AP Calculus AB FRQ Room

Advanced Analysis of a Piecewise Function

Consider the function $$f(x)=\begin{cases} x^2*\sin\left(\frac{1}{x}\right) & x\neq 0 \\ 0 & x=0 \en

Advanced Analysis of an Oscillatory Function

Consider the function $$ f(x)= \begin{cases} \sin(1/x), & x\ne0 \\ 0, & x=0 \end{cases} $$.

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

Arithmetic Sequence in Temperature Data and Continuity Correction

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

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 and Limit Comparison for Two Particle Paths

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

Continuity at Zero for a Trigonometric Function

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

Continuity of Constant Functions

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

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

Factorization and Limit Evaluation

Consider the function $$f(x) = \frac{x^3 - 8}{x - 2}$$. (a) Factor the numerator and simplify the e

Graph-Based Analysis of Discontinuities

Examine the graph of a function $$ f(x) $$ depicted in the stimulus. The graph shows the function fo

Graph-Based Analysis of Discontinuity

Examine the graph of a function that appears to be defined by $$f(x)= 3x - 2$$ for all $$x \neq 2$$,

Graphical Estimation of a Limit

The following graph shows the function $$f(x)$$. Use the graph to answer the subsequent questions re

Horizontal Asymptote of a Rational Function

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

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 in Context

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

Limits at Infinity for Non-Rational Functions

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

Mixed Function with Jump Discontinuity at Zero

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

Oscillatory Behavior and Discontinuity

Consider the function $$f(x)=\begin{cases} x\cos(\frac{1}{x}) & x\neq0 \\ 2 & x=0 \end{cases}$$. Ans

Parameters for Continuity

Consider the function $$f(x)=\begin{cases} a*x^2+3, & x \le 2 \\ b*x+5, & x > 2 \end{cases}$$ Dete

Piecewise Function with Different Expressions

Consider the function $$f(x)=\begin{cases} 3*x-1 & x<2\\ \frac{x^2-4}{x-2} & x>2\\ 5 & x=2 \end{case

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 and Redefinition

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

Removal of Discontinuity by Redefinition

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

Squeeze Theorem Application

Let $$f(x)=x^2\sin(1/x)$$ for \(x\neq 0\) and define \(f(0)=0\). Use the Squeeze Theorem to complete

Squeeze Theorem for an Oscillatory Function

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

Vertical Asymptote and End Behavior

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

Water Tank Inflow-Outflow Analysis

Consider a water tank operating over the time interval $$0 \le t \le 12$$ minutes. The water inflow

Approximating Derivatives Using Secant Lines

For the function $$f(x)=\ln(x)$$, we want to approximate the derivative at $$x=3$$ using secant line

Approximating the Tangent Slope

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

Derivative from First Principles

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

Derivative of a Trigonometric Function

Let \(f(x)=\sin(2*x)\). Answer the following parts.

Derivatives on an Ellipse

The ellipse given by $$\frac{x^2}{9} + \frac{y^2}{4} = 1$$ represents a race track. Answer the follo

Differentiating an Absolute Value Function

Consider the function $$f(x)= |3*x - 6|$$.

Differentiation of Exponential Functions

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

DIY Rainwater Harvesting System

A household's rainwater harvesting system collects rain at a rate of $$f(t)=12-0.5*t$$ (liters/min)

Event Ticket Sales Dynamics

For a popular concert, tickets are sold at a rate of $$f(t)=100-3*t$$ (tickets/hour) while cancellat

Exploring the Difference Quotient for a Trigonometric Function

Consider the trigonometric function $$f(x)= \sin(x)$$, where $$x$$ is measured in radians. Use the d

Exponential Decay Analysis

A radioactive substance decays according to the function $$N(t)=N_0 \cdot e^{-0.03t}$$, where t is m

Finding Derivatives with Product and Quotient Rule

Let $$f(x)=\sin(x)*\frac{x^2+1}{x}$$ for $$x \neq 0$$. Answer the following questions:

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

Graph vs. Derivative Graph

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

Graphical Analysis of Secant and Tangent Slopes

A function $$f(x)$$ is represented by the red curve in the graph below. Answer the following questio

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 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 $$

Interpreting Derivative Graphs and Tangent Lines

A graph of the function $$f(x)=x^2 - 2*x + 1$$ along with its tangent line at $$x=2$$ is provided. A

Inverse Function Analysis: Hyperbolic-Type Function

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

Inverse Function Analysis: Logarithmic Transformation

Consider the function $$f(x)=\ln(2*x+5)$$ defined for $$x> -\frac{5}{2}$$.

Inverse Function Analysis: Sum with Reciprocal

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

Limit Definition for a Quadratic Function

For the function $$h(x)=4*x^2 + 2*x - 7$$, answer the following parts using the limit definition of

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

Piecewise Function and Discontinuities

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

Population Growth 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

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

Quotient Rule Application

Let $$f(x)= \frac{e^{x}}{x+1}$$, a function defined for $$x \neq -1$$, which involves both an expone

Quotient Rule Challenge

For the function $$f(x)= \frac{3*x^2 + 2}{5*x - 7}$$, find the derivative.

Rate of Chemical Reaction

The concentration of a reactant in a chemical reaction is modeled by \(C(t)=10*e^{-0.3*t}\), where \

Rates of Change from Experimental Data

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

Real-World Cooling Process

In an experiment, the temperature (in °C) of a substance as it cools is modeled by $$T(t)= 30*e^{-0.

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

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 Approximation Convergence and the Derivative

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

Slope of a Tangent Line from Experimental Data

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

Tangent Line and Instantaneous Rate at a Point with a Radical Function

Consider the function $$f(x)= (x+4)^{1/2}$$, which represents a physical measurement (with the domai

Temperature Change Analysis

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

Using the Limit Definition to Derive the Derivative

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

Advanced Implicit and Inverse Function Differentiation on Polar Curves

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

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 Inverse Trigonometric Function

Let $$f(x)=\arcsin(\sqrt{x})$$. Answer the following:

Composite Function from an Implicit Equation

Consider the implicit equation $$x^2 + y^2 + x*y = 7$$, which defines $$y$$ as an implicit function

Composite Function with Logarithm and Trigonometry

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

Composite Function with Nested Exponential and Trigonometric Terms

Let $$f(x)= e^{\sin(4*x)}$$. This function combines exponential and trigonometric elements.

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 Log-Exponential Function Analysis

A function is defined by $$f(x)=\ln\left(e^(2*x^2) + 5\right)$$. This function is composed of an exp

Composite Temperature Model

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

Concavity Analysis of an Implicit Curve

Consider the relation $$x^2+xy+y^2=7$$.

Differentiation of a Composite Inverse Trigonometric-Log Function

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

Differentiation of an Inverse Trigonometric Composite Function

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

Differentiation of Inverse Trigonometric Function via Implicit Differentiation

Let $$y=\arcsin(\frac{x}{\sqrt{2}})$$. Answer the following:

Exponential Form and Chain Rule Complexity

Define $$Q(x)=(\cos(x))^{\sin(x)}$$. Hint: Express Q(x) as an exponential function.

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 Differentiation in a Financial Model

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

Implicit Differentiation in a Population Growth Model

Consider the model $$e^{x*y} + x - y = 5$$ that relates time \(x\) to a population scale value \(y\)

Implicit Differentiation with Exponential-Trigonometric Functions

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

Implicit Differentiation with Mixed Trigonometric and Algebraic Components

Consider the equation $$x^2+\sin(y)=y^2$$. 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 $

Intersection of Curves via Implicit Differentiation

Two curves are defined by the equations $$y^2= 4*x$$ and $$x^2+ y^2= 10$$. Consider their intersecti

Inverse Function Derivative in Thermodynamics

A thermodynamic process is modeled by the function $$P(V)= 3*V^2 + 2*V + 5$$, where $$V$$ is the vol

Inverse Function Derivative with Composite Functions

Consider the function $$f(x)=x^3+2*x+1$$, which is one-to-one on its domain. Given that $$f(1)=4$$,

Inverse Function Differentiation for a Log Function

Let $$f(x)=\ln(3*x+2)$$, and assume that $$f$$ is invertible with inverse function $$g$$. Find the d

Inverse Function Differentiation in an Exponential Model

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

Inverse Function Differentiation with an Exponential-Linear Function

Let $$f(x)=e^{2*x}+x$$ and assume it is invertible. Answer the following:

Inverse Function Differentiation with Exponentials and Trigonometry

Let $$f(x)=\sin(e^{x})+e^{x}$$ and assume that it is invertible. Answer the following:

Inverse Function in Currency Conversion

A function converting dollars to euros is given by $$f(d) = 0.9*d + 10\ln(d+1)$$ for $$d > 0$$. Let

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

Related Rates of a Shadow

A 2 m tall lamp post casts a shadow from a person who is 1.8 m tall. The person is moving away from

Second Derivative via Implicit Differentiation

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

Analysis of a Composite Function involving Logarithm

The revenue function is given by $$R(x)= x\ln(100/x)$$ for x > 0, where x is the number of units sol

Bacterial Growth Analysis

The number of bacteria in a culture is given by $$P(t)=500e^{0.2*t}$$, where $$t$$ is measured in ho

Chemical Reaction Rate Analysis

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

Complex Piecewise Function Analysis

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

Cost Analysis through a Rational Function

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

Cost Efficiency in Production

A firm's cost function for producing $$x$$ items is given by $$C(x)=0.1*x^2 - 5*x + 200$$. Analyze t

Defect Rate Analysis in Manufacturing

The defect rate in a manufacturing process is modeled by $$D(t)=100e^{-0.05t}+5$$ defects per day, w

Designing a Flower Bed: Optimal Shape

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

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

Economic Cost Function Linearization

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

Elasticity of Demand Analysis

A product’s demand function is given by $$Q(p) = 150 - 10p + p^2$$, where $$p$$ is the price, and $$

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 Small Changes using Differentials

In an electrical circuit, the resistance is given by $$R(x) = \frac{1}{1+x^2}$$, where x is a parame

Evaluating Indeterminate Limits via L'Hospital's Rule

Scientists are studying the limit of the function $$L(x)=\frac{5x^3-4x^2+1}{7x^3+2x-6}$$ as $$x \to

Exponential Decay in Radioactive Material

A radioactive substance decays according to $$M(t)=M_0e^{-0.07t}$$, where $$M(t)$$ is the mass remai

Falling Object Analysis

An object is dropped from a height and its position is modeled by $$s(t)=100-4.9t^2$$ (in meters), w

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

Kinematics on a Straight Line

A particle moves along a straight line with a position function $$s(t)=t^3 - 6*t^2 + 9*t + 2$$, wher

L'Hôpital's Rule in Analysis of Limits

Consider the limit $$L = \lim_{x\to \infty} \frac{5*x^3 - 4*x^2 + 1}{7*x^3 + 2*x - 6}$$. Use L'Hôpit

Linear Approximation of ln(1.05)

Let $$f(x)=\ln(x)$$. Use linearization at x = 1 to approximate the value of $$\ln(1.05)$$.

Logarithmic Differentiation in Exponential Functions

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

Logarithmic Profit Optimization

A company’s profit is modeled by $$P(x) = 50x \ln(x) - 100x$$, where $$x$$ (in thousands) is the num

Marginal Analysis in Economics

A company’s cost function is given by $$C(x)=0.5*x^3 - 3*x^2 + 5*x + 8$$, where $$x$$ represents the

Motion Analysis from Velocity Function

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

Optimization 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

Optimizing Water Flow in a Tank

A water tank is being filled with water. The volume $$V(t)$$ (in cubic meters) at time $$t$$ (in min

Projectile Motion Analysis

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

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$

Rate of Change in a Population Model

A population model is given by $$P(t)=30e^{0.02t}$$, where $$P(t)$$ is the population in thousands a

Rate of Change of Temperature

The temperature of a cooling liquid is modeled by $$T(t)= 100*e^{-0.5*t} + 20$$, where $$t$$ is in m

Reaction Rates in Chemistry

In a chemical reaction, the concentration of a reactant is modeled by $$C(t)=50e^{-0.3*t}+10$$, wher

Related Rates in a Spherical Balloon

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

Revenue and Cost Analysis

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

Revenue Sensitivity to Advertising

A firm's revenue (in thousands of dollars) is given by $$R(t)=50\sqrt{t+1}$$, where $$t$$ represents

Runner’s Speed Analysis

During a sprint, a runner's distance from the starting line is modeled by $$d(t)=-2t^2+12t$$, where

Savings Account Growth Modeled by a Geometric Sequence

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

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:

Application of Rolle's Theorem to a Trigonometric Function

Consider the function $$f(x) = \sin(x)$$ defined on the interval $$[0, \pi]$$. Answer the following:

Average Rate of Change from Experimental Data

A scientist recorded the temperature of a chemical reactor over time. The table below shows temperat

Behavior Analysis of a Logarithmic Function

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

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

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 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

Concavity and Inflection Points in a Quartic Function

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

Discontinuity and Derivative in a Hybrid Piecewise Function

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

FRQ 3: Relative Extrema for a Cubic Function

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

Inverse Analysis of a Function with Square Root and Linear Term

Consider the function $$f(x)=\sqrt{3*x+1}+x$$. Answer the following questions regarding its inverse.

Investigating Limits and Discontinuities in a Rational Function with Complex Denominator

Consider the function $$ f(x) = \begin{cases} \frac{x^2-9}{x-3}, & x < 3, \\ \frac{x^2-9}{x-3} + 4,

Investigating the Behavior of a Composite Function

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

Liquid Cooling System Flow Analysis

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

Motion Analysis with Acceleration Function

A particle moves along a straight line with an acceleration given by $$a(t) = 6 - 4*t$$ (in m/s²). G

Optimizing a Cylindrical Water Tank

A cylindrical water tank without a top is to be built with a fixed surface area of 100 m². Let $$r$$

Projectile Motion and Derivatives

A projectile is launched so that its height is given by $$h(t) = -4.9*t^2 + 20*t + 1$$, where $$t$$

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

Sand Pile Dynamics

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

Tangent Line and MVT for ln(x)

Consider the function $$f(x)= \ln(x)$$ on the interval $$[1,e]$$.

Temperature Change and the Mean Value Theorem

A temperature model for a day is given by $$T(t)= 2*t^2 - 3*t + 5$$, where $$t$$ is measured in hour

Transcendental Function Analysis

Consider the function $$f(x)= \frac{e^x}{x+1}$$ defined for $$x > -1$$ and specifically on the inter

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

Volume of Solid by Cylindrical Shells

Consider the region bounded by $$f(x)= e^x$$ and $$g(x)= e^{2 - x}$$. Analyze the region and set up

Water Reservoir Net Change

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

Area Between Curves

An engineering design problem requires finding the area of the region enclosed by the curves $$y = x

Area Between Curves: $$y=x^2$$ and $$y=4*x$$

Consider the curves defined by $$f(x)= x^2$$ and $$g(x)= 4*x$$. Answer the following questions to de

Area Under a Curve Using Riemann Sums

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

Average 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

Average Value of a Log Function

Let $$f(x)=\ln(1+x)$$ for $$x \ge 0$$. Find the average value of $$f(x)$$ on the interval [0,3].

Composite Functions and Accumulation

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

Estimating River Flow Volume

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

Evaluating an Integral with U-substitution

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

Exact Area Under a Parabolic Curve

Find the exact area under the curve $$f(x)=3*x^2 - x + 4$$ between $$x=1$$ and $$x=4$$.

Experimental Data Analysis using Trapezoidal Sums

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

Exploring the Fundamental Theorem of Calculus

Let the function $$F(x) = \int_{1}^{x} \frac{1}{t^2+1}\,dt$$ represent an accumulation function. Ans

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

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 \

FRQ11: Inverse Analysis of a Parameterized Function

For a positive constant a, consider the function $$ F(x)=\int_{a}^{x} \frac{1}{t+a}\,dt $$ for x > a

Fuel Consumption: Approximating Total Fuel Use

A car's fuel consumption rate (in liters per hour) is modeled by $$f(t)=0.05*t^2 - 0.3*t + 2$$, wher

Fundamental Theorem and Net Change

A function $$f$$ has a derivative given by $$f'(x)=3\sqrt{x}$$ for $$0\le x\le 9$$, and it is known

Integration by Parts: Evaluating $$\int_1^e \ln(x)\,dx$$

Evaluate the integral $$\int_1^e \ln(x)\,dx$$ using integration by parts.

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

Motion Under Variable Acceleration

A particle moves along the x-axis with acceleration $$a(t) = 6 - 4*t$$ (in m/s²) for $$0 \le t \le 3

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 and Evaporation in a Greenhouse

In a greenhouse, rainfall is modeled by $$R(t)= 8\cos(t)+10$$ mm/hr, while evaporation occurs at a c

Rainwater Collection in a Reservoir

Rainwater falls into a reservoir at a rate given by $$R(t)= 12e^{-0.5t}$$ L/min while evaporation re

Riemann Sum Approximation from a Table

The table below gives values of a function $$f(x)$$ at selected points: | x | 0 | 2 | 4 | 6 | 8 | |

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

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

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 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

Analysis of an Autonomous Differential Equation

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

Bacterial Nutrient Depletion

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

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

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

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

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

Economic Growth with Investment Outflow

A company’s investment fund grows continuously at an annual rate of $$5\%$$, but expenses lead to a

Heating and Cooling in an Electrical Component

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

Implicit Differentiation and Slope Analysis

Consider the function defined implicitly by $$y^2+ x*y = 8$$. Answer the following:

Implicit Differentiation with Trigonometric Functions

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

Implicit IVP with Substitution

Solve the initial value problem $$\frac{dy}{dx}=\frac{y}{x}+\frac{x}{y}$$ with $$y(1)=2$$. (Hint: Us

Implicit Solution of a Differential Equation

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

Linear Differential Equation and Integrating Factor

Consider the differential equation $$\frac{dy}{dx} = y - x$$. Use the method of integrating factor t

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

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

Nonlinear Differential Equation with Powers

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

Oil Spill Cleanup Dynamics

To mitigate an oil spill, a cleanup system is employed that reduces the volume of oil in contaminate

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

Radioactive Material with Continuous Input

A radioactive substance decays at a rate proportional to its amount while being produced continuousl

Reaction Kinetics in a Tank

In a chemical reactor, the concentration $$C(t)$$ of a reactant decreases according to the different

Reversible Chemical Reaction

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

Salt Mixing Problem

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

Separable Differential Equation with Trigonometric Factor

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

Slope Field Analysis and Asymptotics

Consider the differential equation $$\frac{dy}{dx}=\frac{x}{1+y^2}$$. Solve the equation and analyze

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 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 Using the SIPPY Method

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

Tank Draining Differential Equation

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

Tank Mixing with Salt

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

Vehicle Deceleration

A vehicle undergoing braking has its speed $$v$$ (in m/s) recorded over time (in seconds) as shown.

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

Water Temperature Regulation in a Reservoir

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

Area Between Curves: Revenue and Cost Analysis

A company’s revenue and cost are modeled by the functions $$f(x)=10-x^2$$ and $$g(x)=2*x$$, where $$

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 a Reaction

A chemical reaction has its concentration modeled by the function $$C(t)=50e^{-0.2*t}+5$$ (in mg/L)

Average Concentration in Medical Dosage

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

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 $$

Charity Donations Over Time

A charity receives monthly donations that form an arithmetic sequence. The first donation is $$50$$

Comparing Sales Projections

A company’s projected sales (in thousands of dollars) are modeled by the function $$f(x)=5*x-x^2$$ w

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

Manufacturing Output Increase

A factory produces goods with weekly output that increases by a constant number of units each week.

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 Along a Straight Line

A particle moves along a straight line with acceleration given by $$a(t)=6-4*t$$ (in m/s²) for $$t \

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

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 Dynamics in a Wildlife Reserve

A wildlife reserve monitors the change in the number of a particular species. The rate of change of

Probability from a Density Function

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

Projectile Motion: Time of Maximum Height

A projectile is launched vertically upward with an initial velocity of $$50\,m/s$$ and an accelerati

Revenue Optimization via Integration

A company’s revenue is modeled by $$R(t)=1000-50*t+2*t^2$$ (in dollars per hour), where $$t$$ (in ho

Stress Analysis in a Structural Beam

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

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\

Traveling Particle with Piecewise Motion

A particle moves along a line with a piecewise velocity function defined as follows: For $$t \in [0

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 of Revolution: Disc Method

Consider the region R bounded by $$y= \sqrt{x}$$, the x-axis, and the vertical line $$x=4$$. When R

Volume of a Solid Using the Washer Method

Consider the region bounded by $$y=x$$ and $$y=x^2$$ between $$x=0$$ and $$x=1$$. This region is rev

Volume of a Solid with Square Cross Sections

A solid has a base in the xy-plane given by the region bounded by the curves $$y=x$$ and $$y=\sqrt{x

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

Water Reservoir Inflow‐Outflow Analysis

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