AP Calculus AB FRQ Room

Algebraic Manipulation and Limit Evaluation

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

Analysis of a Rational Function with Exponential and Logarithmic Components

Consider the function $$g(x)=\frac{e^{x}-1-\ln(1+x)}{x}$$ for $$x \neq 0$$. Evaluate the limit as $$

Analyzing Limit of an Oscillatory Velocity Function

A particle moves along a line with velocity given by $$v(t)= t*\cos\left(\frac{\pi}{t}\right)$$ for

Analyzing Process Data for Continuity

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

Application of the Intermediate Value Theorem

Let the function $$f(x)= x^3 - 4*x - 1$$ be continuous on the interval $$[0, 3]$$. Answer the follow

Arithmetic Sequence in Temperature Data and Continuity Correction

A temperature sensor records the temperature every minute and the readings follow an arithmetic sequ

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 Limit Comparison for Two Particle Paths

Two particles, A and B, travel along the same line. Their position functions are given by $$s_A(t)=

Continuity of a Composite Function

Let $$g(x) = \sqrt{x+3}$$ and $$h(x) = x^2 - 4$$. Define the composite function $$f(x) = g(h(x))$$.

Continuity of a Sine-over-x Function

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

Continuous Extension and Removable Discontinuity

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

Evaluating a Compound Limit Involving Rational and Trigonometric Functions

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

Exponential Function Limits

Consider the function $$f(x) = \frac{e^x - 1}{x}$$ for $$x \neq 0$$, with the definition $$f(0) = 1$

Graph Analysis of Discontinuities

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

Graphical Analysis of Function Behavior from a Table

A real-world experiment recorded the concentration (in M) of a solution over time (in seconds) as sh

Horizontal Asymptote of a Rational Function

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

Inverse Function and Limit Behavior Analysis

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

Investigating Discontinuities in a Rational Function

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

Investigation of Continuity in a Piecewise Log-Exponential Function

A function is defined by $$ f(x)=\begin{cases} \frac{\ln(e^{2*x}+3)-\ln(5)}{x-1} & x \neq 1, \\ D &

Jump Discontinuity in a Piecewise Function

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

Limits of a Composite Particle Motion Function

A particle moves along a line with velocity function $$v(t)= \frac{\sqrt{t+5}-\sqrt{5}}{t}$$ for $$t

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

Mixed Function with Jump Discontinuity at Zero

Consider the function $$f(x)=\begin{cases} 1+x & x<0\\ 2 & x=0\\ \frac{\sin(x)}{x}+1 & x>0 \end{case

One-Sided Limits and an Absolute Value Function

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

One-Sided Limits and Vertical Asymptotes

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

Piecewise Function Continuity at a Junction

Consider the function defined by: For $$x < 0$$: $$f(x) = 2^x + 1$$. For $$x \ge 0$$: $$f(x) = 1 -

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 Application

Consider the function $$f(x)=x^2\sin(\frac{1}{x})$$ for $$x\neq0$$ and $$f(0)=0$$. Answer the follow

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 for an Oscillatory Function

Define the function $$f(x)= x \cos(\frac{1}{x})$$ for x ≠ 0, and let f(0)= 0.

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 Bounded Function

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

Squeeze Theorem with Trigonometric Function

Consider the function \(h(x)=x^2\cos(1/x)\) for \(x\neq0\) with \(h(0)=0\). Answer the following:

Table Analysis for Estimating a Limit

The table below shows values of the function $$g(x)$$ for x near 4. Use this data to answer the foll

Trigonometric Limits

Consider the functions $$g(x)=\frac{\sin(3*x)}{\sin(2*x)}$$ and $$h(x)=\frac{1-\cos(4*x)}{x^2}$$. An

Analyzing Rates Without a Calculator: Average vs Instantaneous Rates

Consider the function $$f(x)= x^2$$.

Cost Function Analysis: Average and Instantaneous Rates

A company’s cost function is given by $$C(x)=0.5*x^2+10$$, where $$x$$ is the number of items produc

Curve Analysis – Increasing and Decreasing Intervals

Given the function $$f(x)= x^3 - 3*x^2 + 2$$, analyze its behavior.

Derivative from the Limit Definition

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

Differentiability and Continuity

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

Economic Cost Function Analysis

A company’s production cost is modeled by $$C(x)= 0.02*x^3 - 0.5*x^2 + 4*x + 100$$, where $$x$$ repr

Economic Model: Revenue and Rate of Change

The revenue for a product is given by $$R(x)= \frac{x^2 + 10*x}{x+2}$$, where $$x$$ is in hundreds o

Exponential Rate of Change

A population growth model is given by $$P(t)=e^{2*t}$$, where $$t$$ is in years.

Finding Derivatives of Composite Functions

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

Finding the Derivative Using First Principles

Consider the function $$f(x)= 5*x^3 - 4*x + 7$$. Use the definition of the derivative to find the de

Graph vs. Derivative Graph

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

Higher Order Derivatives and Concavity

Let \(f(x)=x^3 - 3*x^2 + 5*x - 2\). Answer the following parts.

Instantaneous and Average Velocity

A particle's position is given by $$s(t)=t^3-6*t^2+9*t+2$$, where $$s(t)$$ is in meters and $$t$$ is

Inverse Function Analysis: Restricted Rational Function

Consider the function $$f(x)=\frac{2*x}{x^2+1}$$ defined on the interval $$0\leq x\leq 1$$.

Investigating the Derivative of a Piecewise Function

The function $$f(x)$$ is defined piecewise by $$f(x)=\begin{cases} x^2 & \text{if } x \le 1, \\ 2*x

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

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

Piecewise Function and Discontinuities

A piecewise function $$f$$ is defined by $$ f(x)=\begin{cases} \frac{x^2-4}{x-2} & \text{if } x\ne

Population Growth and Instantaneous Rate of Change

A town's population is modeled by $$P(t)= 2000*e^{0.05*t}$$, where $$t$$ is in years. Analyze the ch

Product and Chain Rule Combined

Let \(f(x)=(3*x+1)^2 * \cos(x)\). Answer the following parts.

Rainfall-Runoff Model

A reservoir receives water from rainfall at a rate modeled by $$R_{in}(t)=10*\sin\left(\frac{\pi*t}{

Rate of Water Flow in a Rational Function Model

The water flow from a reservoir is modeled by $$F(t)= \frac{3*t}{t+2}$$, where $$t$$ is time in hour

Rates of Change from Experimental Data

A chemical experiment yielded the following measurements of a substance's concentration (in molarity

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

Related Rates: Conical Tank Draining

A conical water tank drains so that its volume is given by $$V=\frac{1}{3}\pi r^2h$$. The radius r o

Tangent Line Equation for an Exponential Function

Consider the function $$f(x)= e^{x}$$ and its graph.

Tangent Line to a Cubic Function

The function $$f(x) = x^3 - 6x^2 + 9x + 1$$ models the height (in meters) of a roller coaster at pos

Temperature Change Analysis

A weather station models the temperature (in °C) with the function $$T(t)=15+2*t-0.5*t^2$$, where $$

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 an Economic Model

In a manufacturing process, the cost function is given by $$C(q)= (\ln(1+3*q))^2$$, where $$q$$ is t

Chain Rule in an Economic Model

In an economic model, the cost function for producing a good is given by $$C(x)=(3*x+1)^5$$, where $

Chain Rule in Temperature Variation

A metal rod's temperature along its length is given by the function $$T(x)= \cos((4*x+2)^2)$$, where

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 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 Kinematics

A particle moves along a straight line with its position given by $$s(t) = (2*t+3)^4$$. Analyze the

Composite Function with Logarithm and Trigonometry

Let $$h(x)=\ln(\sin(2*x))$$.

Composite Inverse Trigonometric Function Evaluation

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

Composite Temperature Model

Consider a temperature function given by $$T(t) = \sin(t^3 - 2*t)$$, where t is measured in seconds.

Economic Equilibrium: Composite and Inverse Functions

In an economic model, the demand function is given by the composite function $$D(p)= f(g(p))$$, wher

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 and Inverse Function Differentiation Combined

Suppose that $$x$$ and $$y$$ are related by the equation $$x^2+y^2-\sin(x*y)=4$$. Answer the followi

Implicit Differentiation and Rate Change in Biology

In an ecosystem, the relationship between two population parameters is given by $$e^y+ x*y= 10$$, wh

Implicit Differentiation in a Trigonometric Context

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

Consider the equation $$\ln(x)+\ln(y)=1$$. Answer the following:

Implicit Differentiation Involving a Logarithm

Consider the equation $$x*\ln(y) + y^2 = x^2$$. Answer the following parts.

Implicit Differentiation Involving a Product

Consider the equation $$x^2*y + \sin(y) = x*y^2$$ which relates the variables $$x$$ and $$y$$ in a n

Implicit Differentiation of a Logarithmic Equation

Given the equation $$\ln(x) + \ln(y) = \ln(10)$$, answer the following parts.

Implicit Differentiation of a Trigonometric Composite Function

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

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 Differentiation with Product Rule

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

Implicit Differentiation with Trigonometric and Logarithmic Terms

Consider the equation $$\sin(x) + \ln(y) + x*y = 0.$$ Solve the following:

Implicitly Defined Inverse Relation

Consider the relation $$y + \ln(y)= x.$$ Answer the following:

Inverse Derivative of a Sum of Exponentials and Linear Terms

Let $$f(x)= e^(x)+ x$$ and let g be its inverse function satisfying $$g(f(x))= x$$. Answer the follo

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 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

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

Inverse Function Differentiation Combined with Chain Rule

Let $$f(x)=\sqrt{x-1}+x^2$$, and assume that it is one-to-one on its domain, with an inverse functio

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 an Exponential Model

Let $$f(x) = e^{2*x} + x$$, and let g be its inverse function. Answer the following parts.

Inverse Trigonometric Function Differentiation

Let $$f(x)=\arcsin\left(\frac{2*x}{5}\right)$$, with the understanding that $$\left|\frac{2*x}{5}\ri

Multilayer Composite Function Differentiation

Let $$y=\cos(\sqrt{5*x+3})$$. Answer 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

Projectile Motion and Composite Function Analysis

A projectile is launched and its height $$h(t)$$ (in meters) is recorded at various times t (in seco

Related Rates via Chain Rule

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

Analysis of Experimental Data

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

Analyzing Speed Changes in a Particle’s Motion

A particle moves along a straight line with a velocity function given by $$v(t) = (t-2)^2(t+1)$$ for

Balloon Inflation Related Rates

A spherical balloon is being inflated, and its volume is increasing at a constant rate of $$12$$ cub

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

Cooling Hot Beverage

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

Cost Analysis through a Rational Function

A company's average cost function is given by $$C(x)= \frac{2*x^3 + 5*x^2 - 20*x + 40}{x}$$, where $

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

Determining the Tangent Line

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

Dynamics of a Car: Stopping Distance and Deceleration

A car traveling at 30 m/s begins to decelerate at a constant rate. Its velocity is modeled by $$v(t)

Error Estimation in Pendulum Period

The period of a simple pendulum is given by $$T=2\pi\sqrt{\frac{L}{g}}$$, where $$L$$ is the length

Estimating Function Change Using Differentials

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

Estimating Instantaneous Rates from Discrete Data

In a laboratory experiment, the concentration of a chemical (in molarity, M) is recorded over time (

Expanding Circular Ripple in a Pond

A circular ripple in a pond has its area increasing at a constant rate of 10 square meters per secon

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

Filling a Conical Tank: Related Rates

Water is being pumped into an inverted conical tank at a rate of $$\frac{dV}{dt}=3\;m^3/min$$. The t

FRQ 5: Coffee Cooling Experiment

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

FRQ 14: Optimizing Box Design with Fixed Volume

A manufacturer wants to design an open-top box with a fixed volume of $$V = x^2*y = 32$$ cubic units

FRQ 18: Chemical Reaction Concentration Changes

During a chemical reaction, the concentrations of reactants A and B are related by $$[A]^2 + 3*[A]*[

FRQ 20: Market Demand Analysis

In an economic market, the demand D (in thousands of units) and the price P (in dollars) satisfy the

Function with Vertical Asymptote

Consider the function $$f(x)=\frac{1}{x-3}+2$$. Analyze its behavior.

Graphing a Function via its Derivative

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

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

Inflating Balloon Rates

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

Inflation of a Balloon: Surface Area Rate of Change

A spherical balloon is being inflated so that its volume increases at a rate of $$\frac{dV}{dt}=50$$

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

Optimizing Road Construction Costs

An engineer is designing a road that connects a point on a highway to a town located off the highway

Particle Acceleration and Direction of Motion

A particle moves along a straight line with a velocity function given by $$v(t)=3*t^2-12*t+9$$, wher

Population Change Rate

The population of a town is modeled by $$P(t)= 50*e^{0.3*t}$$, where $$t$$ is in years and $$P(t)$$

Population Growth Rate Analysis

A town's population is modeled by the exponential function $$P(t) = 500e^{0.03t}$$, where $$t$$ is i

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 in a Spherical Balloon

A spherical balloon is being inflated, and its volume $$V$$ (in cubic inches) is related to its radi

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

Savings Account Growth Modeled by a Geometric Sequence

A savings account has an initial balance of $$B_0=1000$$ dollars. The account earns compound interes

Seasonal Water Reservoir

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

Tangent Line and Linearization Approximation

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

Temperature Rate Change in Cooling Coffee

A cup of coffee cools following the model $$x(t)=70+50e^{-0.1t}$$, where x is in degrees Fahrenheit

Analysis of a Trigonometric Piecewise Function

Consider the function $$ f(x) = \begin{cases} \frac{\sin(x)}{x}, & x \neq 0, \\ 2, & x = 0. \end{ca

Analyzing a Piecewise Function and Differentiability

Let $$f(x)$$ be defined piecewise by $$f(x)= x^2$$ for $$x \le 2$$ and $$f(x) = 4*x - 4$$ for $$x >

Analyzing Critical Points in a Piecewise Function

The function \( f(x) \) is defined piecewise by \( f(x)= \begin{cases} x^2, & x \le 2, \\

Application of Rolle's Theorem

Let $$f(x)$$ be a function that is continuous on $$[0,5]$$ and differentiable on $$(0,5)$$ with $$f(

Applying the Mean Value Theorem and Analyzing Discontinuities

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

Approximating Displacement from Velocity Data

A vehicle's velocity (in $$m/s$$) over time (in seconds) was recorded during a test run. The table b

Behavior Analysis of a Logarithmic Function

Consider the function $$f(x)= \frac{\ln(x)}{x}$$ for $$x>0$$. Analyze the critical points and concav

Chemical Reactor Temperature Optimization

In a chemical reactor, the temperature is controlled by the rate of coolant inflow. The coolant infl

Composite Function with Piecewise Exponential and Logarithmic Parts

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

Cost Minimization in Transportation

A transportation company recorded shipping costs (in thousands of dollars) for different numbers of

Derivative of the Natural Log Function by Definition

Let $$f(x)= \ln(x)$$. Use the definition of the derivative to prove that $$f'(a)= \frac{1}{a}$$ for

Discontinuity in a Rational Function Involving Square Roots

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

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

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 12: Optimization in Manufacturing: Minimizing Cost

A company’s cost function is given by $$C(x)= 0.5*x^2 - 10*x + 125$$ (in dollars), where $$x$$ repre

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

Hydroelectric Dam Efficiency

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

Inverse Analysis: Logarithmic Ratio Function in Financial Context

Consider the function $$f(x)=\ln\left(\frac{x+4}{x+1}\right)$$ with domain $$x > -1$$. This function

Liquid Cooling System Flow Analysis

A specialized liquid cooling system operates with non-linear flow rates. The inflow rate is given by

Logarithmic Transformation of Data

A scientist models an exponential relationship between variables by the equation $$y= A*e^{k*x}$$. T

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 via Derivatives

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

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 of a Fenced Enclosure

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

Optimizing a Box with a Square Base

A company is designing an open-top box with a square base of side length $$x$$ and height $$h$$. The

Sand Pile Dynamics

A sand pile is being formed on a surface where sand is both added and selectively removed. The inflo

Temperature Analysis Over a Day

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

Temperature Regulation in a Greenhouse

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

Volume of Solid with Square Cross-Sections

Consider the region between $$f(x)= \sin(x)$$ and the x-axis on the interval $$[0, \pi]$$. A solid i

Water Cooling Tower Efficiency

In a water cooling tower, water is pumped in at a rate $$R_{in}(t)=10+0.5*t^2$$ L/min and discharged

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:

Antiderivative of a Transcendental Function

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

Area Under a Curve Using Riemann Sums

A function $$f(x)$$ is defined over the interval $$[1,7]$$ and its values are provided in the table

Average Temperature Calculation over 12 Hours

In a city, the temperature over a 12-hour period is modeled by $$T(t) = -2*t + 20$$ (in $$^\circ C$$

Car Fuel Consumption Analysis

A car engine’s fuel dynamics are modeled such that fuel is consumed at a rate of $$f(t)=0.1t^2$$ L/m

Definite Integral and the Fundamental Theorem of Calculus

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

Electric Charge Accumulation

An electrical circuit records the current (in amperes) at various times during a brief experiment. U

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

Estimating Accumulated Water Inflow Using Riemann Sums

A water tank fills at varying rates. The table below shows the inflow rate in liters per second at d

Estimating Displacement with a Midpoint Riemann Sum

A vehicle’s velocity is modeled by the function $$v(t) = -t^{2} + 4*t$$ (in meters per second) over

Estimating River Flow Volume

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

Experimental Data Analysis using Trapezoidal Sums

A chemical reaction is monitored over time, and the reaction rate $$f(t)$$ (in moles per minute) is

FRQ16: Inverse Analysis of an Integral Function via U-Substitution

Let $$ U(x)=\int_{0}^{x} 2*(t-3)^2\,dt $$ for x ≥ 3. Answer the following parts.

FRQ17: Inverse Analysis of a Biologically Modeled Accumulation Function

In a biological study, the net concentration of a chemical is modeled by $$ B(t)=\int_{0}^{t} (0.6*t

Function Transformations and Their Integrals

Let $$f(x)= 2*x + 3$$ and consider the transformed function defined as $$g(x)= f(2*x - 1)$$. Analyze

Growth of Investment with Regular Contributions and Withdrawals

An investment account receives contributions at a rate of $$C(t)= 100e^{0.05t}$$ dollars per year an

Medication Concentration and Absorption Rate

A patient's blood concentration of a drug (in mg/L) is monitored over time before reaching its peak.

Mixed Method Approximation of an Integral

A function $$f(t)$$ that represents a biological rate is recorded over time. Use the table below to

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

Piecewise-Defined Function and Discontinuities

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

Population Growth: Accumulation through Integration

A certain population grows at a rate modeled by $$R(t)= 0.5*t^2 - 3*t + 10$$ (individuals per year),

Rate of Drug Metabolism

Researchers recorded the rate at which a drug is metabolized (in mg/hr) at several time intervals. U

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 Room

The rate of change of the temperature in a room is given by $$\frac{dT}{dt}=0.5*t+1$$, where $$T$$ i

Trapezoidal Rule in Estimating Accumulated Change

A rising balloon has its height measured at various times. A portion of the recorded data is given i

Volume of a Solid of Revolution Using the Disk/Washer Method

Consider the region in the first quadrant bounded by the curve $$y=x^2$$ and the horizontal line $$y

Bacterial Growth with Constant Removal

A bacterial colony (in thousands) grows according to the differential equation $$\frac{dP}{dt}=0.4P-

Bacterial Nutrient Depletion

A nutrient in a bacterial culture is depleting over time according to the differential equation $$\f

Bernoulli Differential Equation

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

Charging a Capacitor in an RC Circuit

In an RC circuit, the charge $$Q$$ on a capacitor satisfies the differential equation $$\frac{dQ}{dt

Chemical Reaction Rate

The concentration $$y$$ (in moles per liter) of a reactant in a chemical reaction is modeled by the

Comparative Population Decline

A population declines according to two models. Model 1 follows simple exponential decay: $$\frac{dN}

Cooling of a Hot Beverage

According to Newton's Law of Cooling, the temperature $$T(t)$$ of a hot beverage satisfies $$\frac{d

Cooling with Variable Ambient Temperature

An object cools in an environment where the ambient temperature varies with time. Its temperature $$

Differential Equation with Substitution using u = y/x

Consider the differential equation $$\frac{dy}{dx}= \frac{y}{x}+\sqrt{\frac{y}{x}}$$. Use the substi

Drug Infusion and Elimination

The concentration of a drug in a patient's bloodstream is modeled by the differential equation $$\fr

Environmental Pollution Model

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

Heating a Liquid in a Tank

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

Heating and Cooling in an Electrical Component

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

Linear Differential Equation using Integrating Factor

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

Mixing in a Tank

A tank initially contains $$200$$ liters of water with $$10$$ kg of dissolved salt. Brine containing

Mixing Tank Problem

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

Newton's Law of Cooling with Temperature Data

A thermometer records the temperature of an object cooling in a room. The object's temperature $$T(t

Particle Motion with Variable Acceleration

A particle moves along a straight line with acceleration $$a(t)=3-2*t$$ (in m/s²). Its initial veloc

Pollutant Concentration in a Reservoir

An urban water reservoir contains 100,000 L of water and initially 2000 kg of pollutant. Polluted wa

Qualitative Analysis of a Nonlinear Differential Equation

Consider the differential equation $$\frac{dy}{dx}=1-y^2$$.

Radioactive Decay and Half-Life

A radioactive substance decays according to the differential equation $$\frac{dN}{dt} = -\lambda * N

Reaction Rate Model: Second-Order Decay

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

Separable Differential Equation with Initial Condition

Consider the differential equation $$\frac{dy}{dx}= \frac{x^2}{2*y}$$ with the initial condition $$y

Separable Differential Equation with Trigonometric Factor

Consider the differential equation $$\frac{dy}{dx}=(2y+3)\cos(x)$$. Answer each part using separatio

Separable Differential Equation: y and x

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

Separable Equation with Trigonometric Functions

Solve the differential equation $$\frac{dy}{dx} = \frac{\tan(x)}{1+y^2}$$ given that $$y(0)=0$$.

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

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

Substitution to Linearize

The differential equation $$\frac{dy}{dx} = \frac{x + y}{1 - x*y}$$ appears non-linear. With the sub

Volumes from Cross Sections of a Bounded Region

The solution to a differential equation is given by $$y(x) = \ln(1+x)$$. This curve, combined with t

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

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

Area Between Cost Functions in a Business Analysis

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

Average Speed from a Velocity Function

A car’s velocity is given by $$v(t)=t^2-4*t+5$$ (in m/s) for $$0 \le t \le 5$$. Assume that $$v(t)$$

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 weather scientist models the temperature during a day by the function $$f(t)=5+2*t-0.1*t^2$$ where

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 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

Cost Analysis: Area Between Quadratic Cost Functions

Two cost functions for production are given by $$C_1(x)=0.5*x^2+3*x+10$$ and $$C_2(x)=0.3*x^2+4*x+5$

Cost Optimization for a Cylindrical Container

A manufacturer wishes to design a closed cylindrical container with a fixed volume $$V_0$$. The cost

Download Speeds Improvement

An internet service provider increases its download speeds as part of a new promotional plan such th

Electrical Charge Calculation

The current in an electrical circuit is given by $$I(t)=6*e^{-0.5*t} - 3*e^{-t}$$ (in amperes) for $

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

Hiking Trail: Position from Velocity

A hiker's velocity is given by $$v(t)=3\cos(t/2)+1$$ (in km/h) for 0 ≤ t ≤ 2π. Assuming the hiker st

Modeling Bacterial Growth

A bacterial culture grows at a rate modeled by $$g(t)=a*e^{0.3*t}$$, where $$t$$ is time in hours an

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

Rebounding Ball

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

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

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 Cylindrical Shells

Consider the region bounded by $$y=x$$, $$y=4$$, and $$x=0$$. This region is revolved about the $$y$

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 bounded by $$y=x$$ and $$y=x^2$$ for $$0 \le x \le 1$$. Every cro

Volume of Solid of Revolution: Bottle Design

A region under the curve $$f(x)=\sqrt{x}$$ on the interval 0 ≤ x ≤ 9 is rotated about the x-axis to

Water Reservoir Inflow‐Outflow Analysis

A water reservoir receives water through an inflow pipe and loses water through an outflow valve. Th