AP Calculus AB FRQ Room

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 $$

Analysis of Three Functions

The table below lists the values of three functions f, g, and h at selected x-values. Use the table

Analyzing a Removable Discontinuity

Consider the function $$f(x) = \frac{x^2 - 4}{x - 2}$$ for $$x \neq 2$$. Notice that f is not define

Asymptotic Behavior of a Logarithmic Function

Consider the function $$w(x)=\frac{\ln(x+e)}{x}$$ for $$x>0$$. Analyze its behavior as $$x \to \inft

Continuous Extension and Removable Discontinuity

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

Determining Horizontal Asymptotes for Rational Functions

Given the rational function $$R(x)= \frac{2*x^3+ x^2 - x}{x^3 - 4}$$, answer the following:

Direct Evaluation of Polynomial Limits

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

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

Graph Analysis: Identify Limits and Discontinuities

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

Graph Reading: Left and Right Limits

A graph of a function f is provided below which shows a discontinuity at x = 2. Use the graph to det

Identifying Discontinuities in a Rational Function

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

Intermediate Value Theorem Application

Suppose a continuous function $$f(x)$$ is defined on the interval $$[1,5]$$, with $$f(1)=-3$$ and $$

Intermediate Value Theorem in Temperature Modeling

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

Limit Analysis in Population Modeling

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

One-Sided Limits and Discontinuity Analysis

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

Oscillatory Function and the Squeeze Theorem

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

Particle Motion with Vertical Asymptote in Velocity

A particle moves along a number line with velocity function $$v(t)= \frac{3*t}{t-1}$$ for $$t > 1$$.

Position from Acceleration and Limit Behavior

A particle moves along a straight line with acceleration function $$a(t)= \frac{6-2*t}{t-3}$$ for $$

Rational Functions with Removable Discontinuities

Examine the function $$f(x)= \frac{x^2 - 5x + 6}{x - 2}$$. (a) Factor the numerator and simplify th

Removable Discontinuity in a Rational Function

Consider the function $$f(x)=\begin{cases} \frac{x^2-16}{x-4} & x\neq4 \\ 3*x+1 & x=4 \end{cases}$$.

Removing Discontinuities

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

Squeeze Theorem Application with Trigonometric Functions

Let $$f(x)= x^2 \sin(1/x)$$ for $$x\neq0$$, and define $$f(0)=0$$.

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:

Vertical Asymptotes and Horizontal Limits

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

Application of the Quotient Rule: Velocity on a Curve

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

Approximating the Tangent Slope

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

Average vs. Instantaneous Rate of Change

Consider the function $$f(x)=2*x^2-3*x+1$$ defined for all real numbers. Answer the following parts

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 Applications in Motion Along a Curve

A particle moves such that its horizontal position is given by $$x(t)= t^2 + 2*t$$ and its vertical

Derivative of an Exponential Decay Function

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

Determining Velocity from a Position Graph

The graph below shows the position of a car over time where position is measured in kilometers and t

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 +

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 Growth Rate

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

Exponential Rate of Change

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

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

Graphical Estimation of a Derivative

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

Implicit Differentiation in Motion

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

Implicit Differentiation of a Circle

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

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

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 and Maximum Acceleration

An object’s position is given by $$s(t)=t^4-4t^3+2t^2$$ (in meters), where t is in seconds. Answer t

Inverse Function Analysis: Cubic Function

Consider the function $$f(x)=x^3+2*x+1$$ defined for all real numbers.

Inverse Function Analysis: Hyperbolic-Type Function

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

Inverse Function Analysis: Rational Decay Function

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

Inverse Function Analysis: Rational Function

Consider the function $$f(x)=\frac{2*x+1}{x+3}$$ defined for all x except $$x=-3$$.

Inverse Function Analysis: Rational Function 2

Consider the function $$f(x)=\frac{x+4}{x+2}$$ defined for $$x\neq -2$$, with the additional restric

Inverse Function Analysis: Sum with Reciprocal

Consider the function $$f(x)=x+\frac{1}{x}$$ defined for $$x\geq1$$.

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 of Production Cost

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

Population Growth Rate

Suppose the population of a species is modeled by $$P(t)= 1000*e^{0.07*t}$$, where $$t$$ is measured

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

Production Rate Analysis Using a Production Function

A factory’s production is modeled by the function $$P(t)=t^2 - 5*t + 10$$, where $$P(t)$$ represents

Proof of Scaling in Derivatives

Let $$f(x)$$ be a differentiable function and let $$k$$ be a constant. Consider $$g(x)= k*f(x)$$. Us

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

River Crossover: Inflow vs. Damming

A river receives water from two tributaries at rates $$f_1(t)=7+0.5*t$$ and $$f_2(t)=9-0.2*t$$ (lite

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

Chain Rule Basics

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

Chain Rule in an Implicitly Defined Function

Consider the equation $$\tan(x+y)=x^2-y^2$$. Answer the following:

Chain Rule with Trigonometric and Exponential Functions

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

Chemical Reaction Rate: Exponential and Logarithmic Model

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

Combining Chain Rule, Implicit, and Inverse Differentiation

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

Composite Functions in Population Dynamics

The population of a species is modeled by the composite function $$P(t) = f(g(t))$$, where $$g(t) =

Composite Trigonometric Function Analysis in Pendulum Motion

A pendulum's angular displacement is modeled by the function $$\theta(t)= \sin(\sqrt{2*t+1})$$.

Differentiating an Inverse Trigonometric Function

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

Differentiation of an Inverse Trigonometric Composite Function

Consider the function $$y = \arctan(\sqrt{3x})$$.

Differentiation Under Implicit Constraints in Physics

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

Implicit and Inverse Function Analysis

Consider the function defined implicitly by $$e^{x*y}+x^2=5$$. Answer the following parts.

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 an Elliptical Orbit

Consider the ellipse defined by $$4*x^2 + 9*y^2 = 36$$, which can model the orbit of a satellite.

Implicit Differentiation in Circular Motion

A runner is moving along a circular track described by the equation $$x^2+y^2=16$$, where $$x$$ and

Implicit Differentiation of an Exponential-Product Equation

Consider the curve defined implicitly by $$e^(x*y) + x - y = 0.$$ Solve the following:

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 Mixed Terms

Consider the equation $$x*y + y^2 = 10$$. Answer the following parts.

Implicit Trigonometric Equation Analysis

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

Inverse Analysis in Exponential Decay

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

Inverse Function Derivative for a Log-Linear Function

Let $$f(x)= x+ \ln(x)$$ for $$x > 0$$ and let g be the inverse of f. Solve the following parts:

Inverse Function 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 a Mechanical System

Consider the function $$f(\theta)= 2\theta + \sin(\theta)$$ used to model an angle transformation in

Multilayer Composite Function Differentiation

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

Particle Motion: Logarithmic Position Function

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

Pendulum Angular Displacement Analysis

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

Related Rates: Shadow Length

A 1.8 m tall person is walking away from a street lamp that is 5 m tall at a speed of 1.2 m/s. Using

Second Derivative via Implicit Differentiation

Consider the ellipse given by $$\frac{x^2}{4}+\frac{y^2}{9}=1$$. Answer the following:

Second Derivative via Implicit Differentiation

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

Temperature Change Model Using Composite Functions

The temperature of an object is modeled by the function $$T(t)=e^{-\sqrt{t+2}}$$, where $$t$$ is tim

Temperature Profile and the Chain Rule

A metal rod has a temperature distribution given by $$T(x)=100*e^{-0.05*x^2}$$ (in °C), where x is t

Analysis of Wheel Rotation

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

Balloon Inflation Related Rates

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

Chemical Reaction Rate

In a chemical reaction, the concentration of a reactant is given by $$C(t)=100e^{-0.05*t}$$ mg/L, wh

Chemical Reaction Rate Analysis

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

Cooling Coffee: Temperature Rate of Change

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

Critical Points and Concavity Analysis

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

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 (

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 17: Water Heater Temperature Change

The temperature of water in a heater is modeled by $$T(t) = 20 + 80e^{-0.05*t}$$, where t is in minu

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

Inflection Points and Concavity in Business Forecasting

A company's profit is modeled by $$P(x)= 0.5*x^3 - 6*x^2 + 15*x - 10$$, where $$x$$ represents a pro

Inverse Trigonometric Analysis for Navigation

A navigation system relates the angle of elevation $$\theta$$ to a mountain with the horizontal dist

L'Hôpital's Rule Application

Evaluate the limit $$\lim_{x\to0}\frac{e^{2*x}-1}{3*x}$$.

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.

Linear Approximation for Function Values

Consider the function $$f(x) = x^3$$. Use linearization at $$x = 4$$ to approximate the value of $$f

Linearization for Approximating Square Roots

Let $$f(x)= \sqrt{x}$$. Use linearization to estimate the value of $$\sqrt{16.4}$$, using $$x=16$$ a

Optimization in a Manufacturing Process

A company designs an open-top container whose volume is given by $$V = x^2 y$$, where x is the side

Optimization of Production Costs

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

Optimization: Minimizing Material for a Box

A company wants to design an open-top box with a square base that holds 32 cubic meters. Let the bas

Optimizing Crop Yield

The yield per acre of a crop is modeled by the function $$Y(p) = 100\,p\,e^{-0.1p}$$, where $$p$$ is

Particle Motion Analysis

A particle moves along a straight line with displacement given by $$s(t)=t^3-6t^2+9t+2$$ for $$0\le

Population Growth Rate Analysis

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

Projectile Motion and Maximum Height

A basketball is thrown such that its height (in meters) is modeled by $$h(t)= -4.9*t^2 + 14*t + 1.5$

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

Temperature Change in Cooling Coffee

A cup of coffee cools according to the model $$T(t) = 90e^{-0.05t} + 20$$, where $$t$$ is the time i

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

Transcendental Function Temperature Change

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

Transformation of Logarithmic Functions

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

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

Application of Rolle's Theorem

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

Biological Growth and the Mean Value Theorem

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

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

Comprehensive Analysis of a Rational Function

Given the rational function $$f(x)= \frac{x^2-4}{x^2+1}$$, perform a comprehensive analysis includin

Concavity and Inflection Points in a Quartic Function

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

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

Determining Intervals of Increase and Decrease with a Rational Function

Consider the function $$f(x) = \frac{x^2}{x+2}$$ defined on the interval $$[0, 4]$$. Answer the foll

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

Estimating Total Revenue via Riemann Sums

A company’s marginal revenue (in thousand dollars per unit) is measured at various levels of units s

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

Exploring a Piecewise Function with Multiple Critical Points and Discontinuities

Consider the function $$ f(x) = \begin{cases} x^2 - 4, & x < -1, \\ 2x + 1, & -1 \le x < 2, \\ \sqr

FRQ 14: Projectile Motion – Determining Maximum Height

The height of a projectile (in meters) is modeled by $$h(t)= -4.9*t^2 + 20*t + 5$$, where $$t$$ is t

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

Inverse Analysis of a Function with an Absolute Value Term

Consider the function $$f(x)=x+|x-2|$$ with the domain restricted to $$x\ge 2$$. Analyze the inverse

Inverse Analysis of a Logarithm-Exponential Hybrid Function

Consider the function $$f(x)=\ln(x+2)+e^(x)$$ defined for $$x>-2$$. Address the following regarding

Investigating a Piecewise Function with a Vertical Asymptote

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

Liquid Cooling System Flow Analysis

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

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$

Pharmaceutical Drug Delivery

A drug is administered intravenously using an infusion device. The drug infusion rate is given by $$

Relationship Between Integration and Differentiation

Let $$F(x)= \int_{0}^{x} (t^2 - t + 1)\,dt$$. Explore the relationship between the integral and its

Revenue Optimization in Economics

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

Speed Limit Analysis using the Mean Value Theorem

The position of a car is given by $$s(t) = t^2 + 6*t + 5$$ (in meters) for $$t$$ in seconds, where $

Traffic Flow Modeling

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

Using Derivatives to Solve a Rate-of-Change Problem

A particle’s displacement is given by $$s(t) = t^3 - 9*t^2 + 24*t$$ (in meters), where \( t \) is in

Accumulated Chemical Concentration

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

Accumulated Rainfall Estimation

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

Accumulated Water Volume in a Tank

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

Accumulation and Total Change in a Population Model

A population grows at a rate given by $$r(t)=0.2*t^2 - t + 5$$ (in thousands per year), where t is i

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

Area Under a Curve with a Discontinuous Function

Consider the function $$h(x)= \begin{cases} x+2 & \text{if } 0 \le x < 3,\\ 7 & \text{if } x = 3,\\

Area Under a Parabola

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

Area Under a Piecewise Function

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

Computing a Definite Integral Using the Fundamental Theorem of Calculus

Let the function be defined as $$f(x) = 2*x$$. Use the Fundamental Theorem of Calculus to evaluate t

Displacement from a Velocity Function

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

Economic Revenue Analysis from Marginal Revenue Data

A company's marginal revenue (in thousands of dollars per hour) is recorded over a 4-hour production

Evaluating an Integral with U-substitution

Evaluate the integral $$\int_{1}^{3} 2*(x-1)^5\,dx$$ using u-substitution. Answer the following ques

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.

FRQ7: Inverse Analysis of an Exponential Accumulation Function

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

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

FRQ20: Inverse Analysis of a Function with a Piecewise Continuous Integrand

Consider the function $$ I(x)= \begin{cases} \int_{0}^{x}\cos(t)\,dt, & 0 \le x \le \pi/2 \\ \int_{0

Fuel Consumption for a Rocket Launch

During a rocket launch, fuel is consumed at a rate $$F_{cons}(t)=50-3t$$ kg/s while additional fuel

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

Implicit Differentiation and Integration Verification

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

Implicit Differentiation of a Conic

Consider the relation $$x^2 + x*y + y^2 = 7.$$ Answer the following parts:

Modeling Accumulated Revenue over Time

A company’s revenue rate is given by $$R(t)=100*e^{0.1*t}$$ dollars per month, where t is measured i

Optimizing Fencing Cost for a Garden Adjacent to a River

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

Rate of Drug Metabolism

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

Roller Coaster Work Calculation

An amusement park engineer recorded the force applied by a roller coaster engine (in Newtons) at var

Seismic Data Analysis: Ground Acceleration

A seismograph records ground acceleration (in m/s²) during an earthquake. Use the data in the table

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

Temperature Change in a Chemical Reaction

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

Trigonometric Integration via U-Substitution

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

Volume Accumulation in a Leaking Tank

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

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

A Separable Differential Equation: Growth Model

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

Analysis of an Autonomous Differential Equation

Consider the autonomous differential equation $$\frac{dy}{dx}=y(4-y)$$ with the initial condition $$

Charging a Capacitor in an RC Circuit

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

Charging of a Capacitor

The voltage $$V$$ (in volts) across a capacitor being charged in an RC circuit is recorded over time

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.

Chemical Reaction Rate

In a chemical reaction, the concentration $$C$$ (in mol/L) of a reactant is recorded over time as sh

Chemical Reaction Rate

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

Chemical Reaction Rate with Second-Order Decay

A chemical reaction follows the rate law $$\frac{d[A]}{dt}=-k[A]^2$$, where $$[A](t)$$ (in M) is the

Chemical Reactor Mixing

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

Drug Concentration Model

The concentration $$C(t)$$ (in mg/L) of a drug in a patient's bloodstream is modeled by the differen

Drug Elimination with Infusion

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

Epidemic Spread Modeling

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

Implicit Differentiation and Tangent Lines of an Ellipse

Consider the ellipse defined by $$4x^2+ 9y^2= 36$$. Answer the following:

Implicit Differentiation of a Circle

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

Implicit Differentiation with Trigonometric Functions

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

Integrating Factor Initial Value Problem

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

Logistic Growth Model Analysis

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

Mixing of a Pollutant in a Lake

A lake with a constant volume of $$10^6$$ m\(^3\) receives polluted water from a river at a rate of

Mixing Problem with Constant Flow

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

Mixing Problem with Time-Dependent Inflow Concentration

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

Modeling Cooling with Newton's Law

An object is cooling in a room where the ambient temperature remains constant at $$20^\circ C$$. The

Modeling Orbital Decay

A satellite’s altitude $$h(t)$$ decreases over time due to atmospheric drag, following $$\frac{dh}{d

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 with Variable Ambient Temperature

An object is cooling according to Newton's Law of Cooling, but the ambient temperature is not consta

Non-Separable to Linear DE

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

Nonlinear Differential Equation

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

Nonlinear Differential Equation with Powers

Consider the differential equation $$\frac{dy}{dx} = 4*y^{3/2}$$ with the initial condition $$y(1)=1

Pollutant Concentration in a Lake

A lake receives a constant pollutant input so that the concentration $$C(t)$$ (in mg/L) satisfies th

Population Model with Harvesting

A fish population in a lake is modeled by the differential equation $$\frac{dP}{dt}=0.3*P\left(1-\fr

Qualitative Analysis of a Nonlinear Differential Equation

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

Radioactive Decay Differential Equation

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

Radioactive Isotope in Medicine

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

RC Circuit Discharge

In an RC circuit, the voltage across a capacitor decays according to $$\frac{dV}{dt}=-\frac{1}{RC}V$

Related Rates: Conical Tank Filling

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

Slope Field Analysis for $$dy/dx = x$$

Consider the differential equation $$dy/dx = x$$. A slope field representing this equation is provid

Solving a Differential Equation Using the SIPPY Method

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

Stability and Phase Line Analysis

Consider the autonomous differential equation $$\frac{dy}{dt}=y(4-y)(y+2)$$.

Tank Mixing and Salt Concentration

A tank initially contains 100 L of solution with 5 kg of dissolved salt. A salt solution with concen

Tumor Growth with Allee Effect

The growth of a tumor is modeled by the differential equation $$\frac{dP}{dt}=rP\left(1-\frac{P}{K}\

Volume by Revolution of a Differential Equation Derived Region

The function $$y(x) = e^{-x} + x$$, which is a solution to a differential equation, and the line $$y

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 a Parabola and a Line

Consider the functions $$f(x)=5*x-x^2$$ and $$g(x)=x$$. A set of experimental data provided the foll

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

Average Concentration Calculation

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

Average Flow Rate in a River

The flow rate of a river (in $$m^3/s$$) is measured over a 12-hour period. Use the data provided in

Average Value and the Mean Value Theorem

For the function $$f(x)=\cos(x)$$ on the interval [0, $$\pi/2$$], compute the average value and find

Average vs. Instantaneous Value of a Function

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

Boat Navigation Across a River with Current

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

Cost Analysis Through Area Between Curves

A company analyzes two different manufacturing cost models represented by the curves $$C_1(x)=50+3*x

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

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

Free Workout Class Attendance

The attendance at a free workout class increases by a fixed number of people each session. The first

Hollow Rotated Solid

Consider the region in the first quadrant bounded by the curves $$y=x$$ and $$y=x^2$$. This region i

Implicit Differentiation in Thermodynamics

In a thermodynamics experiment, the pressure $$P$$ and volume $$V$$ of a gas are related by the equa

Integrated Motion Analysis

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

Investment Compound Interest

An investment account starts with an initial deposit of $$1000$$ dollars and earns $$5\%$$ interest

Loaf Volume Calculation: Rotated Region

Consider the region bounded by $$y=\sqrt{x}$$ and $$y=\frac{x}{2}$$ for 0 ≤ x ≤ 4. This region is ro

Manufacturing Profit with Variable Rates

A manufacturer’s profit rate as a function of time (in hours) is given by $$P(t)=100\left(1-e^{-0.2*

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

Particle Motion and Integrated Functions

A particle has acceleration given by $$a(t)=2+\cos(t)$$ (in m/s²) for $$t \ge 0$$. At time $$t=0$$,

Particle Motion on a Line

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

Population Accumulation through Integration

A town’s rate of population growth is modeled by $$r(t)=500*e^{-0.2*t}$$ (people per year), where $$

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

Population Growth with Variable Growth Rate

A city's population changes with time according to a non-constant growth rate given in thousands per

Rebounding Ball

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

Surface Area of a Rotated Curve

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

Tank 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 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 Rotated about a Line

Consider the region bounded by $$y=x^2$$ and $$y=x$$ for $$x\in [0,1]$$. This region is rotated abou

Washer Method with Logarithmic and Exponential Curves

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

Work in Pumping Water from a Conical Tank

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

Work in Spring Stretching

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