AP Calculus AB FRQ Room

Analyzing a Piecewise Function’s Limits and Continuity

Consider the function $$f(x)= \begin{cases} \frac{x^2-9}{x-3} & \text{if } x \neq 3, \\ k & \text{if

Analyzing a Piecewise Velocity Function for Continuity and Limits

A particle moves along a line with a piecewise velocity function given by $$v(t)= \begin{cases} 2*t+

Analyzing Process Data for Continuity

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

Area and Volume Setup with Bounded Regions

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

Composite Function and Continuity Analysis

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

Continuity Analysis of a Radical Function

Consider the function $$f(x) = \frac{\sqrt{x+4} - 2}{x}$$. (a) Evaluate $$\lim_{x \to 0} f(x)$$. (b

Continuity and Asymptotic Behavior of a Rational Exponential Function

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

Continuity of Composite Functions

Let $$f(x)=x+2$$ for all x, and let $$g(x)=\begin{cases} \sqrt{x}, & x \geq 0 \\ \text{undefined},

Determining Parameters for Continuity in a Piecewise Function

Let the function be defined as $$ f(x)=\begin{cases}ax+3, & x<2,\\ x^2+bx+1, & x\ge2.\end{cases} $$

Discontinuity Analysis in Piecewise Functions

Consider the piecewise function $$f(x)= \begin{cases} \frac{x^2-4}{x-2} & x\neq2 \\ 5 & x=2 \end{cas

Economic Limit and Continuity Analysis

A company's profit (in thousands of dollars) from producing x items is modeled by the function $$P(x

Factoring a Cubic Expression for Limit Evaluation

Consider the function $$f(x)= \frac{x^3 - 8}{x - 2}$$ for x \(\neq\) 2. Answer the following parts.

Graph Analysis of Discontinuities

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

Horizontal Asymptote and End Behavior

Consider the rational function $$f(x)=\frac{3*x^2 + x - 5}{6*x^2 - 4*x + 7}$$. Answer the following

Implicit Differentiation and Tangent Slopes

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

Implicit Differentiation in an Exponential Equation

Consider the function defined implicitly by $$e^{x*y} + x - y = 0$$. Answer the following:

Intermediate Value Theorem Application

Consider the polynomial function $$f(x)=x^3-6*x^2+9*x+1$$ on the closed interval [0, 4].

Intermediate Value Theorem in Temperature Modeling

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

Inverse Function Analysis and Derivative

Let $$f(x)= x^3+2$$, defined for all real numbers.

Limit Evaluation using Conjugate Multiplication

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

Limit Evaluation with a Parameter in a Log-Exponential Function

Consider the function $$r(x)=\frac{e^{a*x} - e^{b*x}}{\ln(1+x)}$$ defined for $$x \neq 0$$, where $$

Limit Involving a Radical Expression via Conjugate

Consider the function $$f(x)=\frac{\sqrt{x+9}-3}{x}$$. Answer the following:

Limits Involving Absolute Value Expressions

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

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

One-Sided Limits and an Absolute Value Function

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

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{\

Oscillatory Function and the Squeeze Theorem

Consider the function $$f(x)=x*\sin(1/x)$$ for x ≠ 0, with f(0)=0.

Piecewise Rational Function and Removable Discontinuity

Consider the function $$f(x)=\frac{x^2-9}{x-3}$$ for $$x\neq3$$ and $$f(3)=2$$. A graph of this func

Related Rates: Expanding Circular Ripple

A circular ripple forms at the center of a pond and expands over time. The radius $$r$$ (in meters)

Removable Discontinuity and Limit Evaluation

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

Removable Discontinuity and Redefinition

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

Return on Investment and Asymptotic Behavior

An investor’s portfolio is modeled by the function $$P(t)= \frac{0.02t^2 + 3t + 100}{t + 5}$$, where

Two-dimensional Particle Motion with Continuous Velocity Functions

A particle moves in the plane with velocity components given by $$v_x(t)= \frac{t^2-9}{t-3}$$ and $

Acceleration Through Successive Differentiation

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

Analyzing Function Behavior Using Its Derivative

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

Analyzing the Derivative of a Trigonometric Function

Consider the function $$f(x)= \sin(x) + \cos(x)$$.

Approximating Derivative using Secant Lines

Consider the function $$f(x)= \ln(x)$$. A student records the following data: | x | f(x) | |---

Approximating the Instantaneous Rate of Change Using Secant Lines

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

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

Chemical Reactor Flow Rates

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

Concavity and the Second Derivative

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

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

Cost Optimization and Marginal Analysis

A manufacturer’s cost function is given by $$C(q)=500+4*q+0.02*q^2$$, where $$q$$ is the quantity pr

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

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

Deriving the Derivative from First Principles for a Reciprocal Square Root Function

Let $$f(x)=\frac{1}{\sqrt{x}}$$ for $$x > 0$$. Using the definition of the derivative, show that $$f

Differentiability of a Piecewise Function

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

Finding the Tangent Line Using the Product Rule

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

Implicit Differentiation in Motion

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

Instantaneous 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

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 Rate of Change in Motion

A particle’s position along a straight line is given by $$s(t)= 4*t^3 - 12*t^2 + 9*t + 5$$, where $$

Inverse Function Analysis: Cubic Transformation

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

Inverse Function Analysis: Logarithmic-Hyperbolic Function

Consider the function $$f(x)=\ln\left(x+\sqrt{x^2+1}\right)$$ defined for all real x. (This function

Inverse Function Analysis: Restricted Rational Function

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

Logarithmic Differentiation of a Composite Function

Let $$f(x)= (x^{2}+1)*\sqrt{x}$$. Use logarithmic differentiation to find the derivative.

Marginal Cost Function in Economics

A company’s cost function is given by $$C(x)=200+8*x+0.05*x^2$$, where $$C(x)$$ is in dollars and $$

Marginal Profit Calculation

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

Particle Motion on a Straight Road

A particle moves along a straight road. Its position at time $$t$$ seconds is given by $$s(t) = t^3

Polynomial Rate of Change Analysis

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

Product Rule Application

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

Related Rates: Expanding Ripple Circle

Water droplets create circular ripples on a surface. The area of a ripple is given by $$A = \pi * r^

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

Relating Average and Instantaneous Velocity in a Particle's Motion

A particle’s position is modeled by $$s(t)=\frac{4}{t+1}$$, where $$s(t)$$ is in meters and $$t$$ is

Secant and Tangent Line Approximation in a Real-World Model

A temperature sensor records data modeled by $$g(t)= 0.5*t^2 - 3*t + 2$$, where $$t$$ is in seconds

Secant and Tangent Lines Approximation

A research experiment records temperature variations over time. The temperature function is approxim

Secant and Tangent Lines for a Trigonometric Function

Let $$f(x)=\sin(x)+x^2$$. Use the definition of the derivative to find $$f'(x)$$ and evaluate it at

Secant and Tangent Lines to a Curve

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

Slope of a Tangent Line from a Table of Values

Given the table below for a differentiable function $$f(x)$$: | x | f(x) | |-----|------| | 1 |

Tangent and Normal Lines in Road Construction

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

Water Treatment Plant's Chemical Dosing

A water treatment plant adds a chemical at a rate of $$f(t)=5+0.2*t$$ (liters/min) while the chemica

Analyzing a Composite Function and Its Inverse

Consider the function $$f(x)= (3*x+2)^2$$. Answer the following questions about the derivative of th

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 with Exponential and Trigonometric Functions

A particle's position is given by $$s(t)=e^{3*t}\sin(2*t)$$. Use appropriate differentiation techniq

Chain Rule with Trigonometric Function

Consider the function $$f(x)=\sin(5*x^2)$$. Answer the following:

Chemical Reaction Rate: Exponential and Logarithmic Model

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

Comparing the Rates between a Function and its Inverse

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

Composite Function Differentiation in a Sand Pile Model

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

Composite Function Kinematics

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

Differentiation of a Composite Inverse Trigonometric-Log Function

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

Differentiation of Inverse Trigonometric Composite Function

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

Differentiation with Rational Exponents

Let $$y=(3*x+2)^{4/3}$$. 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 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 in a Biochemical Reaction

Consider a biochemical reaction modeled by the equation $$x*e^{y} + y*e^{x} = 10$$, where $$x$$ and

Implicit Differentiation in a Cubic Relationship

Examine the curve defined by $$x^3 + y^3 = 6*x*y$$, which represents a complex relationship between

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 Economic Model

In an economic model, the relationship between the quantity supplied $$x$$ and the market price $$y$

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 Quadratic Curve

Consider the curve defined by $$x^2+xy+y^2=7$$. Use implicit differentiation to analyze the behavior

Implicit Differentiation with Exponential-Trigonometric Functions

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

Implicit Differentiation with Logarithmic and Radical Components

Consider the equation $$\ln(x+y)=\sqrt{x*y}$$.

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 Involving a Polynomial

Let $$f(x)= x^3 + 2*x + 1$$. Analyze its invertibility and the derivative of its inverse function.

Manufacturing Optimization via Implicit Differentiation

A manufacturing cost relationship is given implicitly by $$x^2*y + x*y^2 = 1000$$, where $$x$$ repre

Nested Composite Function Differentiation

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

Pendulum Angular Displacement Analysis

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

Second Derivative via Implicit Differentiation

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

Accelerating Car Motion Analysis

A car's velocity is modeled by $$v(t)=4t^2-16t+12$$ in m/s for $$t\ge0$$. Analyze the car's motion.

Analysis of Wheel Rotation

Consider a wheel whose angular position is given by $$\theta(t) = 2t^2 + 3t$$, in radians, where $$t

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

Designing a Flower Bed: Optimal Shape

A landscape designer wants to create a rectangular flower bed with a fixed area of 200 square meters

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

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 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 15: Evaluating Limits with L’Hôpital’s Rule

Evaluate the limit $$\lim_{x\to\infty} \frac{5*x^3 - 4*x^2 + 1}{7*x^3 + 2*x - 6}$$ by using L’Hôpita

FRQ 20: Market Demand Analysis

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

Inflating Balloon

A spherical balloon is being inflated. Its volume increases at a constant rate of 12 in³/sec. The vo

L’Hôpital’s Rule in Limit Evaluation

Evaluate the limit $$\lim_{x\to\infty}\frac{5*x^3 - 4*x + 1}{2*x^3 + 7*x^2 - x + 6}$$ using L’Hôpita

Linearization and Differentials

Given the function $$f(x)=x^4$$, use linear approximation to estimate the value of $$(3.98)^4$$.

Local Linearization Approximation

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

Logarithmic Differentiation in Exponential Functions

Let $$y = (2x^2 + 3)^{4x}$$. Use logarithmic differentiation to find $$y'$$.

Optimization for Minimizing Time in Road Construction

A new road connecting two towns involves two segments with different construction speeds. The travel

Optimization of Production Costs

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

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

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: Inflating Balloon

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

Rocket Thrust: Analyzing Exponential Acceleration

A rocket’s velocity is modeled by $$v(t) = 100(1 - e^{-0.05t})$$, where $$t$$ is in seconds and $$v(

Shadow Length Problem

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

Studying a Bouncing Ball Model

A bouncing ball reaches a maximum height after each bounce modeled by $$h(n)= 100*(0.8)^n$$, where n

Temperature Change Analysis

The temperature of a chemical solution is recorded over time. Use the table below, where $$T(t)$$ (i

Temperature Change in a Cooling Process

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

Transcendental Function Temperature Change

A cooling object has its temperature modeled by $$T(t)= 100 + 50e^{-0.2*t}$$, where t is measured in

Using L'Hospital's Rule to Evaluate a Limit

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

Vehicle Deceleration Analysis

A vehicle’s position is given by $$s(t)=100t-5t^2$$ where $$s(t)$$ is in meters and $$t$$ in seconds

Water Tank Volume Change

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

Absolute Extrema via the Candidate's Test

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

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

Analyzing the Function $$f(x)= x*\ln(x) - x$$

Consider the function $$f(x)= x*\ln(x) - x$$ defined for $$x > 0$$.

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:

Car Speed Analysis via MVT

A car's position is given by $$f(t) = t^3 - 3*t^2 + 2*t$$ (in meters) for $$t$$ in seconds on the cl

Concavity Analysis of a Trigonometric Function

For the function $$f(x)= \sin(x) - \frac{1}{2}\cos(x)$$ defined on the interval $$[0,2\pi]$$, analyz

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

Evaluating Pollution Concentration Changes

A study recorded the concentration of a pollutant (in ppm) in a river over time (in hours). Use the

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 2: Daily Temperature Extremes and the Extreme Value Theorem

A function modeling the ambient temperature (in $$^\circ C$$) during the first 6 hours of a day is g

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

Mean Value Theorem Applied to Car Position Data

A car’s position (in meters) is recorded at various times during a journey. Use the information prov

Mean Value Theorem Applied to Exponential Functions

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

Minimizing Average Cost in Production

A company’s cost function is given by $$C(x)= 0.5*x^3 - 6*x^2 + 20*x + 100$$, where $$x$$ represents

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{

Polynomial Rational Discontinuity Investigation

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

Predicting Fuel Efficiency in Transportation

A vehicle’s performance was studied by recording the miles traveled and the corresponding fuel consu

Rational Function Behavior and Extreme Values

Consider the function $$f(x)= \frac{2*x^2 - 3*x + 1}{x - 2}$$ defined for $$x \neq 2$$ on the interv

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

Relative Extrema of a Rational Function

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

Trigonometric Function Behavior

Consider the function $$f(x)= \sin(x) + \cos(2*x)$$ defined on the interval $$[0,2\pi]$$. Analyze it

Accumulated Change Function Evaluation

Let $$F(x)=\int_{1}^{x} (2*t+3)\,dt$$ for $$x \ge 1$$. This function represents the accumulated chan

Analyzing Bacterial Growth via Riemann Sums

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

Antiderivatives of Trigonometric Functions

Evaluate the integral $$\int \sin(2*t)\,dt$$, and then use your result to compute the definite integ

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

Area Under a Piecewise Function

Consider the piecewise function $$ f(x)= \begin{cases} x & \text{for } 0\le x<3,\\ 9-x & \text{for

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 $$^\circ C$$

Cumulative Solar Energy Collection

A solar panel's power output (in Watts) is recorded during a sunny day at various times. Use the dat

Estimating Work Done Using Riemann Sums

In physics, the work done by a variable force is given by $$W=\int F(x)\,dx$$. A force sensor record

Finding the Area of a Parabolic Arch

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

FRQ4: Inverse Analysis of a Trigonometric Accumulation Function

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

FRQ5: Inverse Analysis of a Non‐Elementary Integral Function

Consider the function $$ P(x)=\int_{0}^{x} e^{t^2}\,dt $$ for x ≥ 0. Answer the following parts.

FRQ12: Inverse Analysis of a Temperature Accumulation Function

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

FRQ13: Inverse Analysis of an Investment Growth Function

An investment's accumulated value is given by $$ G(t)=\int_{0}^{t} \frac{1}{1+u}\,du $$ for t ≥ 0. A

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

Graphical Analysis of an Accumulation Function

Let $$f(t)$$ represent the rate of water flow (in $$m^3/hr$$) into a reservoir, and suppose the grap

Logistically Modeled Accumulation in Biology

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

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

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 $$0\le t\le6$$,

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

Population Change in a Wildlife Reserve

In a wildlife reserve, animals immigrate at a rate of $$I(t)= 10\cos(t) + 20$$ per month, while emig

Population Growth and Accumulation

A rabbit population grows in an enclosed field at a rate given by the differential equation $$P'(t)=

Riemann Sum Approximation of f(x) = 4 - x^2

Consider the function $$f(x)=4-x^2$$ on the interval $$[0,2]$$. Use Riemann sums to approximate the

Sand Pile Dynamics

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

Ski Lift Passengers: Boarding and Alighting Rates

On a ski lift, passengers board at a rate $$B(t)= 3e^{-t/2}$$ persons/min and alight at a constant r

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 Over Time

A region experiences a temperature change over time that is modeled by the derivative function $$T'(

The Accumulation Function for a Linear Rate Model

Consider the accumulation function defined by $$A(t)= \int_0^t (3*t' + 2)\,dt'$$, where $$t'$$ is a

Total Cost Function from Marginal Cost

The marginal cost of production for a company is given by $$MC(q)=6+0.5*q$$ dollars per unit for pro

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

Using Integration to Determine Average Value

A function given by $$f(x)= \ln(1+x)$$ is defined on the interval $$[0,3]$$. Use integration to dete

Volume of a Solid: Exponential Rotation

Consider the region bounded by the curve $$y=e^{-x}$$, the x-axis, and the vertical lines $$x=0$$ an

Volume of Water in a Tank from Inflow Data

A water tank's inflow rate (in L/min) is measured at different times. Use the provided data to estim

A Separable Differential Equation: Growth Model

Consider the differential equation $$\frac{dy}{dx}=3*x*y^2$$ that models a growth process. Use separ

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

Bacterial Population with Time-Dependent Growth Rate

A bacterial population grows according to the differential equation $$\frac{dP}{dt}=\frac{k}{1+t^2}P

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 Reaction in a Vessel

A 50 L reaction vessel initially contains a solution of reactant A at a concentration of 3 mol/L (i.

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 with Variable Ambient Temperature

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

Drug Concentration with Continuous Infusion

A drug is administered intravenously such that its blood concentration $$C(t)$$ (in mg/L) follows th

Implicit Differentiation of a Circle

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

Insulin Concentration Dynamics

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

Investment Growth with Continuous Contributions

An investment account grows continuously with an annual interest rate of 5% while continuous deposit

Logistic Growth Model

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

Logistic Growth Model Analysis

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

Logistic Model with Harvesting

A fishery's population is governed by the logistic model with harvesting: $$\frac{dP}{dt} = 0.5\,P\l

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 a Salt Solution

A mixing tank experiment records the salt concentration $$C$$ (in g/L) at various times $$t$$ (in mi

Mixing Problem: Salt in a Tank

A 100-liter tank initially contains 50 grams of salt. Brine with a salt concentration of $$0.5$$ gra

Newton's Law of Cooling

An object is heated to $$100^\circ C$$ and left in a room at $$20^\circ C$$. According to Newton's l

Non-Separable to Linear DE

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

Nonlinear Cooling of a Metal Rod

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

Nonlinear Differential Equation

Consider the differential equation $$\frac{dy}{dx}=(1-y^2)\cos(x)$$ with the initial condition $$y(0

Radioactive Decay Model

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

Reversible Chemical Reaction

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

Salt Tank Mixing Problem

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

Separable Differential Equation: y' = (2*x)/y

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

Slope Field Analysis for a Linear Differential Equation

Consider the linear differential equation $$\frac{dy}{dx}=\frac{1}{2}*x-y$$ with the initial conditi

Slope Field Exploration

Consider the differential equation $$\frac{dy}{dx} = \sin(x)$$. The provided slope field (see stimul

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

Solving a Differential Equation by Substitution

Solve the differential equation $$\frac{dy}{dx}=\frac{2xy}{x^2-4}$$. (a) Separate the variables to

Solving a Differential Equation Using the SIPPY Method

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

Tank Mixing with Salt

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

Traffic Flow Dynamics

On a highway, the density of cars, \(D(t)\) (in cars), changes over time due to a constant inflow of

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

Analysis of an Inverse Function

Consider the function $$f(x)=(x-1)^3+2$$, defined for all real numbers. Analyze its inverse function

Area Between Curves in an Ecological Study

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

Area Between Curves: 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 Curves: River Cross-Section

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

Average Concentration Calculation

In a continuous stirred-tank reactor (CSTR), the concentration of a chemical is given by $$c(t)=5+3*

Average Concentration in Medical Dosage

A patient’s bloodstream concentration of a medication is modeled by the function $$C(t)=\frac{100}{1

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 Drug Concentration in the Bloodstream

The concentration of a drug in the bloodstream is modeled by $$C(t)=\frac{20*t}{1+t^2}$$ (in mg/L) f

Average Value of a Trigonometric Function

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

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

Discontinuities in a Piecewise Function

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

Discounted Cash Flow

A company projects that its annual cash flow will grow according to a geometric sequence. The initia

Economic Profit Analysis via Area Between Curves

A company's revenue and cost are modeled by the linear functions $$R(x)=50*x$$ and $$C(x)=20*x+1000$

Graduated Rent Increase

An apartment’s yearly rent increases by a fixed amount. The initial annual rent is $$1200$$ dollars

Implicit Differentiation in an Economic Equilibrium Model

In an economics model, the relationship between price $$p$$ and quantity $$q$$ is given implicitly b

Particle Motion on a Parametric Path

A particle moves along a path given by the parametric equations $$x(t)= t^2 - t$$ and $$y(t)= 3*t -

Pollutant Accumulation in a River

Along a 20 km stretch of a river, a pollutant enters the water at a rate described by $$p(x)=0.5*x+2

Population Model Using Exponential Function

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

Position and Velocity Relationship in Car Motion

A car's position along a highway is modeled by $$s(t)=t^3-6*t^2+9*t+2$$ (in kilometers) with time $$

Probability from a Density Function

Let a continuous random variable $$X$$ be defined on $$[0,20]$$ with the probability density functio

Radioactive Decay Accumulation

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

Rebounding Ball

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

Stress Analysis in a Structural Beam

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

Temperature Average Calculation

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

Viral Video Views

A viral video’s daily views form a geometric sequence. On day 1, the video is viewed 1000 times, and

Voltage and Energy Dissipation Analysis

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

Volume by the Washer Method: Solid of Revolution

Consider the region bounded by the curves $$y=x$$ and $$y=x^2$$ for $$0 \le x \le 1$$. This region i

Volume of a Solid with Semicircular Cross Sections

A solid has a base in the xy-plane given by the region bounded by $$y=4-x^2$$ and the x-axis for $$0

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 Flow in a River: Average Velocity and Flow Rate

A river has a width of 10 meters. The velocity of the water at a distance $$x$$ (in meters) from one

Work Done by a Variable Force

A variable force is applied along a straight line such that $$F(x)=6-0.5*x$$ (in Newtons). The force

Work Done Stretching a Spring with Variable Constant

A spring does not follow Hooke's law exactly: its effective spring constant varies with displacement

Work in Spring Stretching

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