AP Calculus AB FRQ Room

Advanced Analysis of an Oscillatory Function

Consider the function $$ f(x)= \begin{cases} \sin(1/x), & x\ne0 \\ 0, & x=0 \end{cases} $$.

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

Asymptotic Analysis of a Rational Function

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

Capstone Problem: Continuity and Discontinuity in a Compound Piecewise Function

Consider the function $$f(x)=\begin{cases} \frac{x^2-1}{x-1} & x<2 \\ \frac{x^2-4}{x-2} & x\ge2 \end

Continuity Analysis of a Piecewise Function

Consider the function defined by $$ f(x)=\begin{cases}2x+1, & x<1,\\ x^2, & 1\le x\le 3,\\ 7-x, & x

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 in a Cost Function for a Manufactured Product

A company's cost function for producing $$n$$ items (with $$n > 0$$) is given by $$C(n)= \frac{50}{n

Continuity of a Composite Log-Exponential Function

Let $$f(x)=\begin{cases} \frac{\ln(1+e^{x})-\ln(2)}{x} & \text{if } x\neq 0, \\ C & \text{if } x=0.

Continuity of a Radical Function

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

Continuity of a Sine-over-x Function

Consider the function $$f(x)=\begin{cases} \frac{\sin(x)}{x}, & x \neq 0 \\ 1, & x=0 \end{cases}$$.

Determining Parameters for a Continuous Log-Exponential Function

Suppose a function is defined by $$ v(x)=\begin{cases} \frac{\ln(e^{p*x}+x)-q*x}{x} & \text{if } x \

Estimating Limits from a Data Table

A function f(x) is studied near x = 3. The table below shows selected values of f(x):

Evaluating Limits Involving Square Roots

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

Graph Analysis of Discontinuities

A graph of a function f(x) shows a jump discontinuity at x = 1 and a removable discontinuity (a hole

Graph Transformations and Continuity

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

Horizontal Asymptote of a Rational Function

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

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 with Square Root and Removable Discontinuity

Consider the function $$f(x)=\begin{cases} \frac{\sqrt{4*x+8}-4}{x-2} & x\neq2 \\ 1 & x=2 \end{cases

Limits at Infinity and Horizontal Asymptotes

Examine the function $$f(x)=\frac{3x^2+2x-1}{6x^2-4x+5}$$ and answer the following:

Logarithmic Function Continuity

Consider the function $$g(x)=\frac{\ln(2*x+3)-\ln(5)}{x-1}$$ for $$x \neq 1$$. To make $$g(x)$$ cont

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

Oscillatory Behavior and Continuity

Consider the function $$f(x)=\begin{cases} x*\sin\left(\frac{1}{x}\right) & x\neq 0 \\ 0 & x=0 \end{

Oscillatory Behavior and Discontinuity

Consider the function $$f(x)=\begin{cases} x\cos(\frac{1}{x}) & x\neq0 \\ 2 & x=0 \end{cases}$$. Ans

Piecewise Function with Different Expressions

Consider the function $$f(x)=\begin{cases} 3*x-1 & x<2\\ \frac{x^2-4}{x-2} & x>2\\ 5 & x=2 \end{case

Rational Function and Removable Discontinuity

Consider the function $$f(x)=\frac{x^2-4}{x-2}$$ for $$x \neq 2$$. Answer the following:

Rational Function Limits and Removable Discontinuities

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

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

Removing a Removable Discontinuity

Consider the function $$f(x)=\begin{cases} \frac{x^2 - 1}{x - 1} & x \neq 1 \\ 3 & x=1 \end{cases}$$

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

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 $$

Squeeze Theorem with Bounded Function

Suppose that for all x in some interval around 0, the function $$f(x)$$ satisfies $$-x^2 \le f(x) \l

Vertical Asymptote and End Behavior

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

Analyzing a Projectile's Motion

A projectile is launched vertically, and its height (in feet) at time $$t$$ seconds is given by $$s(

Analyzing Rate of Change in Economics

The cost function for producing $$x$$ units of product (in dollars) is given by $$C(x)= 0.5*x^2 - 8*

Approximating Derivatives Using Secant Lines

For the function $$f(x)=\ln(x)$$, we want to approximate the derivative at $$x=3$$ using secant line

Approximating the Tangent Slope

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

Car's Position and Velocity

A car’s position is modeled by \(s(t)=t^3 - 6*t^2 + 9*t\), where \(s\) is in meters and \(t\) is in

Derivative from Definition for a Piecewise Function

Consider the piecewise function $$ f(x)= \begin{cases} e^{x} & x < 0 \\ \ln(x+1) + 1 & x \ge

Derivative from First Principles

Let $$f(x)=\sqrt{x}$$. Using the limit definition of the derivative, answer the following:

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

Difference Quotient and Derivative of a Rational Function

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

Differentiability and Continuity

A function is defined piecewise as follows: $$f(x)= \begin{cases} x^2 & \text{if } x \le 1, \\ 2*x +

Finding the Derivative using the Limit Definition

Let $$h(x)= 5*x^2 + 3*x - 7$$. Use the limit definition of the derivative to determine $$h'(x)$$.

Graphical Analysis of a Function's Increasing/Decreasing Intervals

A graph of the function $$f(x)=x^3 - 3*x$$ is provided. Analyze the function based on its graph.

Graphical Analysis of Secant and Tangent Slopes

A function $$f(x)$$ is represented by the red curve in the graph below. Answer the following questio

Graphical Estimation of a Derivative

Consider the graph provided which plots the position $$s(t)$$ (in meters) of an object versus time $

Instantaneous Growth in a Population Model

In a laboratory experiment, the growth of a bacterial population is modeled by $$P(t)= e^{0.3*t}$$,

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 $$

Inverse Function Analysis: Exponential-Linear Function

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

Motion Analysis with Acceleration and Direction Change

A particle moves along a straight line with acceleration given by $$a(t)=12-4*t$$, where $$t$$ is in

Optimization in Revenue Models

A company's revenue function is given by $$R(x)= x*(50 - 2*x)$$, where $$x$$ represents the number o

Optimization of Production Cost

A manufacturer’s cost function is given by $$C(x)=x^3-15x^2+60x+200$$, where x represents the produc

Product and Quotient Rule Combination

Given $$u(x)=3*x^2+2$$ and $$v(x)=x-4$$, consider the function $$F(x)=\frac{u(x)*v(x)}{x+1}$$. Answe

Projectile Motion Analysis

A projectile is launched with its height (in meters) modeled by the function $$f(t)= -5*t^2 + 20*t +

Radioactive Decay Analysis

The amount of a radioactive substance is modeled by $$N(t)= 200*e^{-0.03*t}$$, where $$t$$ is in yea

Real-World Cooling Process

In an experiment, the temperature (in °C) of a substance as it cools is modeled by $$T(t)= 30*e^{-0.

Tangent Line to a Hyperbola

Consider the hyperbola defined by $$xy=20$$. Answer the following:

Using the Quotient Rule for a Rational Function

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

Water Tank Inflow-Outflow Analysis

A water tank receives water at a rate given by $$f(t)=3*t+2$$ (liters/min) and loses water at a rate

Advanced Composite Function Differentiation with Multiple Layers

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

Chain Rule Basics

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

Composite Function and Inverse Analysis via Graph

Consider the function $$f(x)= \sqrt{4*x-1}$$, defined for $$x \geq \frac{1}{4}$$. Analyze the functi

Composite Function Chain Reaction

A chemist models the concentration of a reacting solution at time $$t$$ (in seconds) with the compos

Composite Function in a Real-World Fuel Consumption Problem

A company models its fuel consumption with the function $$C(t)=\ln(5*t^2+7)$$, where $$t$$ represent

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 with Inverse Trigonometric Components

Let $$f(x)= \sin^{-1}\left(\frac{2*x}{1+x^2}\right)$$. This function involves an inverse trigonometr

Composite Inverse Trigonometric Function Evaluation

Let $$f(x)= \tan(2*x)$$, defined on a restricted domain where it is invertible. Analyze this functio

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

Concavity Analysis of an Implicit Curve

Consider the relation $$x^2+xy+y^2=7$$.

Differentiation of a Composite Inverse Trigonometric-Log Function

Let $$f(x)= \ln\left(\arctan(e^(x))\right)$$. Differentiate and evaluate as required:

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 Under Implicit Constraints in Physics

A particle moves along a path defined by the equation $$\sin(x*y)=x-y$$. This equation implicitly de

Implicit Curve Analysis: Horizontal Tangents

Consider the curve defined implicitly by $$x^2+ e^(y)= 5$$. Answer the following:

Implicit Differentiation in an Economic Demand-Supply Model

In an economic model, the relationship between supply (\(S\)) and demand (\(D\)) is given by the equ

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 Sine

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

Implicit Differentiation with Exponential-Trigonometric Functions

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

Implicit Differentiation with Logarithmic and Trigonometric Combination

Consider the equation $$\ln(x+y)+\cos(x*y)=0$$, where $$y$$ is an implicit function of $$x$$. Find $

Implicit Differentiation with Mixed Trigonometric and Polynomial Terms

Consider the equation $$x*\cos(y) + y^2 = x^2$$, which mixes trigonometric and polynomial expression

Implicit Trigonometric Equation Analysis

Consider the equation $$x \sin(y) + \cos(y) = x$$. Answer the following parts.

Inverse Analysis in Exponential Decay

A radioactive substance decays according to $$N(t)= N_0*e^(-0.5*t)$$, where N(t) is the quantity at

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 Differentiation

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

Inverse Function Differentiation

Let $$f(x)=x^3+x$$ and assume it is invertible. Answer the following:

Inverse Function Theorem in a Composite Setting

Let $$f(x)=x+\sin(x)$$ with inverse function $$g(x)$$.

Inverse Trigonometric Differentiation

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

Nested Composite Function Differentiation

Consider the function $$f(x)= \sqrt{\ln(3*x^2+2)}$$, where $$\sqrt{\ }$$ denotes the square root. So

Related Rates of a Shadow

A 2 m tall lamp post casts a shadow from a person who is 1.8 m tall. The person is moving away from

Second Derivative via Implicit Differentiation

Consider the curve defined by $$x^2+x*y+y^2=7$$. Answer the following parts.

Second Derivative via Implicit Differentiation

Given the ellipse $$\frac{x^2}{9}+\frac{y^2}{4}=1$$, find the second derivative $$\frac{d^2y}{dx^2}$

Tangent Lines on an Ellipse

Consider the ellipse given by $$\frac{x^2}{9} + \frac{y^2}{4} = 1$$. Use the graph provided to aid i

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 a Piecewise Function with Discontinuities

Consider the function $$f(x)=\begin{cases} \frac{x^2-4}{x-2} & \text{if } x<2 \\ x+1 & \text{if } x\

Analyzing Rate of Change in a Compound Interest Model

The amount in a bank account is modeled by $$A(t)= P e^{rt}$$, where $$P = 1000$$, r = 0.05 (per yea

Balloon Inflation Analysis

A spherical balloon inflates such that its volume increases at a constant rate of 10 cubic inches pe

Chemical Reaction Rate Analysis

The concentration of a reactant in a chemical reaction is modeled by $$C(t)=\frac{10}{1+e^{0.5t}}$$,

Cooling Coffee: Exponential Decay Model

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

Deceleration with Air Resistance

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

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

Estimating Function Change Using Differentials

Let $$f(x)=x^{1/3}$$. Use differentials to approximate the change in $$f(x)$$ when $$x$$ increases f

Evaluating an Indeterminate Limit using L'Hospital's Rule

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

Evaluating Indeterminate Limits via L'Hospital's Rule

Scientists are studying the limit of the function $$L(x)=\frac{5x^3-4x^2+1}{7x^3+2x-6}$$ as $$x \to

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

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

FRQ 9: Production Efficiency Analysis

A factory’s production efficiency is modeled by the relation $$L^2 + L*Q + Q^2 = 1500$$, where L rep

FRQ 10: Chemical Kinetics Analysis

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

Implicit Differentiation and Related Rates in Conic Sections

A point moves along the ellipse defined by $$\frac{x^2}{9} + \frac{y^2}{16} = 1$$. At a certain inst

Interpretation of the Derivative from Graph Data

The graph provided represents the position function $$s(t)$$ of a particle moving along a straight l

Kinematics on a Straight Line

A particle moves along a straight line with a position function $$s(t)=t^3 - 6*t^2 + 9*t + 2$$, wher

L'Hôpital's Rule in Analysis of Limits

Consider the limit $$L = \lim_{x\to \infty} \frac{5*x^3 - 4*x^2 + 1}{7*x^3 + 2*x - 6}$$. Use L'Hôpit

Linearization and Differentials Approximation

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

Linearization of a Machine Component's Length

A machine component's length is modeled by $$L(x)=x^4$$, where x is a machine setting in inches. Use

Maximizing Enclosed Area

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

Modeling a Bouncing Ball with a Geometric Sequence

A ball is dropped from a height of 10 m. Each time it bounces, it reaches 70% of the height of the p

Rate of Change in a Freefall Problem

An object is dropped from a height. Its height (in meters) after t seconds is modeled by $$h(t)= 100

Related Rates in Shadows: A Lamp and a Tree

A lamp post 5 m tall casts a shadow of a tree. At a certain moment, the tree is 3 m from the lamp an

Related Rates: Expanding Oil Spill

An oil spill on calm water forms a perfect circle. The area of the spill is increasing at a constant

Related Rates: Inflating Balloon

A spherical balloon is being inflated such that its volume increases at a rate of $$15\;cm^3/s$$. Th

Related Rates: Inflating Spherical Balloon

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

Train Motion Analysis

A train’s acceleration is given by $$a(t)=\sin(t)+0.5$$ (m/s²) for $$0 \le t \le \pi$$ seconds. The

Absolute Extrema for a Transcendental Function

Examine the function $$f(x)= e^{-x}*(x-2)$$ on the closed interval $$[0,3]$$ to determine its absolu

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

Analysis of an Exponential-Logarithmic Function

Consider the function $$f(x)= e^{x} - 3*\ln(x)$$ defined for $$x>0$$. Answer the following:

Chemical Mixing in a Tank

A 200-liter tank initially contains pure water. A salt solution with a concentration of 0.5 kg/L flo

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

Discontinuity and Derivative in a Hybrid Piecewise Function

Consider the function $$ f(x) = \begin{cases} x^2, & x \le 1, \\ 3x - 2, & x > 1. \end{cases} $$ A

Exploration of a Removable Discontinuity in a Rational Function

Consider the function $$ f(x) = \begin{cases} \frac{x^2 - 16}{x - 4}, & x \neq 4, \\ 7, & x = 4. \e

Extrema in a Cost Function

A company's cost function is given by $$C(x) = x^3 - 6*x^2 + 12*x + 7$$, where $$x$$ represents the

FRQ 1: Car's Motion and the Mean Value Theorem

A car’s position along a straight road is modeled by $$s(t) = t^3 - 6*t^2 + 9*t + 5$$ (in meters) fo

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

Graphical Analysis and Derivatives

A function \( f(x) \) is represented by the graph provided below. Answer the following based on the

Hydroelectric Dam Efficiency

A hydroelectric dam experiences water inflow and outflow that affect its efficiency. The inflow is g

Identification of Extrema and Critical Points

Let $$f(x)= x^3 - 6*x^2 + 9*x + 1$$ be defined on the interval $$[0,4]$$. Use your understanding of

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 Cubic Function

Consider the function $$f(x)=x^3+1$$. Answer all parts regarding its inverse below.

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 Quadratic Function (Restricted Domain)

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

Inverse Analysis of a Rational Function

Consider the function $$f(x)=\frac{2*x-1}{x+3}$$. Perform the following analysis regarding its inver

Inverse Analysis: Transformation Geometry of a Parabolic Function

Consider the function $$f(x)=4-(x-3)^2$$ with the domain $$x\le 3$$. Analyze its inverse function as

Modeling Disease Spread with an Exponential Model

In an epidemic, the number of infected individuals is modeled by $$I(t)= I_0 * e^{r*t}$$, where $$t$

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

Motion Analysis: A Runner's Performance

A runner’s distance (in meters) is recorded at several time intervals during a race. Analyze the run

Optimization in a Physical Context with the Mean Value Theorem

A car's velocity is modeled by $$v(t) = t^2 - 4*t + 5$$ (in m/s) for time $$t$$ in seconds on the in

Optimization of a Fenced Enclosure

A farmer wants to construct a rectangular garden using 120 meters of fencing along three sides, with

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

Related Rates in an Evaporating Reservoir

A reservoir’s volume decreases due to evaporation according to $$V(t)= V_0*e^{-a*t}$$, where $$t$$ i

Reservoir Sediment Accumulation

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

Temperature Change and the Mean Value Theorem

A temperature model for a day is given by $$T(t)= 2*t^2 - 3*t + 5$$, where $$t$$ is measured in hour

Volume of Solid by Cylindrical Shells

Consider the region bounded by $$f(x)= e^x$$ and $$g(x)= e^{2 - x}$$. Analyze the region and set up

Accumulated Bacteria Growth

A laboratory observes a bacterial colony whose rate of growth (in bacteria per hour) is modeled by t

Accumulated Chemical Concentration

A scientist observes that the rate of change of chemical concentration in a solution is given by $$r

Accumulated Rainfall Estimation

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

Accumulated Water Volume in a Tank

A water tank is being filled at a rate given by $$R(t) = 4*t$$ (in cubic meters per minute) for $$0

Accumulation and Inflection Points

Suppose a function's rate of change is given by $$f'(x)=3*x^2-12*x+9.$$ 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 Between Curves

An engineering design problem requires finding the area of the region enclosed by the curves $$y = x

Calculating Total Distance Traveled from a Changing Velocity Function

A particle moves with a velocity given by $$v(t)=t^2 - 4*t + 3$$ (in m/s) for $$0 \le t \le 5$$. Not

Chemical Accumulation in a Reactor

A chemical reactor has a net accumulation rate given by $$R(t)=5*\cos(t) + 2$$ (in kg/hour), where $

Definite Integral as an Accumulation Function

A generator consumes fuel at a rate given by $$f(t)=t*e^{-t}$$ (in liters per second). Answer the fo

Definite Integral Evaluation via U-Substitution

Consider the integral $$\int_{2}^{6} 3*(x-2)^4\,dx$$ which arises in a physical experiment. Evaluate

Estimating River Flow Volume

A river's flow rate (in cubic meters per second) has been measured at various times during an 8-hour

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

Evaluating Total Rainfall Using Integral Approximations

During a storm, the rainfall rate (in inches per hour) was recorded at several times. The table belo

Exact Area Under a Parabolic Curve

Find the exact area under the curve $$f(x)=3*x^2 - x + 4$$ between $$x=1$$ and $$x=4$$.

FRQ7: Inverse Analysis of an Exponential Accumulation Function

Define the function $$ Q(x)=\int_{1}^{x} \left(\ln(t)+\frac{1}{t}\right)\,dt $$ for x > 1. Answer th

FRQ15: Inverse Analysis of a Quadratic Accumulation Function

Consider the function $$ Q(x)=\int_{0}^{x} (4*t+1)\,dt $$. Answer the following parts.

Integration of a Piecewise Function

A function $$f(x)$$ is defined piecewise as follows: $$f(x)= \begin{cases} x^2, & x < 3, \\ 2*x+1,

Medication Infusion in Bloodstream

A patient receives medication through an IV at a rate $$I(t)= 5\sqrt{t+1}$$ mg/min, while the body m

Modeling Savings with a Geometric Sequence

A person makes annual deposits into a savings account such that the first deposit is $100 and each s

Modeling Water Volume in a Tank via Integration

A tank is being filled with water at a rate given by $$R(t)= \frac{50}{t+2}$$ cubic meters per minut

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

Net Change in Population Growth

A town's population grows at a rate given by $$P'(t)=0.1*t+50$$ (in individuals per year) where $$t$

Net Change in Salaries: An Accumulation Function

A company models its annual bonus savings with the rate function $$B'(t)= 500*(1+\sin(t))$$ dollars

Particle Motion on a Road with Varying Speed

A particle moves along a straight road with velocity $$v(t)=4-0.5*t^2$$ (in m/s) for $$0\le t\le6$$,

Population Growth in a Bacterial Culture

A bacterial culture grows at a rate given by $$r(t)=0.5*e^{-0.3*t}$$ (in thousands of bacteria per h

Analyzing Direction Fields for $$dy/dx = y-1$$

Consider the differential equation $$dy/dx = y - 1$$. A slope field for this equation is provided. A

Analyzing Slope Fields for $$dy/dx=x\sin(y)$$

Consider the differential equation $$\frac{dy}{dx}=x\sin(y)$$. A corresponding slope field is provid

Area Under a Differential Equation Curve

Consider the differential equation $$\frac{dy}{dx} = y(1-y)$$, with a particular solution given by $

Bernoulli Differential Equation Challenge

Consider the nonlinear differential equation $$\frac{dy}{dt} - y = -y^3$$ with the initial condition

Boat Crossing a River with Current

A boat is attempting to cross a river flowing at a constant speed of $$2$$ m/s. The boat is directed

Chemical Reactor Mixing

In a continuously stirred chemical reactor, a reactant is added at a rate that results in an inflow

Cooling with a Time-Dependent Coefficient

A substance cools according to $$\frac{dT}{dt} = -k(t)(T-25)$$ where the cooling coefficient is give

Drug Elimination with Infusion

A drug is administered continuously to a patient. Its blood concentration $$C(t)$$ (in mg/L) satisfi

Economic Decay Model

An asset depreciates in value according to the model $$\frac{dC}{dt}=-rC$$, where $$C$$ is the asset

Economic Growth with Investment Outflow

A company’s investment fund grows continuously at an annual rate of $$5\%$$, but expenses lead to a

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 (Simplified Logistic Model)

In a simplified model of an epidemic, the number of infected individuals $$I(t)$$ (in thousands) is

Epidemic Spread Modeling

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

Exact Differential Equation

Consider the differential equation $$ (2xy+y^2)\,dx+(x^2+2xy)\,dy=0$$. Answer the following question

Exponential Growth and Doubling Time

A bacterial culture grows according to the differential equation $$\frac{dy}{dt} = k * y$$ where $$y

Fishery Harvesting Model

The fish population in a lake is modeled by the differential equation $$\frac{dP}{dt} = 0.8P\left(1-

Heating and Cooling in an Electrical Component

An electronic component experiences heating and cooling according to the differential equation $$\fr

Integrating Factor Initial Value Problem

Solve the initial value problem $$\frac{dy}{dx}+\frac{2}{x}y=x^2$$ for $$x>0$$ with $$y(1)=3$$.

Linear Differential Equation and Integrating Factor

Consider the differential equation $$\frac{dy}{dx} = y - x$$. Use the method of integrating factor t

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 Population Growth

A population of organisms is modeled by the logistic differential equation $$\frac{dP}{dt}=rP\left(1

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 Problem with Constant Flow

A tank initially contains 200 liters of water with 10 kg of dissolved salt. Brine with a salt concen

Mixing Problem with Time-Dependent Inflow Concentration

A tank initially contains 100 liters of water with 8 kg of dissolved salt. Brine enters the tank at

Mixing Problem with Time-Dependent Inflow Rate

A tank initially holds 200 L of water with 10 kg of salt. Brine containing 0.2 kg/L of salt flows in

Motion Under Gravity with Air Resistance

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

Population Growth in a Bacterial Culture

A bacterial culture has its population measured (in thousands) at various times (in hours). The tabl

Population Growth with Harvesting

A fish population in a lake grows at a rate proportional to its current size, but fishermen harvest

Radioactive Decay

A radioactive substance is measured over time. The activity $$A$$ (in grams) is recorded at several

Radioactive Decay Differential Equation

A radioactive substance decays according to the differential equation $$\frac{dM}{dt}=-k*M$$. If the

Radioactive Decay Model

A radioactive substance decays according to the differential equation $$\frac{dN}{dt} = -kN$$. At ti

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 $$A$$

Reaction Rate Model: Second-Order Decay

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

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: 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: $$dy/dx = x*y$$

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

Sketching Solution Curves on a Slope Field

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

Slope Field and Solution Curve Analysis

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

Tank Mixing with Salt

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

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 Cost Functions in a Business Analysis

A company analyzes its cost structure using two functions: the fixed-plus-variable cost function $$C

Area Between Curves: Complex Polynomial vs. Quadratic

Consider the functions $$f(x)= x^3 - 6*x^2 + 9*x+1$$ and $$g(x)= x^2 - 4*x+5$$. These curves interse

Area Between Transcendental Functions

Consider the curves $$f(x)=\cos(x)$$ and $$g(x)=\sin(x)$$ on the interval $$[0,\frac{\pi}{4}]$$.

Average Concentration in a Chemical Reaction

A chemical reaction in a laboratory setting is monitored by recording the concentration (in moles pe

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 meteorologist recorded the temperature (in $$^\circ C$$) over a 24-hour period at different times.

Average Temperature Analysis

A research facility recorded the temperature in a greenhouse over a period of 5 hours. The temperatu

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 vs. Instantaneous Value of a Function

Consider the function $$f(x)= \sin(x) + 1$$ on the interval $$[0, \pi]$$.

Boat Navigation Across a River with Current

A boat aims to cross a river that is 100 m wide. The boat moves due north at a constant speed of 5 m

Combining Position, Area, and Average Concepts in River Navigation

A boat navigates a river where its speed relative to the water is given by $$v(t)=4+\sin(t)$$ km/h.

Distance Traveled Analysis from a Velocity Graph

An object’s motion is recorded, and its velocity is shown on the graph provided for $$t \in [0,10]$$

Estimating Instantaneous Velocity from Position Data

A car's position along a straight road is recorded over a 10-second interval as shown in the table b

Particle Motion Along a Straight Line

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

Piecewise Function Analysis

Consider a piecewise function defined by: $$ f(x)=\begin{cases} 3 & \text{for } 0 \le x < 2, \\ -x+5

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

Reconstructing Position from Acceleration Data

A particle traveling along a straight line has its acceleration given by the values in the table bel

Resource Consumption in an Ecosystem

The rate of consumption of a resource in an ecosystem is given by $$C(t)=50*\ln(1+t)$$ (in units per

Retirement Savings Auto-Increase

A person contributes to a retirement fund such that the monthly contributions form an arithmetic seq

Shaded Area between $$y=\sqrt{x}$$ and $$y=\frac{x}{2}$$

Consider the curves $$f(x)=\sqrt{x}$$ and $$g(x)=\frac{x}{2}$$. Use integration to determine the are

Stress Analysis in a Structural Beam

A beam in a building experiences a stress distribution along its length given by $$\sigma(x)=100-15*

Surface Area of a Rotated Curve

Consider the function $$f(x)=\sqrt{x}$$ on the interval $$[0,4]$$. This curve is rotated about the $

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

Total Distance from a Runner's Variable Velocity

A runner’s velocity (in m/s) is modeled by the function $$v(t)=t^2-10*t+16$$ for $$0 \le t \le 10$$

Volume by Rotation using the Disc Method

Consider the region bounded by $$y=\sqrt{x}$$, the $$x$$-axis, and the vertical lines $$x=0$$ and $$

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 Solid of Revolution: Disc Method

Consider the region R bounded by $$y= \sqrt{x}$$, the x-axis, and the vertical line $$x=4$$. When R

Water Pumping from a Parabolic Tank

A water tank has ends shaped by the region bounded by $$y=x^2$$ and $$y=4$$, and the tank has a unif

Water Tank Filling with Graduated Inflow

A water tank is filled daily by adding a certain amount of water that increases by a fixed amount ea

Work in Pumping Water

A water pump is used to empty a reservoir. The force required to pump water out at a depth $$y$$ (in

Work in Pumping Water from a Conical Tank

A water tank is in the shape of an inverted right circular cone with height $$10\,m$$ and top radius

Work in Spring Stretching

A spring obeys Hooke's law, where the force required to stretch the spring a distance $$x$$ from its