represents the quan

Derivatives of Trigonometric Functions

Let $$f(x)=\sin(x)+\cos(x)$$, where $$x$$ is measured in radians. This function may represent a comb

Derivatives on an Ellipse

The ellipse given by $$\frac{x^2}{9} + \frac{y^2}{4} = 1$$ represents a race track. Answer the follo

Differentiation of a Composite Motion Function

A particle’s position is given by $$s(t) = t^2 * \ln(t)$$ for $$t > 0$$. Use differentiation to anal

Differentiation of a Log-Linear Function

Consider the function $$f(x)= 3 + 2*\ln(x)$$ which might model a process with a logarithmic trend.

Evaluating Derivative of a Composite Function using the Definition

Consider the function $$h(x)=\sqrt{4+x}$$. Answer the following questions:

Finding Derivatives of Composite Functions

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

Finding the Tangent Line Using the Product Rule

For the function $$f(x)=(3*x^2-2)*(x+5)$$, which models a physical quantity's behavior over time (in

Graph vs. Derivative Graph

A graph of a function $$f(x)$$ and a separate graph of its derivative $$f'(x)$$ are provided in the

Implicit Differentiation of a Circle

Consider the equation $$x^2 + y^2 = 25$$ representing a circle with radius 5. Answer the following q

Inverse Function Analysis: Cubic Transformation

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

Inverse Function Analysis: Hyperbolic-Type Function

Consider the function $$f(x)=\sqrt{x^2+1}$$ defined for $$x\geq0$$.

Inverse Function Analysis: Quadratic Function

Consider the function $$f(x)=x^2$$ restricted to $$x\geq0$$.

Limit Definition for a Quadratic Function

For the function $$h(x)=4*x^2 + 2*x - 7$$, answer the following parts using the limit definition of

Marginal Cost from Exponential Cost Function

A company’s cost function is given by $$C(x)= 500*e^{0.05*x} + 200$$, where $$x$$ represents the num

Marginal Profit Calculation

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

Mountain Stream Flow Adjustment

A mountain stream receives additional water from snowmelt at a rate of $$f(t)=4*t$$ (cubic feet/seco

Polynomial Rate of Change Analysis

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

Product Rule Application in Economics

A company's cost function for producing $$x$$ units is given by $$C(x)= (3*x+2)*(x^2+5)$$ (cost in d

Rate of Chemical Reaction

The concentration of a reactant in a chemical reaction is modeled by \(C(t)=10*e^{-0.3*t}\), where \

RC Circuit Voltage Decay

An RC circuit's capacitor voltage is modeled by $$V(t)= V_{0}*e^{-t/(R*C)}$$, where $$V_{0}$$ is the

Related Rates: Balloon Surface Area Change

A spherical balloon has volume $$V=\frac{4}{3}\pi r^3$$ and surface area $$S=4\pi r^2$$. If the volu

Riemann Sums and Derivative Estimation

A car’s position $$s(t)$$ in meters is recorded in the table below at various times $$t$$ in seconds

Sand Pile Growth with Erosion Dynamics

A sand pile is growing as sand is added at a rate of $$f(t)=8+0.3*t$$ (kg/min) and simultaneously lo

Secant and Tangent Lines Analysis

Consider the function $$g(t)=t^3-6*t^2+9*t+2$$ modeling the height (in meters) of a ball at time $$t

Secant and Tangent Lines for a Cubic Function

Consider the function $$f(x)= x^3 - 4*x$$.

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 Slope from Tabulated Data

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

Tangent Line Approximation for a Cubic Function

Let $$f(x)=2*x^3 - 7*x + 1$$. At $$x=1$$, determine the equation of the tangent line and use it to a

Tangent to an Implicit Curve

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

Using the Limit Definition of the Derivative

Consider the function $$g(x)=3*x^3-2*x+5$$, which models the cost (in dollars) of manufacturing $$x$

Analyzing a Function and Its Inverse

Consider the invertible function $$f(x)= \frac{x^3+1}{2}$$.

Chain Rule with Trigonometric and Exponential Functions

Let $$y = \sin(e^{3*x})$$. Answer the following:

Chemical Reaction Rate: Exponential and Logarithmic Model

The concentration of a chemical reaction is modeled by $$C(t)= \ln\left(3*e^(2*t) + 7\right)$$, wher

Combining Chain Rule, Implicit, and Inverse Differentiation

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

Composite Differentiation of an Inverse Trigonometric Function

Let $$H(x)= \arctan(\sqrt{x+3})$$.

Composite Function Differentiation in a Sand Pile Model

Sand is added to a pile at an inflow rate of $$A(t)= 4 + t^2$$ (kg/min) and removed at an outflow ra

Composite Function with Inverse Trigonometric Outer Function

Consider the function $$H(x)=\arctan(\sqrt{x^2+1})$$. Answer the following parts.

Composite Log-Exponential Function Analysis

A function is defined by $$f(x)=\ln\left(e^(2*x^2) + 5\right)$$. This function is composed of an exp

Differentiation of Inverse Function with Polynomial Functions

Let \(f(x)= x^3+2*x+1\) be a one-to-one function. Its inverse is denoted by \(f^{-1}\).

Differentiation of Inverse Trigonometric Function via Implicit Differentiation

Let $$y=\arcsin(\frac{x}{\sqrt{2}})$$. Answer the following:

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

Expanding Spherical Balloon

A spherical balloon has its volume given by $$V=\frac{4}{3}\pi r^3$$. The radius of the balloon incr

Implicit Differentiation in Circular Motion

Consider the circle defined by the equation $$x^2+y^2=100$$, which could represent the track of an o

Implicit Differentiation Involving Logarithms

Consider the equation $$\ln(x) + x*y = \ln(y) + x$$ which relates $$x$$ and $$y$$. Use implicit diff

Implicit Differentiation of an Ellipse in Navigation

A flight path is modeled by the ellipse $$\frac{x^2}{16}+\frac{y^2}{9}=1$$.

Implicit Differentiation on a Circle

Consider the circle defined by $$x^2+y^2=25$$. Use implicit differentiation to find the slope of the

Implicit Differentiation with Mixed Functions

Consider the relation $$x\cos(y)+y^3=4*x+2*y$$.

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: Combined Product and Chain Rules

Consider the equation $$x^2*y + \sin(x*y) = 0$$. 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 Derivative in Thermodynamics

A thermodynamic process is modeled by the function $$P(V)= 3*V^2 + 2*V + 5$$, where $$V$$ is the vol

Inverse Function Differentiation for a Log Function

Let $$f(x)=\ln(3*x+2)$$, and assume that $$f$$ is invertible with inverse function $$g$$. Find the d

Inverse Function Differentiation in Logarithmic Functions

Let $$f(x)=\ln(x+2)$$, which is one-to-one and has an inverse function $$g(y)$$. Answer the followin

Inverse Function in Currency Conversion

A function converting dollars to euros is given by $$f(d) = 0.9*d + 10\ln(d+1)$$ for $$d > 0$$. Let

Inverse Trigonometric Differentiation

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

Optimization in a Container Design Problem

A manufacturer is designing a closed cylindrical container with a fixed volume of $$1000\,cm^3$$. Th

Particle Motion: Logarithmic Position Function

The position of a particle moving along a line is given by $$s(t)= \ln(3*t+2)$$, where s is in meter

Pendulum Angular Displacement Analysis

A pendulum's angular displacement is modeled by the function $$\theta(t)=\sin(2t^2+3)$$, where t is

Analyzing a Nonlinear Rate of Revenue Change

A company's revenue in thousands of dollars is modeled by the function $$R(x)=100\ln(x+1) + 0.5x$$,

Analyzing Position Data with Table Values

A moving object’s position, given by $$x(t)$$ in meters, is recorded in the table below. Use the dat

Critical Points and Concavity Analysis

Consider the polynomial function $$f(x)= x^4 - 4*x^3 + 6*x^2 - 4*x + 1$$ modeling the position of an

Defect Rate Analysis in Manufacturing

The defect rate in a manufacturing process is modeled by $$D(t)=100e^{-0.05t}+5$$ defects per day, w

Determining the Tangent Line

Consider the function $$f(x)=\ln(x)+ x$$. The graph of the function is provided for reference.

Differentiability of a Piecewise Function

Consider the piecewise function $$ f(x)=\begin{cases} x^2, & x \leq 2 \\ 4x-4, & x>2 \end{cases} $$

Drainage Analysis in a Conical Tank

Water is draining from a conical tank at a constant rate of 3 cubic meters per minute. The tank has

Error Approximation in Engineering using Differentials

The cross-sectional area of a circular pipe is given by $$A=\pi r^2$$. If the radius is measured as

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 10: Chemical Kinetics Analysis

In a chemical reaction, the concentration of reactant A, denoted by [A], and time t (in minutes) are

Graphing a Function via its Derivative

Consider the function $$f(x) = x^{1/3}$$ defined for all real numbers.

Inflating Spherical Balloon

A spherical balloon is being inflated such that its volume increases at a constant rate of $$\frac{d

Inverse Function Analysis in a Real-World Model

Consider the function $$f(x)=x^3+1$$ which models a certain transformation of measurable quantities.

L'Hôpital's Rule in Action

Evaluate the following limit by applying L'Hôpital's Rule as necessary: $$\lim_{x \to \infty} \frac{

Linear Approximation of ln(1.05)

Let $$f(x)=\ln(x)$$. Use linearization at x = 1 to approximate the value of $$\ln(1.05)$$.

Linearization and Differentials Approximation

A function $$f(x)= x^3$$ models the volume (in cubic centimeters) of liquid in a container as a func

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

Maximizing Enclosed Area

A rancher has 120 meters of fencing to enclose a rectangular pasture along a straight river (the sid

Minimizing Materials for a Cylindrical Can

A manufacturer aims to design a closed cylindrical can that holds exactly $$500$$ cubic centimeters

Particle Motion with Changing Acceleration

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

Population Growth Model and Asymptotic Limits

A population is modeled by $$P(t) = \frac{5000e^{0.1t}}{1 + e^{0.1t}}$$, where $$P(t)$$ is the popul

Projectile Motion: Maximum Height

A ball is thrown upward and its height is modeled by $$h(t)=-5t^2+20t+2$$ (in meters). Analyze its m

Related Rates: Expanding Circular Ripple

A ripple in a still pond expands in the shape of a circle. The area of the ripple is given by $$A=\p

Related Rates: Inflating Spherical Balloon

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

Seasonal Water Reservoir

A reservoir's water volume (in million m³) changes with the seasons according to $$V(t)=5+2\sin\left

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

Volume Change Analysis in a Swimming Pool

The volume of a pool is given by $$V(t)=8t^2-32t+4$$, where V is in gallons and t in hours. Analyze

Absolute Extrema via the Candidate's Test

Consider the function $$f(x)= \sqrt{x} - x$$ on the closed interval $$[0,4]$$. Use the candidate's t

Asymptotic Behavior in an Exponential Decay Model

Consider the model $$f(t)= 100*e^{-0.3*t}$$ representing a decaying substance over time. Answer the

Bacterial Culture Growth: Identifying Critical Points from Data

A microbiologist records the population of a bacterial culture (in millions) at different times (in

Biological Growth and the Mean Value Theorem

In a bacterial culture, the population is modeled by $$P(t)= 4*t^2 + 3*t + 7$$ for $$t$$ in hours on

Capacitor Discharge in an RC Circuit

The voltage across a capacitor during discharge is given by $$V(t)= V_0*e^{-t/(RC)}$$, where $$t$$ i

Concavity Analysis of a Cubic Function

Consider the function $$f(x)= x^3 - 6*x^2 + 9*x + 2$$. Use the second derivative to investigate the

Continuity Analysis of a Rational Piecewise Function

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

Cubic Polynomial Analysis

Consider the cubic function $$f(x)= x^3 - 6*x^2 + 9*x + 2$$ defined on the interval $$[0,4]$$. Analy

Economic Demand and Revenue Optimization

The demand for a product is modeled by $$D(p) = 100 - 2*p$$, where $$p$$ is the price in dollars. Th

Finding Local Extrema Using the First Derivative Test

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

FRQ 6: Particle Motion with Variable Acceleration

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

FRQ 9: Extreme Value Analysis for a Rational Function

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

FRQ 10: First Derivative Test for a Cubic Profit Function

A company’s profit function is given by $$P(x)= x^3 - 9*x^2 + 24*x + 1$$, where $$x$$ represents the

FRQ 17: Analysis of a Trigonometric Function for Extrema and Inflection Points

Let $$f(x)= \sin(x) - 0.5*x$$ for $$x \in [0, 2\pi]$$.

FRQ 20: Profit Analysis Combining MVT and Optimization

A company’s profit function is given by $$P(x)= -2*x^3 + 18*x^2 - 48*x + 40$$, where $$x$$ (in thous

Increase and Decrease Analysis of a Polynomial Function

Consider the function $$f(x)= x^4 - 4*x^3 + 6*x^2 - 4*x + 1$$ on the interval $$[-1, 3]$$.

Inverse Analysis of a Composite Function

Consider the function $$f(x)=e^(x)+x$$. Although its inverse cannot be written in closed form, answe

Investigating the Behavior of a Composite Function

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

Oil Spill Cleanup

In an oil spill scenario, oil continues to enter an affected area while cleanup efforts remove oil.

Optimization of a Rectangle Inscribed in a Semicircle

A rectangle is inscribed in a semicircle of radius 5 (with the base along the diameter). The top cor

Optimization of an Open-Top Box

A company is designing an open-top box with a square base. The volume of the box is modeled by the f

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

Population Growth Analysis via the Mean Value Theorem

A country's population data over a period of years is given in the table below. Use the data to anal

Profit Function Concavity Analysis

A company’s profit is modeled by $$P(x) = -2*x^3 + 18*x^2 - 48*x + 10$$, where $$x$$ is measured in

Radioactive Substance Decay

A radioactive substance decays according to the model $$A(t)= A_0 * e^{-\lambda*t}$$, where $$t$$ is

Relative Extrema of a Rational Function

Examine the function $$f(x)= \frac{x+1}{x^2+1}$$ and determine its relative extrema using derivative

Revenue Optimization in Economics

A company's revenue is modeled by the function $$R(x)= x*e^{-0.1*x}$$, where $$x$$ (in thousands) re

Sign Analysis of f'(x)

The first derivative $$f'(x)$$ of a function is known to have the following behavior on $$[-2,2]$$:

Tangent Line to an Implicitly Defined Curve

The curve is defined by the equation $$x^2 + x*y + y^2 = 7$$.

Temperature Regulation in a Greenhouse

A greenhouse is regulated by an inflow of warm air and an outflow of cooler air. The inflow temperat

Transcendental Function Analysis

Consider the function $$f(x)= \frac{e^x}{x+1}$$ defined for $$x > -1$$ and specifically on the inter

Verifying the Mean Value Theorem for a Polynomial Function

Consider the function $$f(x) = x^3 - 3*x^2 + 4$$ defined on the interval $$[0, 3]$$. Answer the foll

Analyzing Bacterial Growth via Riemann Sums

A biologist measures the instantaneous growth rate of a bacterial population (in thousands of cells

Application of the Fundamental Theorem in a Discounted Cash Flow Model

A continuous cash flow is given by $$C(t)=500(1+0.05*t)$$ dollars per year. Using a continuous disco

Approximating the Area with Riemann Sums

Consider the linear function $$f(x) = 2*x + 1$$ on the interval $$[1,5]$$. Use Riemann sums to appro

Comparing Riemann Sum Methods for $$\int_1^e \ln(x)\,dx$$

Consider the function $$f(x)= \ln(x)$$ on the interval $$[1,e]$$. A table of approximate values is p

Comparing Riemann Sum Methods for a Complex Function

Consider the function $$f(x)=\frac{1}{1+x^2}$$ on the interval [0,1]. Answer the following:

Elevation Profile Analysis on a Hike

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

Environmental Modeling: Pollution Accumulation

The pollutant enters a lake at a rate given by $$P(t)=5*e^{-0.3*t}$$ (in kg per day) for $$t$$ in da

Evaluating the Accumulated Drug Concentration

In a pharmacokinetics study, a drug is infused into a patient's bloodstream at a rate given by $$R(t

FRQ4: Inverse Analysis of a Trigonometric Accumulation Function

Let $$ H(x)=\int_{0}^{x} (\sin(t)+2)\,dt $$ for $$ x \in [0,\pi] $$, representing a displacement fun

Fuel Consumption Analysis

A truck's fuel consumption rate (in L/hr) is recorded at various times during a 12-hour drive. Use t

Integration by Parts: Evaluating $$\int_1^e \ln(x)\,dx$$

Evaluate the integral $$\int_1^e \ln(x)\,dx$$ using integration by parts.

Motion Under Variable Acceleration

A particle moves along the x-axis with acceleration $$a(t) = 6 - 4*t$$ (in m/s²) for $$0 \le t \le 3

Optimizing Fencing Cost for a Garden Adjacent to a River

A farmer plans to fence a rectangular garden adjacent to a river, so that no fence is required along

Particle Motion with Variable Acceleration

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

Piecewise-Defined Function and Discontinuities

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

Rainfall Accumulation via Integration

A region experiences rain where the rate of rainfall (in inches per hour) is given by $$r(t)=0.5+0.2

Tabular Riemann Sums for Electricity Consumption

A household's daily electricity consumption (in kWh) over 5 consecutive days is recorded in the tabl

Temperature Change in a Chemical Reaction

During an exothermic chemical reaction, the temperature (in °C) is recorded over a 15-minute period.

Volume Accumulation in a Leaking Tank

Water leaks from a tank at a rate given by $$R(t)=3-0.5*t$$ (in liters per minute) for t in [0,6]. I

Work Done by a Variable Force

A force acting on an object is given by $$F(x)=3*x^2+2*x$$ where x is in meters and F in Newtons. Th

Bernoulli Differential Equation

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

Chemical Reaction Rate and Concentration Change

The rate of a chemical reaction is described by the differential equation $$\frac{dC}{dt}=-0.3*C^2$$

Direction Fields for an Autonomous Equation

Consider the differential equation $$\frac{dy}{dx}=y^2-9$$. Analyze the behavior of its solutions.

Environmental Pollution Model

Pollutant concentration in a lake is modeled by the differential equation $$\frac{dC}{dt}=\frac{R}{V

Epidemic Spread with Limited Capacity

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

Implicit Differentiation of a Circle

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

Implicit Differentiation of a Transcendental Equation

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

Linear Differential Equation using Integrating Factor

Solve the linear differential equation $$\frac{dy}{dx} + 2y = x$$ with the initial condition $$y(0)=

Logistic Growth in a Population

A population $$P(t)$$ grows according to the logistic differential equation $$\frac{dP}{dt}=0.5P\lef

Logistic Population Growth

A population grows according to the logistic model $$\frac{dP}{dt}= r * P\left(1-\frac{P}{K}\right)$

Mixing Problem with Variable Inflow Concentration

A tank initially contains 50 L of water with 5 kg of dissolved salt. A solution enters the tank at a

Mixing Tank Problem

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

Mixing with Variable Inflow Rate

A 50-liter tank initially contains water with 1 kg of dissolved salt. Water containing 0.2 kg of sal

Modeling Continuous Compound Interest

An account accrues interest continuously according to the differential equation $$\frac{dA}{dt}=rA$$

Modeling Cooling with Newton's Law of Cooling

A hot beverage cools according to Newton's Law of Cooling, modeled by the differential equation $$\f

Motion Under Gravity with Air Resistance

An object is falling vertically under the influence of gravity and air resistance. Its velocity $$v(

Newton's Law of Cooling

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

Oil Spill Cleanup Dynamics

To mitigate an oil spill, a cleanup system is employed that reduces the volume of oil in contaminate

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

Related Rates: Conical Tank Filling

Water is pumped into a conical tank at a rate of $$3$$ m$^3$/min. The tank has a height of $$4$$ m a

Related Rates: Expanding Balloon

A spherical balloon is inflated such that its radius increases at a constant rate of $$\frac{dr}{dt}

Salt Tank Mixing Problem

A tank initially contains 100 liters of pure water. A salt solution with concentration 0.5 kg/L is p

Separable Differential Equation: $$dy/dx = x*y$$

Consider the differential equation $$dy/dx = x*y$$ with the initial condition $$y(0)=2$$. Solve the

Separable Differential Equation: Growth Model

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

Slope Field and General Solution

Consider the differential equation $$\frac{dy}{dx}=x$$. The attached slope field shows the slopes at

Solving a Differential Equation Using the SIPPY Method

Solve the differential equation $$\frac{dy}{dx}=4*x^3*e^{-y}$$ with the initial condition $$y(1)=0$$

Soot Particle Deposition

In an environmental study, the thickness $$P$$ (in micrometers) of soot deposited on a surface is me

Tank Mixing with Salt

In a mixing problem, a tank contains salt that is modeled by the differential equation $$\frac{dS}{d

Temperature Regulation in a Greenhouse

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

Accumulated Electrical Charge from a Current Function

An electrical device charges according to the current function $$I(t)= 10*e^{-0.3*t}$$ amperes, wher

Accumulated Nutrient Intake from a Drip

A medical nutrient drip administers a nutrient at a variable rate given by $$N(t)=-0.03*t^2+1.5*t+20

Area Between \(\ln(x+1)\) and \(\sqrt{x}\)

Consider the functions $$f(x)=\ln(x+1)$$ and $$g(x)=\sqrt{x}$$ over the interval $$[0,3]$$.

Average Temperature Analysis

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

Average Temperature of a Cooling Liquid

The temperature of a cooling liquid is modeled by $$T(t)=50*e^{-0.1*t}+20$$ (in $$^\circ C$$) for $$

Average Temperature Over a Day

In a city, the temperature (in $$^\circ C$$) is modeled by $$T(t)=10+5*\cos\left(\frac{\pi*t}{12}\ri

Average Voltage in a Physics Experiment

In a physics experiment, the voltage across a resistor is modeled by $$V(t)=5+3*\cos\left(\frac{\pi*

Center of Mass of a Lamina with Variable Density

A thin lamina occupies the interval $$[0,4]$$ along the x-axis and has a variable density $$\delta(x

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=

Distance Traveled by a Jogger

A jogger increases her daily running distance by a fixed amount. On the first day she runs $$2$$ km,

Exponential Decay Function Analysis

A lab experiment models the decay of a chemical concentration with the function $$f(t)=8*e^{-0.5*t}$

Ice Rink Design: Volume and Area

An ice rink is designed with a cross-sectional profile given by $$y=4-x^2$$ (with y=0 as the base).

Integrated Motion Analysis

A particle moving along a straight line has an acceleration given by $$a(t)= 4 - 6*t$$ (in m/s²) for

Net Change in Biological Population

A species' population changes at a rate given by $$P'(t)=0.5e^{-0.2*t}-0.05$$ (in thousands per year

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 Rate Analysis

Suppose the instantaneous growth rate of a population is given by $$r(t)=0.04 - 0.002*t$$ for $$t \i

Radioactive Decay Accumulation

A radioactive substance decays at a rate given by $$r(t)= C*e^{-k*t}$$ grams per day, where $$C$$ an

Rebounding Ball

A ball is dropped from a height of $$16$$ meters. Each time the ball bounces, its maximum height is

River Discharge Analysis

The flow rate of a river is modeled by $$Q(t)=20+5*\sin\left(\frac{\pi*t}{12}\right)$$ (in cubic met

Tank Filling Process Analysis

Water flows into a tank at a rate modeled by $$R(t)=5+0.5*t$$ (in liters per minute) for $$0 \le t \

Temperature Average Calculation

A scientist records the temperature in a lab using a continuous function $$T(t)=3*t^2 - 4*t + 5$$, w

Temperature Increase in a Chemical Reaction

During a chemical reaction, the rate of temperature increase per minute follows an arithmetic sequen

Volume by the Disc Method for a Rotated Region

Consider a function $$f(x)$$ that represents the radius (in cm) of a region rotated about the x-axis

Volume of a Rotated Region by the Disc Method

Consider the region bounded by the curve $$f(x)=\sqrt{x}$$ and the line $$y=0$$ for $$0 \le x \le 4$

Volume of a Solid of Revolution Using the Washer Method

The region bounded by the curves $$x=\sqrt{y}$$ and $$x=\frac{y}{2}$$ for $$y\in[0,4]$$ is revolved

Volume of a Solid with Square Cross Sections

A solid has a base in the xy-plane given by the region bounded by the curves $$y=x$$ and $$y=\sqrt{x

Volume with Semicircular Cross-Sections

A solid has a base on the interval $$[0,3]$$ along the x-axis, and its cross-sectional slices perpen

Volume with Square Cross-Sections

Consider the region bounded by the curve $$y=x^2$$ and the line $$y=4$$ for $$0 \le x \le 2$$. Squar

Washer Method with Logarithmic and Exponential Curves

Consider the region bounded by the curves $$f(x)=\ln(x+1)$$ and $$g(x)=e^{-x}$$ on the interval $$[0