AP Calculus AB FRQ Room

Algebraic Manipulation and Limit Evaluation

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

Analyzing a Piecewise Function for Continuity

Consider the piecewise function $$ f(x)=\begin{cases} 2x+1, & x<2 \\ x^2-1, & x\geq2 \end{cases}$$.

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 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 Multiple Discontinuities in a Rational Function

Let $$f(x)= \frac{(x^2-9)(x+4)}{(x-3)(x^2-16)}$$.

Application of the Squeeze Theorem

Consider the function defined by $$h(x)=\begin{cases} x^2 \sin\left(\frac{1}{x}\right) & \text{if }

Area and Volume Setup with Bounded Regions

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

Asymptotic Analysis of a Radical Rational Function

Consider the function $$f(x)=\sqrt{4x^2+x}-2x$$ for \(x>0\). Answer the following:

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 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 of Constant Functions

Consider the constant function $$f(x)=7$$ for all x. Answer the following parts.

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:

Determining Parameters for a Continuous Log-Exponential Function

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

Determining Parameters for Continuity in a Piecewise Function

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

End Behavior of Rational Functions

Examine the rational function $$f(x)=\frac{3*x^3-2*x+1}{6*x^3+4*x^2-5}$$. Determine its behavior as

Evaluating Limits Involving Square Roots

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

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

Limit Involving a Square Root and Removable Discontinuity

Consider the function $$h(x)=\frac{\sqrt{x+4}-2}{x}$$ for $$x\neq0$$ and $$h(0)=1$$. Answer the foll

Limits at Infinity and Horizontal Asymptotes

Consider the rational function $$R(x) = \frac{2x^2 - 3x + 4}{x^2 + 5}$$. Analyze its behavior as x a

Limits at Infinity for Non-Rational Functions

Consider the function $$ h(x)=\frac{2*x+3}{\sqrt{4*x^2+7}} $$.

Limits Involving a Removable Discontinuity

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

Limits Near Vertical Asymptotes

Consider the function $$f(x) = \frac{1}{x - 4}$$. (a) Determine $$\lim_{x \to 4^-} f(x)$$. (b) Dete

Limits of Absolute Value Functions

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

Modeling Bacterial Growth with a Geometric Sequence

A particular bacterial colony doubles in size every hour. The population at time $$n$$ hours is give

One-Sided Limits and an Absolute Value Function

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

Optimization and Continuity in a Manufacturing Process

A company designs a cylindrical can (without a top) for which the cost function in dollars is given

Oscillatory Behavior and Non-Existence of Limit

Let \(f(x)=\sin(1/x)\) for \(x\neq0\). Answer the following:

Particle Motion with Squeeze Theorem Application

A particle moves along a line with velocity given by $$v(t)= t^2 \sin(1/t)$$ for $$t>0$$ and is defi

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

Squeeze Theorem Application

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

Squeeze Theorem for an Oscillatory Function

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

Vertical Asymptote Analysis

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

Analysis of Motion in the Plane

A particle moves in the plane with its position given by $$\mathbf{s}(t)=\langle t^2 - 4*t,\, 3*t +

Analyzing the Derivative of a Trigonometric Function

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

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

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

Derivative from First Principles

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

Differentiating a Product of Linear Functions

Let $$f(x) = (2*x^2 + 3*x)\,(x - 4)$$. Use the product rule to find $$f'(x)$$.

Finding the Derivative using the Limit Definition

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

Graphical Estimation of a Derivative

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

Higher-Order Derivatives in Motion

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

Instantaneous Acceleration from a Velocity Function

A runner's velocity is given by $$v(t)= 3*t^2 - 12*t + 9$$, where $$t$$ is in seconds. Analyze the r

Inverse Function Analysis: Cubic Transformation

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

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: Trigonometric Function with Linear Term

Consider the function $$f(x)=x+\sin(x)$$ defined on the interval $$\left[-\frac{\pi}{2},\frac{\pi}{2

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 $$

Medication Infusion with Clearance

A patient receives medication via an IV at a rate of $$f(t)=5*e^{-0.1*t}$$ mg/min, while the body cl

Optimizing Car Speed: Rate of Change Analysis

A car’s speed in km/h is modeled by the function $$s(t)=50+2*t^2-0.1*t^3$$ for $$0 \leq t \leq 10$$

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

Product Rule Application in Economics

A company's cost function for producing $$x$$ units is given by $$C(x)= (3*x+2)*(x^2+5)$$ (cost in d

Projectile Motion Analysis

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

RC Circuit Voltage Decay

An RC circuit's capacitor voltage is modeled by $$V(t)= V_{0}*e^{-t/(R*C)}$$, where $$V_{0}$$ is the

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

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

Secant Slope from Tabulated Data

A table below gives values of a function $$f(x)$$ representing the concentration of a solution at di

Tangent Line Equation for an Exponential Function

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

Using Derivative Rules on a Trigonometric Function

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

Using the Difference Quotient with a Polynomial Function

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

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 and Product Rules in a Rate of Reaction Process

In a chemical reaction, the rate is modeled by the function $$R(t)= \sqrt{t+3}*\cos(2*t)$$, where $$

Chain Rule in an Economic Model

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

Chain Rule in Population Modeling

A biologist models the population of a species with the function $$P(t)= f(g(t))$$, where $$g(t)=25*

Chain Rule in Temperature Model

A scientist models the temperature in a laboratory experiment by the function $$T(t)=\sqrt{3*t^2+2}$

Chain Rule with Trigonometric and Exponential Functions

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

Chain Rule with Trigonometric Function

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

Composite Function in Biomedical Model

The concentration C(t) (in mg/L) of a drug in the bloodstream is modeled by $$C(t) = \sin(3*t^2)$$,

Differentiation of an Inverse Trigonometric Composite Function

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

Differentiation of Inverse Function with Polynomial Functions

Let \(f(x)= x^3+2*x+1\) be a one-to-one function. Its inverse is denoted by \(f^{-1}\).

Estimating Derivatives Using a Table

An experiment measures a one-to-one function $$f$$ and its inverse $$g$$, yielding the following dat

Implicit Differentiation for an Ellipse

Consider the ellipse defined by the equation $$\frac{x^2}{9} + \frac{y^2}{4} = 1$$. This equation re

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 a Financial Model

An implicit relationship between revenue $$R$$ (in thousands of dollars) and price $$p$$ (in dollars

Implicit Differentiation in an Economic Model

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

Implicit Differentiation in an Ellipse

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

Implicit Differentiation in Circular Motion

Given the circle defined by $$x^2 + y^2 = 16$$, analyze its differential properties.

Implicit Differentiation Involving Logarithms

Consider the equation $$\ln(x) + x*y = \ln(y) + x$$ which relates $$x$$ and $$y$$. Use implicit diff

Implicit Differentiation of a Logarithmic-Exponential Equation

Consider the equation $$\ln(x+y) + e^{x*y} = 7$$, which implicitly defines $$y$$ as a function of $$

Implicit Differentiation of an Exponential-Product Equation

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

Implicit Differentiation on a Circle

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

Implicit Differentiation with Product Rule

Consider the implicit equation $$x*y+y^2=6$$ which defines $$y$$ as a function of $$x$$. Use implici

Implicit Differentiation with Product Rule

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

Implicit Differentiation with Trigonometric Components

Consider the equation $$\sin(x) + \cos(y) = x*y$$, which implicitly defines $$y$$ as a function of $

Implicitly Defined Inverse Relation

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

Inverse Derivative of a Sum of Exponentials and Linear Terms

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

Inverse Function 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 Trigonometric Function Differentiation

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

Pendulum Angular Displacement Analysis

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

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

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

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.

Chemical Reaction Rate Analysis

A chemical reaction follows the concentration model $$c(t)=\frac{100}{1+5e^{-0.3t}}$$, where c is in

Cost Function Optimization

A company’s cost is modeled by the function $$C(x)=0.5x^3-6x^2+20x+100$$, where x (in hundreds of un

Dynamics of a Car: Stopping Distance and Deceleration

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

Economic Cost Analysis Using Derivatives

A company’s cost function for producing $$x$$ units is given by $$C(x)=0.05*x^3 - 2*x^2 + 40*x + 100

Expanding Circular Ripple in a Pond

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

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 16: Implicit Differentiation in Orbital Mechanics

A satellite’s orbit is described by the equation $$x^2 + 2*x*y + y^2 = 25$$, where x and y represent

FRQ 20: Market Demand Analysis

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

Graphing a Function via its Derivative

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

Growth Rate Estimation in a Biological Experiment

In a biological experiment, the mass $$M(t)$$ (in grams) of a bacteria colony is recorded over time

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

Inflation of a Balloon: Surface Area Rate of Change

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

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

Interpretation of the Derivative from Graph Data

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

Linearization and Differentials

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

Linearization of a Nonlinear Function

Suppose $$f(x)=\ln(x)$$. Use linearization about $$x=4$$ to approximate $$\ln(4.1)$$. Answer the fol

Local Linearization Approximation

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

Minimizing Materials for a Cylindrical Can

A manufacturer aims to design a closed cylindrical can that holds exactly $$500$$ cubic centimeters

Optimization: Minimizing Surface Area of a Box

An open-top box with a square base is to have a volume of 500 cubic inches. The surface area (materi

Particle Acceleration and Direction of Motion

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

Population Change Rate

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

Population Growth Rate Analysis

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

Reaction Rate and Temperature

The rate of a chemical reaction is modeled by $$r(T)= 0.5*e^{-0.05*T}$$, where $$T$$ is the temperat

Related Rates in Shadows: A Lamp and a Tree

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

Related Rates: Expanding Circle

A circular pool is being filled such that its surface area increases at a constant rate of $$10$$ sq

Related Rates: Expanding Circular Ripple

A circular ripple on a calm water surface is expanding such that its area is increasing at a rate of

Related Rates: Shadow Length

A 1.8-meter tall person is walking away from a 4.5-meter tall streetlight at a constant speed of 1.2

Revenue and Cost Analysis

A company’s revenue is modeled by $$R(t)=200e^{0.05t}$$ and its cost by $$C(t)=10t^3-30t^2+50t+200$$

Route Optimization for a Rescue Boat

A rescue boat must travel from a point on the shore to an accident site located 2 km along the shore

Seasonal Water Reservoir

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

Temperature Cooling in a Cup of Coffee

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

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

Analyzing a Piecewise Function and Differentiability

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

Approximating Displacement from Velocity Data

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

Area Bounded by $$\sin(x)$$ and $$\cos(x)$$

Consider the functions $$f(x)= \sin(x)$$ and $$g(x)= \cos(x)$$ on the interval $$[0, \frac{\pi}{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:

Chemical Mixing in a Tank

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

Chemical Reaction Rate and Exponential Decay

In a chemical reaction, the concentration of a reactant declines according to $$C(t)= C_0* e^{-k*t}$

Concavity and Inflection Points of a Cubic Function

Consider the cubic function $$f(x)=x^3-6*x^2+9*x+2$$. Answer the following questions regarding its d

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

Drag Force and Rate of Change from Experimental Data

Drag force acting on an object was measured at various velocities. The table below presents the expe

Exponential Bacterial Growth

A bacterial culture grows according to $$P(t)= P_0 * e^{k*t}$$, where $$t$$ is in hours. The culture

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 11: Particle Motion with Non-Constant Acceleration

A particle moves along a straight line with acceleration given by $$a(t)= 12*t - 6$$ (in m/s²). If t

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 16: Finding Relative Extrema for a Logarithmic Function

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

FRQ 19: Analysis of an Exponential-Polynomial Function

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

Garden Fence Optimization Problem

A rectangular garden is to be built adjacent to a building. Fencing is required on only three sides

Graphical Analysis and Derivatives

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

Implicit Differentiation and Tangent Lines

Consider the curve defined implicitly by the equation $$x^2 + x*y + y^2= 7$$.

Inverse Analysis of a Linear Function

Consider the function $$f(x)=3*x+2$$. Analyze its inverse function by answering all parts below.

Inverse Analysis: Transformation Geometry of a Parabolic Function

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

Limit Analysis of a Piecewise Function Involving a Rational Expression

Consider the function $$ f(x) = \begin{cases} \frac{2x^2-8}{x-2}, & x < 2, \\ x+2, & x \ge 2. \end{

Liquid Cooling System Flow Analysis

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

Logarithmic Transformation of Data

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

Logistic Population Model Analysis

Consider the logistic model $$P(t)= \frac{500}{1+ 9e^{-0.4t}}$$, where $$t$$ is in years. Answer the

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{

Polynomial Rational Discontinuity Investigation

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

Population Growth Analysis via the Mean Value Theorem

A country's population data over a period of years is given in the table below. Use the data to anal

Production Cost Optimization and the Extreme Value Theorem

A company monitored its production cost as a function of units produced. The following table gives e

Profit Analysis and Inflection Points

A company's profit is modeled by $$P(x)= -x^3 + 9*x^2 - 24*x + 10$$, where $$x$$ represents thousand

Profit Function Concavity Analysis

A company’s profit is modeled by $$P(x) = -2*x^3 + 18*x^2 - 48*x + 10$$, where $$x$$ is measured in

Radioactive Substance Decay

A radioactive substance decays according to the model $$A(t)= A_0 * e^{-\lambda*t}$$, where $$t$$ is

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

Relative Extrema of a Rational Function

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

Sand Pile Dynamics

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

Traffic Flow Modeling

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

Urban Water Supply Management

An urban water supply system receives water from two sources. The inflow rates are $$R_1(t)=15+2*t$$

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

Accumulated Change Function Evaluation

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

Accumulation Function and Its Derivative

Define the function $$F(x)= \int_0^x \Big(e^{t} - 1\Big)\,dt$$. Answer the following parts related t

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

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

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 Value of a Function

The temperature over the first 8 hours of a day is modeled by $$T(t)=-0.5*t^2 + 4*t + 10$$, where t

Chemical Reactor Conversion Process

In a chemical reactor, the instantaneous reaction rate is given by $$R(t)=4t^2-t+3$$ mol/min, while

Definite Integral Approximation Using Riemann Sums

Consider the function $$f(x)= x^2 + 3$$ defined on the interval $$[2,6]$$. A table of sample values

Electric Charge Accumulation

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

Elevation Profile Analysis on a Hike

A hiker records the elevation (in meters) along a trail at various distances. Use this data to analy

Evaluating a Trigonometric Integral Using U-Substitution

Evaluate the integral $$\int_{0}^{\frac{\pi}{2}} \sin(2*x)\,dx$$ using u-substitution.

Evaluating an Integral with a Trigonometric Function

Evaluate the definite integral $$\int_{0}^{\frac{\pi}{2}} \cos(x)*\sin(x)\,dx$$ using an appropriate

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

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 \

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

Implicit Differentiation and Integration Verification

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

Integration by U-Substitution in Physics

Consider the integral $$I=\int_0^4 \frac{t}{\sqrt{4+t^2}}\,dt.$$ This integral arises in determining

Net Surplus Calculation

A consumer's satisfaction is given by $$S(x)=100-4*x^2$$ and the marginal cost is given by $$C(x)=30

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 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

Roller Coaster Work Calculation

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

Total Distance from Velocity Data

A car’s velocity, in meters per second, is recorded over time as given in the table below: | Time (

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

Trigonometric Integral with U-Substitution

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

Volume of a Solid by Washer Method

A region is bounded by the curves $$y=\sqrt{x}$$ and $$y=\frac{x}{2}$$. This region, between the cur

Volume of a Solid with Square Cross Sections

A solid has a base in the xy-plane bounded by the curves $$y=x$$ and $$y=x^2$$. Cross sections perpe

Water Accumulation in a Tank

Water flows into a tank at a rate given by $$R(t)=2*\sqrt{t}$$ (in m³/min) for t in minutes. Answer

Water Tank: Accumulation and Maximum Level

A water tank is being filled with water at a rate $$r_{in}(t) = 4 + \sin(t)$$ L/min and is simultane

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

Carbon Dating and Radioactive Decay

Carbon dating is based on the radioactive decay model given by $$\frac{dC}{dt}=-kC$$. Let the initia

Chemical Reaction Rate

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

Chemical Reaction Rate and Concentration Change

The rate of a chemical reaction is described by the differential equation $$\frac{dC}{dt}=-0.3*C^2$$

Cooling with Variable Ambient Temperature

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

Drug Concentration Model

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

Drug Concentration with Continuous Infusion

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

Falling Object with Air Resistance

A falling object experiences air resistance proportional to its velocity. Its motion is modeled by t

Fishery Harvesting Model

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

Implicit Solution of a Differential Equation

The differential equation $$\frac{dy}{dx} = \frac{2x}{1+y^2}$$ requires an implicit solution.

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

A population $$P(t)$$ grows according to the logistic model $$\frac{dP}{dt} = rP\left(1-\frac{P}{K}\

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

Motion Under Gravity with Air Resistance

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

Particle Motion in the Plane

A particle moving in the plane has a constant x-component velocity of $$v_x(t)=2$$ m/s, and its y-co

Radioactive Decay

A radioactive substance decays according to $$\frac{dN}{dt} = -\lambda N$$. Initially, there are 500

Radioactive Decay

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

Radioactive Decay

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

Radioactive Decay and Half-Life

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

Radioactive Material with Constant Influx

A laboratory receives radioactive waste material at a constant rate of $$3$$ g/day. Simultaneously,

Separable Differential Equation with Initial Condition

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

Separable Differential Equation with Trigonometric Component

Solve the differential equation $$\frac{dy}{dx}=\frac{3x^2}{1+\sin(y)}$$ with the initial condition

Separable Differential Equation: $$dy/dx = x*y$$

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

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 $$dy/dx = x$$

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

Temperature Regulation in a Greenhouse

The temperature $$T$$ (in °F) inside a greenhouse is recorded over time (in hours) as shown. The war

Traffic Flow Dynamics

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

Water Temperature Regulation in a Reservoir

A reservoir’s water temperature adjusts according to Newton’s Law of Cooling. Let $$T(t)$$ (in \(^{\

Accumulated Rainfall Calculation

During a 12-hour storm, the rainfall rate is modeled by $$r(t)=3+\sin(t)$$ (in mm/hr) for $$0 \le t

Area Between Cost Functions in a Business Analysis

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

Area Between Curves 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 with Variable Limits

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

Average and Instantaneous Rates in a Cooling Process

A cooling process is modeled by the function $$T(t)= 100*e^{-0.05*t}$$ (in degrees Fahrenheit), wher

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 Speed from a Velocity Function

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

Average Temperature Analysis

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

Average Temperature Analysis

A local weather station recorded the temperature throughout a day using the model $$T(t)=-0.5*t+35$$

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 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 Value of a Function in a Production Process

A factory machine's temperature (in $$^\circ C$$) during a production run is modeled by $$T(t)= 5*t

Car Braking Analysis

A car decelerates with acceleration given by $$a(t)=-4e^{-t/2}$$ (in m/s²) and has an initial veloci

Cooling Process Analysis

A cup of coffee cools in a room, and its temperature (in °C) is modeled by $$T(t)=30*e^{-0.1*t}+5$$

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\

Cost Optimization for a Cylindrical Container

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

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

Medication Dosage Increase

A patient receives a daily medication dose that increases by a fixed amount each day. The first day'

Motion Analysis Using Integration of a Sinusoidal Function

A car has velocity given by $$v(t)=3*\sin(t)+4$$ (in $$m/s$$) for $$t \ge 0$$, and its initial posit

Particle Motion with Exponential Acceleration

A particle moves along a straight line with acceleration given by $$a(t)=2*e^{-t} - 1$$ (in m/s²) fo

Population Growth Rate Analysis

Suppose the instantaneous growth rate of a population is given by $$r(t)=0.04 - 0.002*t$$ for $$t \i

Population Growth with Variable Growth Rate

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

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 $$

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

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

Temperature Average Calculation

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

Volume with Square Cross-Sections

Consider the region bounded by the curve $$y=x^2$$ and the line $$y=4$$ for $$0 \le x \le 2$$. Squar

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