AP Calculus AB FRQ Room

Analyzing a Piecewise Velocity Function for Continuity and Limits

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

Analyzing Process Data for Continuity

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

Area and Volume Setup with Bounded Regions

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

Continuity in a Cost Function for a Manufactured Product

A company's cost function for producing $$n$$ items (with $$n > 0$$) is given by $$C(n)= \frac{50}{n

Determining Horizontal Asymptotes of a Log-Exponential Function

Examine the function $$s(x)=\frac{e^{x}+\ln(x+1)}{x}$$, which is defined for $$x > 0$$. Determine th

Direct Evaluation of Polynomial Limits

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

Discontinuity in Acceleration Function and Integration

A particle’s acceleration is defined by the piecewise function $$a(t)= \begin{cases} \frac{1-t}{t-2}

Evaluating Sequential Limits in Particle Motion

A particle’s velocity is given by the function $$v(t)= \frac{(t-2)(t+4)}{t-2}$$ for $$t \neq 2$$, an

Factoring a Cubic Expression for Limit Evaluation

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

Implicit Differentiation in an Exponential Equation

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

Intermediate Value Theorem and Root Existence

Consider the function $$f(x)= x^3 - 6*x + 1$$ on the interval [1, 3].

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

Inverse Function and Limit Behavior Analysis

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

Investigation of Continuity in a Piecewise Log-Exponential Function

A function is defined by $$ f(x)=\begin{cases} \frac{\ln(e^{2*x}+3)-\ln(5)}{x-1} & x \neq 1, \\ D &

Jump Discontinuity in a Piecewise Function

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

Limit Analysis in a Population Growth Model

Consider the function $$y(t)=\frac{e^{2*t}-e^{t}}{t}$$ for $$t \neq 0$$, and define $$y(0)=L$$ so th

Limit Evaluation in a Parametric Particle Motion Context

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

Limits and the Squeeze Theorem Application

Consider two scenarios: (1) A function f(x) satisfying $$ -|x| \le f(x) \le |x| $$ for all x near 0,

Limits at Infinity and Horizontal Asymptotes

Examine the function $$f(x)=\frac{3x^2+2x-1}{6x^2-4x+5}$$ and answer the following:

Limits Involving Absolute Value Expressions

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

Limits Involving Trigonometric Functions in Particle Motion

A particle moves along a line with velocity given by $$v(t)= \frac{\sin(2*t)}{t}$$ for $$t > 0$$. An

Long-Term Behavior of Particle Motion: Horizontal Asymptotes

For a particle, the velocity function is given by $$v(t)= \frac{4*t^2-t+1}{t^2+2*t+3}$$. Answer the

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

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 in a Function Involving Logarithms

Define the function $$f(x)=\frac{e^{x}-1}{\ln(1+x)}$$ for $$x \neq 0$$ with a continuous extension g

One-Sided Limits of a Piecewise Function

Let $$f(x)$$ be defined by $$f(x) = \begin{cases} x + 2 & \text{if } x < 3, \\ 2x - 3 & \text{if }

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

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 $$t > 1$$.

Rational Function Limits and Removable Discontinuities

Consider the function $$f(x)=\frac{(x+3)(x-2)}{(x+3)(x+5)}$$. Answer the following:

Real-World Analysis of Vehicle Deceleration Using Data

A study measures the speed of a car (in m/s) as it approaches a stop sign. The recorded speeds at di

Real-World Application: Temperature Sensor Calibration

A temperature sensor in a lab records temperatures (in °C) according to the function $$f(t)= \frac{t

Trigonometric Limit Evaluation

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

Vertical Asymptote Analysis

Consider the function $$f(x)=\frac{1}{x-3}$$. Answer the following parts.

Acceleration Through Successive Differentiation

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

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 Function Behavior Using Its Derivative

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

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

Derivation of $$h(x)= \ln(2*x+3)$$ Using the Chain Rule

Let $$h(x)= \ln(2*x+3)$$, a composition of a logarithmic and a linear function.

Derivative Estimation from Experimental Data

The table below shows the concentration $$h(x)$$ (in molarity, M) of a chemical substance at various

Differentiability of a Piecewise Function

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

Economic Marginal Revenue

A company's revenue function is given by \(R(x)=x*(50-0.5*x)\) dollars, where \(x\) represents the n

Identifying Horizontal Tangents

A continuous function $$f(x)$$ has a derivative $$f'(x)$$ such that $$f'(4)=0$$ and $$f'(x)$$ change

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 Acceleration from a Velocity Function

An object's velocity is given by $$v(t)=3*t^2 - 12*t + 9$$, where $$t$$ is in seconds. Answer the fo

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: Exponential Transformation

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

Inverse Function Analysis: Quadratic Transformation

Consider the function $$f(x)=x^2+2*x+2$$ with the domain restricted to $$x\geq -1$$ so that f is one

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

Marginal Cost from Exponential Cost Function

A company’s cost function is given by $$C(x)= 500*e^{0.05*x} + 200$$, where $$x$$ represents the num

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

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 $$0 \leq t \leq 10$$

Position Function from a Logarithmic Model

A particle’s position in meters is modeled by $$s(t)= \ln(t+1)$$ for $$t \geq 0$$ seconds.

Quotient Rule Challenge

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

Rates of Change in Chemical Concentration

In a chemical reaction, the concentration $$C(t)$$ of a substance in a tank is modeled by $$C(t)=\fr

Real World Application: Rate of Change in River Depth

The depth of a river (in meters) across its width (in kilometers) is given by $$d(x)= 10 - 0.5*x^2$$

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 Lines

Consider the function $$f(x)= x^2$$. Use graphical and algebraic methods to examine the behavior of

Slope of a Tangent Line from a Table of Values

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

Social Media Followers Dynamics

A social media account gains followers at a rate $$f(t)=150-10*t$$ (followers/hour) and loses follow

Tangent Line Approximation

Suppose a continuous function $$f(x)$$ is differentiable with $$f(2)=8$$ and $$f'(2)=5$$. Use this i

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 of the Derivative

Consider the function $$g(x)=3*x^3-2*x+5$$, which models the cost (in dollars) of manufacturing $$x$

Water Tank Inflow-Outflow Analysis

A water tank receives water at a rate given by $$f(t)=3*t+2$$ (liters/min) and loses water at a rate

Analyzing a Composite Function and Its Inverse

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

Chain Rule and Product Rule Combination

Let $$f(x)=x^2\sqrt{1+x^3}$$. Answer the following:

Chain Rule with Trigonometric Function

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

Composite Differentiation with Nested Functions

Differentiate the function $$F(x)=\sqrt{\cos(4*x^2+1)}$$ using the chain rule. Your answer should re

Composite Function Differentiation in a Sand Pile Model

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

Composite Function in Biomedical Model

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

Composite Function in Finance

An account balance is modeled by the function $$B(t)=(2*t+1)^{3/2}$$ dollars, where $$t$$ represents

Composite Function Involving Exponential and Cosine

Consider the function $$f(x)= e^(\cos(x^2))$$. Address the following:

Composite Function with Logarithm and Trigonometry

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

Composite Temperature Model

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

Composite Trigonometric Differentiation in Sound Waves

The sound intensity in a room is modeled by the function $$I(t)= \cos(3*t^2+\sin(t))$$, where $$t$$

Concavity Analysis of an Implicit Curve

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

Derivative of an Inverse Function

Let $$f$$ be a differentiable function with an inverse function $$g$$ such that $$f(2)=5$$ and $$f'(

Differentiation of a Complex Implicit Equation

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

Differentiation of a Log-Exponential-Trigonometric Composite

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

Implicit and Inverse Function Analysis

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

Implicit Differentiation Involving a Product

Consider the equation $$x^2*y + \sin(y) = x*y^2$$ which relates the variables $$x$$ and $$y$$ in a n

Implicit Differentiation Involving Exponential Functions

Let the equation $$x*e^{y}+y*e^{x}=10$$ define $$y$$ implicitly as a function of $$x$$. Use implicit

Implicit Differentiation with Logarithmic and Trigonometric Combination

Consider the equation $$\ln(x+y)+\cos(x*y)=0$$, where $$y$$ is an implicit function of $$x$$. Find $

Implicitly Defined Inverse Relation

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

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 Differentiation

Let $$f(x)=x^3+x+1$$, a one-to-one function, and let $$g$$ be the inverse of $$f$$. Use inverse func

Inverse Function Differentiation Combined with Chain Rule

Let $$f(x)=\sqrt{x-1}+x^2$$, and assume that it is one-to-one on its domain, with an inverse functio

Inverse Function Differentiation in Mixing Solutions

Let the function $$f(x)=2*x^3+x-5$$ model the concentration of a solution as a function of a paramet

Inverse Function Differentiation: Composite Inversion

Let $$f(x) = \frac{x}{1-x}$$ for x < 1, and let g denote its inverse function. Answer the following

Inverse Function Theorem in a Composite Setting

Let $$f(x)=x+\sin(x)$$ with inverse function $$g(x)$$.

Inverse Trigonometric Differentiation

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

Inverse Trigonometric Function Differentiation

Consider the function $$y=\arctan(2*x)$$. Answer the following:

Nested Trigonometric Function Analysis

A physics experiment produces data modeled by the function $$h(x)=\cos(\sin(3*x))$$, where $$x$$ is

Optimization in an Implicitly Defined Function

The curve defined by $$x^2y + \sin(y) = 10$$ implicitly defines $$y$$ as a function of $$x$$ near $$

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 and Composite Functions

A 10-foot ladder is leaning against a wall such that its bottom moves away from the wall according t

Related Rates in a Circular Colony

A circular microorganism colony expands such that its radius at time $$t$$ (in seconds) is given by

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

Analysis of a Piecewise Function with Discontinuities

Consider the function $$f(x)=\begin{cases} \frac{x^2-4}{x-2} & \text{if } x<2 \\ x+1 & \text{if } x\

Analysis of Experimental Data

The graph below shows the displacement of an object moving in a straight line. Analyze the object's

Analyzing Experimental Motion Data

The table below shows the position (in meters) of a moving object at various times (in seconds):

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

Deceleration with Air Resistance

A car’s velocity is modeled by $$v(t) = 30e^{-0.2t} + 5$$ (in m/s) for $$t$$ in seconds.

Depth of a Well: Related Rates Problem

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

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

Expanding Oil Spill

The area of an oil spill is modeled by $$A(t)=\pi (2+t)^2$$ square kilometers, where $$t$$ is in hou

Expanding Oil Spill: Related Rates Problem

An oil spill forms a circular patch on the water with area $$A = \pi r^2$$. The area is increasing a

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 4: Revenue and Cost Implicit Relationship

A company’s revenue (R) and cost (C) are related by the equation $$R^2 + 3*R*C + C^2 = 1000$$. Treat

FRQ 5: Coffee Cooling Experiment

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

FRQ 14: Optimizing Box Design with Fixed Volume

A manufacturer wants to design an open-top box with a fixed volume of $$V = x^2*y = 32$$ cubic units

Function with Vertical Asymptote

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

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

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 of Natural Logarithm

Estimate $$\ln(1.05)$$ using linear approximation for the function $$f(x)=\ln(x)$$ at $$a=1$$.

Linearization and Differentials Approximation

A function $$f(x)= x^3$$ models the volume (in cubic centimeters) of liquid in a container as a func

Logarithmic Differentiation in Exponential Functions

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

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

Multi‐Phase Motion Analysis

A car's motion is described by a piecewise velocity function. For $$0 \le t < 2$$ seconds, the veloc

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{

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

Particle Motion Analysis

A particle moves along a straight line with an acceleration given by $$a(t) = 6 - 4*t$$, where $$t$$

Population Growth Model and Asymptotic Limits

A population is modeled by $$P(t) = \frac{5000e^{0.1t}}{1 + e^{0.1t}}$$, where $$P(t)$$ is the popul

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: The Expanding Ripple

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

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 Function and Marginal Revenue Analysis

A company's revenue is modeled by $$R(x)= -0.5*x^3 + 20*x^2 + 15*x$$, where $$x$$ represents the num

Revenue Sensitivity to Advertising

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

Seasonal Water Reservoir

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

Shadow Length: Related Rates

A 15-foot tall lamp post casts light on a 6-foot tall man walking away from the lamp at 4 ft/sec. Le

Transcendental Function Temperature Change

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

Vehicle Deceleration Analysis

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

Vehicle Deceleration Analysis

A car's position function is given by $$s(t)= 3*t^3 - 12*t^2 + 5*t + 7$$, where $$s(t)$$ is measured

Water Tank Volume Change

The volume of water in a tank is given by $$V(r) = \frac{4}{3}\pi r^3$$, where $$r$$ (in m) is the r

Analysis of a Parametric Curve

Consider the parametric equations $$x(t)= t^2 - 3*t$$ and $$y(t)= 2*t^3 - 9*t^2 + 12*t$$. Analyze th

Analyzing Critical Points in a Piecewise Function

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

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

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

Application of Rolle's Theorem for a Quadratic Function

Let $$f(x)= x^2 - 4$$ be defined on the interval $$[-2,2]$$. In this problem, you will verify the co

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:

Application of the Trapezoidal Rule in a Chemical Reaction

In a chemistry experiment, the concentration of a reactant (in molarity, M) is measured at various t

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:

Behavior Analysis of a Logarithmic Function

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

Capacitor Discharge in an RC Circuit

The voltage across a capacitor during discharge is given by $$V(t)= V_0*e^{-t/(RC)}$$, where $$t$$ i

Concavity and Inflection Points in a Quartic Function

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

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

Continuity Analysis of a Rational Piecewise Function

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

Cost Function and the Mean Value Theorem in Economics

An economic model gives the cost function as $$C(x)= 100 + 20*x - 0.5*x^2$$, where x represents the

Economic Demand and Revenue Optimization

The demand for a product is modeled by $$D(p) = 100 - 2*p$$, where $$p$$ is the price in dollars. Th

Extrema in a Cost Function

A company's cost function is given by $$C(x) = x^3 - 6*x^2 + 12*x + 7$$, where $$x$$ represents the

FRQ 8: Mean Value Theorem and Non-Differentiability

Examine the function $$f(x)=|x|$$ on the interval [ -1, 1 ].

FRQ 9: Extreme Value Analysis for a Rational Function

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

FRQ 17: Analysis of a Trigonometric Function for Extrema and Inflection Points

Let $$f(x)= \sin(x) - 0.5*x$$ for $$x \in [0, 2\pi]$$.

Graph Analysis: Exponentially Modified Function

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

Increase and Decrease Analysis of a Polynomial Function

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

Instantaneous Velocity Analysis via the Mean Value Theorem

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

Inverse Analysis of a Function with Parameter

Consider the function $$f(x)=x^3+a*x$$ where a is a real parameter. Analyze the invertibility of f a

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 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,

Logistic Growth Model and Derivative Interpretation

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

Logistic Population Model Analysis

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

Mean Value Theorem for a Logarithmic Function

Consider the function $$f(x)= \ln(x)$$ defined on the interval $$[1, e^2]$$. Use the Mean Value Theo

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

Optimization in Production with Exponential Cost Function

A manufacturer’s cost function is modeled by $$C(x)= 200 + 50*x + 100*e^{-0.1*x}$$ where $$x$$ repre

Rate of Heat Loss in a Cooling Process

In a cooling experiment, the temperature of an object is recorded over time as it loses heat. Use th

Relationship Between Integration and Differentiation

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

Tangent Line and MVT for ln(x)

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

Accumulation and Inflection Points

Suppose a function's rate of change is given by $$f'(x)=3*x^2-12*x+9.$$ Answer the following parts:

Application of the Fundamental Theorem of Calculus

Let $$f(x)=\ln(x)$$. Use the Fundamental Theorem of Calculus to evaluate the definite integral $$\in

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 Polynomial Curve

Consider the function $$f(x)=2*x^2-3*x+1$$ defined on the interval $$[0,4]$$. Answer the following p

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

Comparing Riemann Sum and the Fundamental Theorem

Let $$f(x)=3*x^2$$ on the interval $$[1,4]$$.

Composite Functions and Accumulation

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

Electric Charge Accumulation

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

Estimating Accumulated Water Inflow Using Riemann Sums

A water tank fills at varying rates. The table below shows the inflow rate in liters per second at d

Estimating Area Under a Curve Using Riemann Sums

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

Estimating Displacement with a Midpoint Riemann Sum

A vehicle’s velocity is modeled by the function $$v(t) = -t^{2} + 4*t$$ (in meters per second) over

Evaluating an Integral with a Trigonometric Function

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

Evaluating the Accumulated Drug Concentration

In a pharmacokinetics study, a drug is infused into a patient's bloodstream at a rate given by $$R(t

FRQ14: Inverse Analysis of a Logarithmic Accumulation Function

Let $$ L(x)=\int_{1}^{x} \frac{1}{t}\,dt $$ for x > 0. Answer the following parts.

FRQ18: Inverse Analysis of a Square Root Accumulation Function

Consider the function $$ R(x)=\int_{1}^{x} \sqrt{t+1}\,dt $$. Answer the following parts.

Function Transformations and Their Integrals

Let $$f(x)= 2*x + 3$$ and consider the transformed function defined as $$g(x)= f(2*x - 1)$$. Analyze

Improving Area Approximations with Increasing Subintervals

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

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

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

Integration to Determine Work Done by a Variable Force

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

Medication Concentration and Absorption Rate

A patient's blood concentration of a drug (in mg/L) is monitored over time before reaching its peak.

Motion Analysis with Variable Acceleration

A particle moves along a straight line with acceleration given by $$a(t)=8-2*t$$ (in m/s²). The part

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) for $$

Particle Motion with Variable Acceleration

A particle moves along a straight line with an acceleration given by $$a(t)=4*t - 2$$ (in m/s²). Giv

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

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.

Total Fuel Used Over a Trip

A car consumes fuel at a rate modeled by $$r(t) = 0.2*t + 1.5$$ (in gallons per hour) during a long

Trigonometric Integration via U-Substitution

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

U-Substitution in a Trigonometric Integral

Evaluate the integral $$\int \sin(2*x) * \cos(2*x)\,dx$$ using u-substitution.

Vehicle Distance Estimation from Velocity Data

A car's velocity (in m/s) is recorded at several time points during a trip. Use the table below for

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

Work Done by a Variable Force

A force acting on an object is given by $$F(x)=3*x^2+2*x$$ where x is in meters and F in Newtons. Th

Area Under a Differential Equation Curve

Consider the differential equation $$\frac{dy}{dx} = y(1-y)$$, with a particular solution given by $

Bacterial Culture with Antibiotic Treatment

A bacterial culture grows at a rate proportional to its size, but an antibiotic is administered cont

Chemical Reactor Temperature Profile

In a chemical reactor, the temperature $$T$$ (in °C) is recorded over time (in minutes) as shown. Th

Combined Cooling and Slope Field Problem

A cooling process is modeled by the equation $$\frac{d\theta}{dt}=-0.07\,\theta$$ where $$\theta(t)=

Cooling of Electronic Components

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

Epidemic Spread (Simplified Logistic Model)

In a simplified model of an epidemic, the number of infected individuals $$I(t)$$ (in thousands) is

Exact Differential Equation

Consider the differential equation written in differential form: $$(2*x*y + y^2)\,dx + (x^2 + 2*x*y)

Heating a Liquid in a Tank

A liquid in a tank is being heated by mixing with an incoming fluid whose temperature oscillates ove

Implicit IVP with Substitution

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

Investment Growth with Continuous Deposits

An investment account accrues interest continuously at an annual rate of 0.05 and receives continuou

Mixing Problem in a Salt Solution Tank

A 100-liter tank initially contains a solution with 10 kg of salt. Brine with a salt concentration o

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 Variable Inflow Concentration

A tank initially contains 50 L of water with 5 kg of dissolved salt. A solution enters the tank at a

Motion Under Gravity with Air Resistance

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

Population with Constant Harvesting

A fish population in a lake grows according to the differential equation $$\frac{dy}{dt} = r*y - H$$

Radioactive Decay with Production

A radioactive substance decays while being produced at a constant rate, and its mass $$M(t)$$ (in kg

RC Circuit Charging

In a resistor-capacitor (RC) circuit, the charge $$Q(t)$$ on the capacitor is modeled by the differe

Reaction Kinetics in a Tank

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

Separable Differential Equation with Trigonometric Factor

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

Separable Differential Equation: y and x

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

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 $$\frac{dy}{dx}=\frac{y}{x}$$

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

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

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

Tank Mixing with Salt

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

Water Tank Flow Analysis

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

Water Temperature Regulation in a Reservoir

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

Analysis of an Inverse Function

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

Arch of a Bridge

An arch of a bridge is modeled by the function $$y=10-0.5*(x-5)^2$$, where $$x$$ is in meters and th

Area Between a Parabolic Curve and a Line

Consider the functions $$f(x)=5*x-x^2$$ and $$g(x)=x$$ on the interval $$[0,4]$$. The table below sh

Area Between Transcendental Functions

Consider the curves $$f(x)=\cos(x)$$ and $$g(x)=\sin(x)$$ on the interval $$[0,\frac{\pi}{4}]$$.

Average Growth Rate in a Biological Process

In a biological study, the instantaneous growth rate of a bacterial colony is modeled by $$k(t)=0.5*

Average Speed Over a Journey

A car travels along a straight road and its speed (in m/s) is modeled by the function $$v(x)=2*x^2-3

Center of Mass of a Lamina with Variable Density

A thin lamina occupies the interval $$[0,4]$$ along the x-axis and has a variable density $$\delta(x

Charity Donations Over Time

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

Determining Velocity and Position from Acceleration

A particle moves along a line with acceleration given by $$a(t)=4-2*t$$ (in $$m/s^2$$). At time $$t=

Distance Traveled by a Jogger

A jogger increases her daily running distance by a fixed amount. On the first day she runs $$2$$ km,

Download Speeds Improvement

An internet service provider increases its download speeds as part of a new promotional plan such th

Implicit Differentiation in an Economic Equilibrium Model

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

Particle Motion 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 \

Sales Increase in a Store

A store experiences an increase in weekly sales such that the sales figures form a geometric sequenc

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

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 Filling Process Analysis

Water flows into a tank at a rate modeled by $$R(t)=5+0.5*t$$ (in liters per minute) for $$0 \le t \

Technology Adoption Growth

A new technology is being adopted in a community such that the number of new users each day forms a

Temperature Increase in a Chemical Reaction

During a chemical reaction, the rate of temperature increase per minute follows an arithmetic sequen

Volume of a Solid with Rectangular Cross Sections

A solid has a base on the x-axis from $$x=0$$ to $$x=3$$. The cross-sectional areas (in m²) perpendi

Volume of a Solid with Semicircular Cross Sections

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

Volume of a Solid with Square Cross-Sections

A solid has a base in the xy-plane bounded by $$y=x$$ and $$y=x^2$$ for $$0 \le x \le 1$$. Every cro