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 a Removable Discontinuity in a Log-Exponential Function

Consider the function $$p(x)= \frac{e^{x}-e}{\ln(x)-\ln(1)}$$ for $$x \neq 1$$. The function is unde

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 a Discontinuous Function with a Sequence Component

The function is given by $$f(x) = \frac{\sin(\pi x)}{\pi (x - 1)}$$ for $$x \neq 1$$ (with f(1) unde

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 Squeeze Theorem in Trigonometric Limits

Consider the function $$f(x) = x^2 * \sin(1/x)$$ for $$x \neq 0$$ with $$f(0)=0$$. Answer the follow

Area and Volume Setup with Bounded Regions

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

Continuity Analysis 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 at Zero for a Trigonometric Function

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

Determining Parameters for Continuity

Consider the function $$f(x)= \begin{cases} 2*x + k, & x < 1 \\ x^2, & x \geq 1 \end{cases}$$, where

Discontinuities in a Rational-Exponential Function

Consider the function $$ f(x) = \begin{cases} \frac{e^{x} - 1}{x}, & x \neq 0 \\ k, & x = 0. \en

Epsilon-Delta Analysis of a Limit

Consider the linear function $$f(x) = 3*x + 1$$. For $$\epsilon = 0.5$$, answer the following:

Error Analysis in Limit Calculation

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

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 a Limit with Radical Expressions

Evaluate the limit $$\lim_{x \to 9} \frac{\sqrt{x} - 3}{x - 9}$$. Answer the following:

Graph Analysis of Discontinuities

Examine the provided graph of a function f(x) that displays both a removable discontinuity and a jum

Intermediate Value Theorem in Context

Let $$f(x) = x^3 - 6x^2 + 9x + 2$$, which is continuous on the interval [0, 4]. Answer the following

Intermediate Value Theorem in Temperature Modeling

A continuous function $$ f(x) $$ describes the temperature (in °C) throughout a day, with $$f(0)=15$

Jump Discontinuity Analysis

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

Limit Evaluation in a Parametric Particle Motion Context

A particle’s position in the plane is given by the parametric equations $$x(t)= \frac{t^2-4}{t-2}, \

Limits from Data in Chemical Reaction Rates

In a chemical reaction, the concentration of a reactant (in M) is monitored over time (in seconds).

Limits Involving Absolute Value Functions

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

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 Continuity of a Piecewise Function

Consider the piecewise function $$w(x)= \begin{cases} \frac{e^{x}-1}{x} & \text{if } x<0, \\ \frac{\

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 Function and the Squeeze Theorem

Consider the function $$f(x)= x*\sin(1/x)$$ for $$x \neq 0$$, with $$f(0)=0$$. Answer the following:

Rational Function Analysis

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

Rational Function and Removable Discontinuity

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

Removal of Discontinuity by Redefinition

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

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 $$

Trigonometric Limit Evaluation

Examine the function $$ f(x)= \frac{\sin(3*x)}{x} $$ for $$x\ne0$$.

Vertical Asymptote Analysis

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

Analyzing Rates Without a Calculator: Average vs Instantaneous Rates

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

Application of Product Rule

Differentiate the function $$f(x)=(3x^2+2x)(x-4)$$ by two methods. Answer the following:

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 Small Changes with Differentials

Let $$f(x)= x^3 - 5*x + 2$$. Use differentials to approximate small changes in the value of $$f(x)$$

Behavior of the Derivative Near a Vertical Asymptote

Consider the function \(f(x)=\frac{1}{x+2}\) defined for \(x \neq -2\). Answer the following parts.

Chain Rule Application

Consider the composite function $$f(x)=\sqrt{1+4*x^2}$$, which may describe a physical dimension.

Comparing Average vs. Instantaneous Rates

Consider the function $$f(x)= x^3 - 2*x + 1$$. Experimental data for the function is provided in the

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 +

Differentiation Using the Quotient Rule

Consider the function \(q(x)=\frac{3*x^2+5}{2*x-1}\). Answer the following parts.

Exponential Growth Rate

Consider the function $$f(x)= 3*e^{2*x}$$ which models a quantity growing exponentially over time.

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

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 $$h'(x)$$.

Graph Interpretation of the Derivative

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

Graphical Estimation of a Derivative

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

Highway Traffic Flow Analysis

Vehicles enter a highway ramp at a rate given by $$f(t)=60+4*t$$ (vehicles/min) and exit the highway

Implicit Differentiation with Trigonometric Functions

Let the relationship between x and y be given by the equation $$\sin(x*y) = x + y$$. Answer the foll

Inverse Function Analysis: Cubic with Linear Term

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

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

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 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

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

Related Rates: Shadow Length Change

A person 1.8 m tall is walking away from a streetlight that is 5 m high. Let x represent the distanc

Secant Approximation Convergence and the Derivative

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

Slope of a Tangent Line from Experimental Data

Experimental data recording the distance traveled by an object over time is provided in the table be

Tangent Line and Differentiability

Let $$h(x)=\frac{1}{x+2}$$, modeling the concentration of a substance in a chemical solution over ti

Using Derivative Rules on a Trigonometric Function

Consider the function $$f(x)=3*\sin(x)+\cos(2*x)$$. Answer the following questions:

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$

Advanced Implicit and Inverse Function Differentiation on Polar Curves

Consider the curve defined implicitly by $$x^2+y^2= \sin(x*y)$$. Although not a typical polar curve,

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 Angular Motion

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

Chain Rule in Particle Motion

A particle's position is given by $$s(t)=\cos(5*t^2)$$ in meters, where time $$t$$ is measured in se

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

Chain Rule with Exponential and Polynomial Functions

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

Combining Composite and Implicit Differentiation

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

Composite and Rational Function Differentiation

Let $$P(x)=\frac{x^2}{\sqrt{1+x^2}}$$.

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: Engineering Stress-Strain Model

In an engineering context, the stress σ as a function of strain ε is given by $$\sigma(\epsilon) = \

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 Log-Exponential-Trigonometric Composite

Consider the function $$f(x)= \ln\left(e^(\cos(x)) + x^2\right)$$. Solve the following:

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

Finding Second Derivative via Implicit Differentiation

Given the curve defined by $$x^2+y^2+ x*y=7$$, answer the following:

Graph Analysis of a Composite Motion Function

A displacement function representing the motion of an object is given by $$s(t)= \ln(2*t+3)$$. The g

Implicit Curve Analysis: Horizontal Tangents

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

Implicit Differentiation in a Trigonometric Context

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

Implicit Differentiation in Elliptical Orbits

Consider an elliptical orbit described by the equation $$\frac{x^2}{16}+\frac{y^2}{9}=1$$, where the

Implicit Trigonometric Equation Analysis

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

Inverse Function Differentiation

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

Inverse Function Differentiation 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 in a Biological Growth Curve

A biological measurement is modeled by the function $$f(t)= \frac{4*t}{t+2}$$, which is one-to-one o

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 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 Differentiation in Temperature Conversion

Consider the function $$f(x)= \frac{1}{1+e^{-0.5*x}}$$, which converts a temperature reading in Cels

Inverse Trigonometric and Logarithmic Function Composition

Let $$y=\arctan(\ln(x))$$. Answer the following:

Rate of Change in a Circle's Shadow

The equation of a circle is given by $$x^2 + y^2 = 36$$. A point \((x,y)\) on the circle corresponds

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 Flow Analysis using Composite Functions

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

Airplane Altitude Change

An airplane's altitude (in meters) as a function of time is modeled by $$A(t)= 500*t - 4.9*t^2 + 100

Chemical Reaction Rate Analysis

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

Complex Piecewise Function Analysis

Consider the function $$f(x)=\begin{cases}\frac{\sin(x)}{x} & x<\pi \\ 2 & x=\pi \\ 1+\cos(x-\pi) &

Depth of a Well: Related Rates Problem

A bucket is being lowered into a well, and its depth is modeled by $$d(t)= \sqrt{t + 4}$$, where $$t

Economic Cost Function Linearization

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

Economic Efficiency in Speed

A vehicle’s fuel consumption per mile (in dollars) is modeled by the function $$C(v)=0.05*v^2 - 3*v

Estimating Instantaneous Rates from Discrete Data

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

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

Friction and Motion: Finding Instantaneous Rates

A block slides down an inclined plane. The height of the plane at a horizontal distance $$x$$ is giv

FRQ 2: Balloon Inflation Analysis

A spherical balloon is being inflated. Its volume is given by $$V = \frac{4}{3}\pi r^3$$, and the ra

FRQ 6: Particle Motion Analysis on a Straight Line

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

FRQ 13: Cost Function Linearization

A company’s cost function is given by $$C(x) = 5*x^3 - 60*x^2 + 200*x + 1000$$, where x represents t

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

Inflating Balloon Rates

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

L'Hôpital's Rule in Action

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

L'Hôpital’s Rule in Limits with Contextual Application

Consider the function $$f(x)= \frac{e^{2*x} - 1}{5*e^{2*x} - 5}$$, which models a growth phenomenon.

Modeling Coffee Cooling

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

Motion Analysis from Velocity Function

A particle moves along a straight line with a velocity given by $$v(t) = t^2 - 4t + 3$$ (in m/s). Th

Optimization of Production Costs

A manufacturing company’s cost function for producing $$x$$ units per hour is given by $$C(x)=\frac{

Projectile Motion Analysis

A projectile is launched vertically, and its height (in meters) as a function of time is given by $$

Projectile Motion with Velocity Components

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

Projectile Motion: Evaluating Maximum Height

A projectile is launched vertically with its height given by $$h(t)= -4.9*t^2 + 19.6*t + 3$$, where

Related Rates in a Conical Tank

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

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: The Expanding Ripple

Ripples form in a pond such that the radius of a circular ripple increases at a constant rate. Given

Shadow Length Problem

A 10-meter tall streetlight casts a shadow of a 1.8-meter tall person. If the person walks away from

Transformation of Logarithmic Functions

Consider the function $$f(x)=\ln(3x-2)$$. Analyze the function and its transformation:

Analyzing Critical Points in a Piecewise Function

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

Area and Volume: Polynomial Boundaries

Let $$f(x)= x^2$$ and $$g(x)= 4 - x^2$$. Consider the region bounded by these two curves.

Area Growth of an Expanding Square

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

Average Value of a Function and Mean Value Theorem for Integrals

Consider the function $$f(x)= e^{-x}$$ on the interval $$[0, 2]$$. Answer the following:

Bacterial Culture Growth: Identifying Critical Points from Data

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

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

Chemical Reactor Temperature Optimization

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

Concavity and Inflection Points in a Quartic Function

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

Evaluating Rate of Change in Electric Current Data

An electrical engineer recorded the current (in amperes) in a circuit over time. The table below sho

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 3: Relative Extrema for a Cubic Function

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

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 19: Analysis of an Exponential-Polynomial Function

Consider the function $$f(x)= e^{-x}*x^2$$ defined for $$x \ge 0$$.

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

Graph Analysis: Exponentially Modified Function

Consider the function $$f(x)= 2e^{x}-5\ln(x+1)$$ defined for $$x> -1$$. Answer the following:

Inverse Analysis of a Cooling Temperature Function

A cooling process is described by the function $$f(t)=20+80*e^{-0.05*t}$$, where t is the time in mi

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

Investigating the Behavior of a Composite Function

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

Logistic Growth Model and Derivative Interpretation

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

Mean Value Theorem for a Cubic Function

Consider the function $$f(x)= x^3 - 2*x^2 + x$$ on the closed interval $$[0,2]$$. In this problem, y

Mean Value Theorem for a Piecewise Function

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

Optimization of a Rectangular Enclosure

A rectangular pen is to be constructed along the side of a barn so that only three sides require fen

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

Piecewise Function and the Mean Value Theorem

Consider the piecewise function $$f(x)= \begin{cases} x^2 & \text{if } x \le 1, \\ 2*x - 1 & \text{

Solving an Exponential Equation

Solve for $$x$$ in the equation $$e^{2x}= 5*e^{x}$$. Answer the following:

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

Traffic Flow Modeling

A highway segment experiences varying traffic flows. Cars enter at a rate $$I(t)=50+10*\sin(\frac{\p

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

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 Droplet Free Fall Analysis

A water droplet is released from a ceiling, and its height (in meters) above the ground is modeled b

Water Reservoir Net Change

A water reservoir receives water from a river at a rate $$R_{in}(t)=5+0.5*t$$ and discharges water a

Accumulation and Flow Rate in a Tank

Water flows into a tank at a rate given by $$R(t)=3*t^2-2*t+1$$ (in m³/hr) for $$0\le t\le2$$. The t

Analyzing Work Done by a Variable Force

An object is acted on by a force given by $$F(x)= 3*x^2 - x + 2$$ (in newtons), where $$x$$ is the d

Antiderivative via U-Substitution

Evaluate the antiderivative of the function $$f(x)=(3*x+5)^4$$ and use it to compute a definite inte

Antiderivatives with Initial Conditions: Temperature

The rate of temperature change in a chemical reaction is given by $$T'(t)=-0.2*t+3$$ (in °C/min), wi

Approximating Area Under a Curve with Riemann Sums

Consider a function $$f(x)$$ whose values are tabulated below for different values of $$x$$. Use the

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

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:

Composite Functions and Accumulation

Let the accumulation function be defined by $$F(x)=\int_{2}^{x} \sqrt{t+1}\,dt.$$ Answer the followi

Computing Accumulated Volume from a Filling Rate Function

A small pond is being filled at a rate given by $$r(t)=2*t + 3$$ (in $$m^3/hr$$), where $$t$$ is in

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

Economic Analysis: Consumer Surplus

In a competitive market, the demand function is given by $$D(p)=100-2*p$$ and the supply function is

Economic Cost Function Analysis

A company's marginal cost (in dollars per unit) is recorded at various production levels. Use the da

Error Estimates in Numerical Integration

Suppose a function $$f(x)$$ defined on an interval $$[a,b]$$ is known to be concave downward. Consid

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 Area Under a Curve Using Riemann Sums

Consider the function whose values are given in the table below. Use the table to estimate the area

Estimating River Flow Volume

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

Estimating the Area Under a Curve Using Riemann Sums

A function $$f(x)$$ is defined on the interval $$[0,6]$$ and its values are listed in the table belo

Evaluating a Definite Integral Using U-Substitution

Compute the integral $$\int_{0}^{3} (2*t+1)^5\,dt$$ using u-substitution.

Evaluating a Radical Integral via U-Substitution

Evaluate the integral $$\int_{1}^{9}\sqrt{2*x+1}\,dx$$ using U-substitution. Answer the following pa

FRQ8: Inverse Analysis of a Piecewise-Defined Accumulation Function

Let $$ R(x)=\begin{cases} \int_{1}^{x} t\,dt, & 1 \le x \le 3 \\ \int_{1}^{x} (2*t-1)\,dt, & x > 3 \

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

General Antiderivatives and the Constant of Integration

Given the function $$f(x)= 4*x^3$$, address the following questions about antiderivatives.

Improving Area Approximations with Increasing Subintervals

Consider the function $$f(x)= \sqrt{x}$$ on the interval $$[0,4]$$. Explore how Riemann sums approxi

Integration of Exponential Functions with Shifts

Evaluate the integral $$\int_{0}^{2} e^{2*(x-1)}\,dx$$ using an appropriate substitution.

Integration to Determine Work Done by a Variable Force

A variable force along a straight line is given by $$F(x)=4*x^2 - 3*x + 2$$ (in Newtons). Determine

Motion Along a Line: Changing Velocity

A particle moves along a line with a velocity given by $$v(t)=12-2*t$$ (in m/s) for $$0\le t\le8$$,

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

Particle Trajectory in the Plane

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

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),

Rainfall Accumulation Analysis

The rainfall intensity at a location is modeled by the function $$i(t) = 0.5*t$$ (inches per hour) f

Sand Pile Dynamics

A sand pile is being formed by delivering sand at a rate of $$r_{in}(t) = 3t$$ kg/min while erosion

Total Water Volume from a Flow Rate Function

A river’s flow rate (in cubic meters per second) is modeled by the function $$Q(t)=4+2*t$$, where $$

Trapezoidal Rule Application with Population Growth

A biologist recorded the instantaneous growth rate (in thousands per year) of a species over several

Trigonometric Integral with U-Substitution

Evaluate the definite integral $$\int_{0}^{\frac{\pi}{4}} \sec^2(t)\tan(t)\,dt$$.

Trigonometric Integration via U-Substitution

Evaluate the integral $$I=\int_{0}^{\frac{\pi}{4}} \tan(x)*\sec^2(x)\,dx.$$ Answer the following par

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

Cooling of a Cup of Coffee

Newton's Law of Cooling states that the rate of change of temperature of an object is proportional t

Cooling of Electronic Components

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

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

Fishery Harvesting Model

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

Homogeneous Differential Equation

Consider the differential equation $$\frac{dy}{dx} = \frac{x+y}{x-y}$$ with the initial condition $$

Insulin Concentration Dynamics

The concentration $$I$$ (in μU/mL) of insulin in the blood follows the model $$\frac{dI}{dt}=-k(I-I_

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)$

Logistic Population Model Analysis

A population $$P$$ grows according to the logistic equation $$\frac{dP}{dt}=0.4P\left(1-\frac{P}{100

Mixing Problem in a Tank

A tank initially contains 100 liters of brine with 10 kg of dissolved salt. Brine with a concentrati

Mixing Tank Problem

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

Motion Along a Curve with Implicit Differentiation

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

Newton's Law of Cooling

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

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

Newton's Law of Cooling with Variable Ambient Temperature

An object cools according to Newton's Law of Cooling, but the ambient temperature is not constant. I

Population Growth in a Bacterial Culture

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

Radioactive Isotope in Medicine

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

Reversible Chemical Reaction

In a reversible chemical reaction, the concentration $$C(t)$$ of a product is governed by the differ

Salt Mixing in a Tank

A tank initially contains 100 L of water with 5 kg of salt dissolved. Brine with a concentration of

Salt Mixing Problem

A tank initially contains $$100$$ kg of salt dissolved in $$1000$$ L of water. A salt solution with

Sand Erosion in a Beach Model

During a storm, a beach loses sand. Let $$S(t)$$ (in tons) be the amount of sand on a beach at time

Slope Field Analysis for $$\frac{dy}{dx}=\frac{y}{x}$$

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

Slope Field Interpretation for $$\frac{dy}{dx} = y-x$$

A slope field for the differential equation $$\frac{dy}{dx}=y-x$$ is provided in the stimulus. Use t

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

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

Tank Draining Differential Equation

Water drains from a tank at a rate that depends on the square root of the volume, according to $$\fr

Water Tank Flow Analysis

A water tank receives an inflow of water at a rate $$Q_{in}(t)=50+10*\sin(t)$$ (liters/min) and an o

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

Analysis of a Rational Function's Average Value

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

Area Between Curves with Variable Limits

Consider two functions, $$f(x)$$ and $$g(x)$$, whose values are tabulated below. The functions inter

Area Between Curves: River Cross-Section

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

Area Between Parabolic Curves

Consider the curves $$f(x)=x^2$$ and $$g(x)=4*x-x^2$$. Determine the area of the region bounded by t

Average Density of a Rod

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

Average Reaction Rate Determination

A chemical reaction’s rate is modeled by the function $$r(t)=k*e^{-t}$$, where $$t$$ is in seconds a

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 Velocity from Position Data

The position of a vehicle moving along a straight road is given in the table below. Use these data t

Bloodstream Drug Concentration

A drug enters the bloodstream at a rate given by $$R(t)= 5*e^{-0.5*t}$$ mg/min for $$t \ge 0$$. Simu

Car Motion: Position, Velocity, and Acceleration

A car moving along a straight eastbound road has an acceleration given by $$a(t)=4-0.5*t$$ (in m/s²)

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 with Discontinuous Pricing

A utility company’s billing is modeled by the function $$C(q)=\begin{cases} 3*q & \text{if } 0\le q\

Designing a Water Slide

A water slide is designed along the curve $$y=-0.1*x^2+2*x+3$$ (in meters) over the interval $$[0,10

Displacement from a Velocity Graph

A moving object has its velocity given as a function of time. A velocity versus time graph is provid

Environmental Impact: Average Pollutant Concentration

The pollutant concentration in a river is modeled by $$h(x)=0.01*x^3-0.5*x^2+5*x$$ (in mg/L) over a

Funnel Design: Volume by Cross Sections

A funnel is designed by rotating the region bounded by the curve $$y=4-x^2$$ for -2 ≤ x ≤ 2 about th

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

Population Growth and Average Rate

A town's population is modeled by the function $$P(t)=1000*e^{0.03*t}$$, where $$t$$ is measured in

Tank Draining with Variable Flow Rates

A water tank is undergoing simultaneous inflow and outflow. The inflow rate is given by $$I(t)=10+2\

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 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 of Revolution: Curve Raised to a Power

Consider the region under the curve $$f(x)=x^{1/3}$$ for $$x\in [0,8]$$. This region is revolved abo

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

Volume of a Solid with a Hole Using the Washer Method

Consider the region bounded by $$y=x^2$$ and $$y=4$$. This region is revolved about the $$x$$-axis t

Volume with Semicircular Cross‐Sections

A region in the first quadrant is bounded by the curve $$y=x^2$$ and the x-axis for $$0 \le x \le 3$

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 Done by a Variable Force

A force acting along a straight line is given by $$F(x)=3*x^2+2$$ (in Newtons) when an object is dis

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