AP Calculus AB FRQ Room

Analyzing End Behavior and Asymptotes

Consider the function $$f(x)= \frac{5x - 7}{\sqrt{x^2 + 1}}$$. Answer the following:

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

Application of the Squeeze Theorem

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

Combined Limit Analysis of a Piecewise Function

Consider the function $$f(x)= \begin{cases} \frac{x^2-1}{x-1} & \text{if } x \neq 1, \\ c & \text{if

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

Discontinuities in a Rational-Exponential Function

Consider the function $$ f(x) = \begin{cases} \frac{e^{x} - 1}{x}, & x \neq 0 \\ k, & x = 0. \en

Estimating Limits from a Data Table

A function f(x) is studied near x = 3. The table below shows selected values of f(x):

Exponential and Logarithmic Limits

Consider the functions $$f(x)=\frac{e^{2*x}-1}{x}$$ and $$g(x)=\frac{\ln(1+x)}{x}$$. Evaluate the li

Factorization and Removable Discontinuity

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

Graph Reading: Left and Right Limits

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

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 Analysis of Function Behavior from a Table

A real-world experiment recorded the concentration (in M) of a solution over time (in seconds) as sh

Implicit Differentiation Involving Logarithms

Consider the curve defined implicitly by $$\ln(x) + \ln(y) = \ln(5)$$. Answer the following:

Intermediate Value Theorem with an Exponential-Logarithmic Function

Consider the function $$u(x)=e^{x}-\ln(x+2)$$, defined for $$x > -2$$. Since $$u(x)$$ is continuous

Investigating Discontinuities in a Rational Function

Consider the function $$ h(x)=\frac{x^2-4}{x-2} $$ for $$x\ne2$$.

Jump Discontinuity in a Piecewise Function

Consider the function $$f(x)=\begin{cases} x+2, & x < 1 \\ 3, & x = 1 \\ 2*x, & x > 1 \end{cases}$$.

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 and Integration in Non-Polynomial Particle Motion

A particle moves along a line with velocity defined by $$v(t)= \frac{e^{2*t}-e^{4}}{t-2}$$ for \(t \

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 Involving Absolute Value

Consider the function $$f(x) = \frac{|x - 3|}{x - 3}$$. (a) Evaluate $$\lim_{x \to 3^-} f(x)$$ and

Limits Involving Composition and Square Roots

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

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

Logarithm Transformation and Limit Evaluation

Consider the function $$Y(x)=\ln\left(\frac{e^{2*x}+5}{e^{2*x}-5}\right)$$. Investigate the limits a

Modeling Bacterial Growth with a Geometric Sequence

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

Oscillatory Function and the Squeeze Theorem

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

Parameter Determination from a Logarithmic-Exponential Limit

Let $$v(x)=\frac{\ln(e^{2*x}+1)-2*x}{x}$$ for $$x\neq0$$. Find the value of the limit $$\lim_{x \to

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

Piecewise-Defined Function Continuity Analysis

Let $$f(x)$$ be defined as follows: For $$x < 2$$: $$f(x)= 3x - 1$$. For $$2 \leq x \leq 5$$: $$f(x

Removable Discontinuity and Limit

Consider the function $$ f(x)=\frac{x^2-9}{x-3} $$ for $$ x\ne3 $$, which is not defined at $$ x=3 $

Removable Discontinuity Through Cancellation

Consider the function $$f(x)=\begin{cases} \frac{(x-3)(x+5)}{x-3} & x\neq3 \\ 7 & x=3 \end{cases}$$.

Removal of Discontinuity by Redefinition

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

Table Analysis for Estimating a Limit

The table below shows values of the function $$g(x)$$ for x near 4. Use this data to answer the foll

Vertical Asymptote Analysis

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

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 a Projectile's Motion

A projectile is launched vertically, and its height (in feet) at time $$t$$ seconds is given by $$s(

Analyzing Rates Without a Calculator: Average vs Instantaneous Rates

Consider the function $$f(x)= x^2$$.

Approximating Small Changes with Differentials

Let $$f(x)= x^3 - 5*x + 2$$. Use differentials to approximate small changes in the value of $$f(x)$$

Approximating the Tangent Slope

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

Average and Instantaneous Rates of Change

A function $$f$$ is defined by $$f(x)=x^2+3*x+2$$, representing the height (in meters) of a projecti

Bacterial Culture Growth with Washout

In a bioreactor, bacteria grow at a rate of $$f(t)=50*e^{0.05*t}$$ (cells/min) while simultaneous wa

Car Fuel Consumption vs. Refuel

A car is being refueled at a constant rate of $$4$$ liters/min while it is being driven. Simultaneou

Critical Points of a Log-Quotient Function

Let $$f(x)= \frac{\ln(x)}{x}$$ for $$x > 0$$. This function is used to analyze efficiency in logarit

Derivative of a Logarithmic Function

Let $$g(x)= \ln(x)$$, a fundamental function in many natural logarithm applications.

Derivative of an Absolute Value Function

Consider the function \(f(x)=|x|\). Answer the following parts, restricting your analysis to \(x\ne

Difference Quotient for a Cubic Function

Let \(f(x)=x^3\). Using the difference quotient, answer the following parts.

Differentiating an Absolute Value Function

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

Estimating Instantaneous Slope of a Logarithmic Function

Consider the function \(f(x)=\ln(x)\). Without directly using the derivative rules, estimate the ins

Finding Derivatives of Composite Functions

Let $$f(x)= (3*x+1)^4$$.

Finding the Second Derivative

Given $$f(x)= x^4 - 4*x^2 + 7$$, compute its first and second derivatives.

Graph Interpretation of the Derivative

Consider the function $$f(x)= x^3 - 6*x^2 + 9*x + 1$$. A graph of this function is provided below.

Implicit Differentiation in Demand Analysis

Consider the implicitly defined demand function $$x^2 + x*y + y^2 = 100$$, where x represents the pr

Instantaneous Rate of Change from a Graph

A graph of a smooth curve (representing a function $$f(x)$$) is provided with a tangent line drawn a

Integrating Graph and Table Data for Revenue Analysis

A company’s revenue function $$R(x)$$ (in thousand dollars) appears to be linear. Data from a recent

Inverse Function Analysis: Restricted Rational Function

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

Logarithmic Differentiation of a Composite Function

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

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

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

Physical Motion with Variable Speed

A car's velocity is given by $$v(t)= 2*t^2 - 3*t + 1$$, where $$t$$ is in seconds.

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

Polynomial Rate of Change Analysis

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

Population Growth and Instantaneous Rate of Change

A town's population is modeled by $$P(t)= 2000*e^{0.05*t}$$, where $$t$$ is in years. Analyze the ch

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.

Product Rule Application in a Real-World Setting

A company's revenue is modeled by $$R(x)= x*e^{-x}$$, where $$x$$ (in thousands of units) represents

Real-World Application: Temperature Change in a Chemical Reaction

The temperature (in $$\degree C$$) during a chemical reaction is modeled by $$T(t)= 25 - 2*t + \frac

Related Rates: Balloon Surface Area Change

A spherical balloon has volume $$V=\frac{4}{3}\pi r^3$$ and surface area $$S=4\pi r^2$$. If the volu

Related Rates: Shadow Length Change

A person 1.8 m tall is walking away from a streetlight that is 5 m high. Let x represent the distanc

River Pollution Dynamics

A factory discharges pollutants into a river at a rate of $$f(t)=20+3*t$$ (kg/hour), while the river

Secant and Tangent Lines Approximation

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

Secant and Tangent Lines for a Trigonometric Function

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

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

Tangent Line to a Cubic Function

The function $$f(x) = x^3 - 6x^2 + 9x + 1$$ models the height (in meters) of a roller coaster at pos

Using the Limit Definition to Derive the Derivative

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

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 Function and Its Inverse

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

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 Rule with Exponential and Polynomial Functions

Let $$h(x)= e^{3*x^2+2*x}$$ represent a function combining exponential and polynomial elements.

Chain Rule with Logarithmic and Radical Functions

Let $$R(x)=\sqrt{\ln(1+x^2)}$$.

Chain Rule with Logarithms

Let $$h(x)=\ln(\sqrt{4*x^2+1})$$. Answer the following:

Chain Rule: Exponential-Log Composite Function

Consider the function $$f(x) = e^{(\ln(3*x^2+1))^2}$$. Answer the following parts.

Differentiation of Inverse Trigonometric Composite Function

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

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 in a Trigonometric Context

Consider the equation $$\sin(x*y)+x-y=0$$. Answer the following:

Implicit Differentiation in an Economic Demand-Supply Model

In an economic model, the relationship between supply (\(S\)) and demand (\(D\)) is given by the equ

Implicit Differentiation in an Elliptical Orbit

The orbit of a satellite is given by the ellipse $$\frac{x^2}{16} + \frac{y^2}{9} = 1$$. Answer the

Implicit Differentiation in Circular Motion

Consider the circle defined by the equation $$x^2+y^2=100$$, which could represent the track of an o

Implicit Differentiation Involving a Logarithm

Consider the equation $$x*\ln(y) + y^2 = x^2$$. Answer the following parts.

Implicit Differentiation of an Exponential-Product Equation

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

Implicit Differentiation of Quadratic Curve

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

Implicit Differentiation on a Circle

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

Implicit Differentiation with Exponential Terms

Consider the equation $$e^{x} + y = x + e^{y}$$ which relates $$x$$ and $$y$$ via exponential functi

Implicit Differentiation with Mixed Functions

Consider the relation $$x\cos(y)+y^3=4*x+2*y$$.

Implicitly Defined Inverse Relation

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

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 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 Function Differentiation in a Piecewise Scenario

Consider the piecewise function $$f(x)=\begin{cases} x^2+1, & x \geq 0 \\ -x+1, & x<0 \end{cases}$$

Inverse Function Differentiation in Logarithmic Functions

Let $$f(x)=\ln(x+2)$$, which is one-to-one and has an inverse function $$g(y)$$. Answer the followin

Inverse Trigonometric and Logarithmic Function Composition

Let $$y=\arctan(\ln(x))$$. Answer the following:

Inverse Trigonometric Differentiation

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

Inverse Trigonometric Differentiation

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

Optimization in a Container Design Problem

A manufacturer is designing a closed cylindrical container with a fixed volume of $$1000\,cm^3$$. Th

Related Rates via Chain Rule

A spherical balloon is being inflated so that its volume increases at a rate of $$\frac{dV}{dt}=150\

Analysis of Particle Motion

A particle moves along a horizontal line with velocity function $$v(t)=4t-t^2$$ (m/s) for $$t \geq 0

Analyzing Position Data with Table Values

A moving object’s position, given by $$x(t)$$ in meters, is recorded in the table below. Use the dat

Demand Function Inversion and Analysis

The product demand is modeled by $$p(q)=\frac{100}{q+1}+20$$, where p is the price (in dollars) and

Economic Efficiency in Speed

A vehicle’s fuel consumption per mile (in dollars) is modeled by the function $$C(v)=0.05*v^2 - 3*v

Economics and Marginal Analysis: Revenue and Cost Differentiation

A company’s revenue and cost functions are given by $$R(x)=100*x - 0.5*x^2$$ and $$C(x)=20*x + 50$$,

Estimating Instantaneous Rates from Discrete Data

In a laboratory experiment, the concentration of a chemical (in molarity, M) is recorded over time (

Evaluating an Indeterminate Limit using L'Hospital's Rule

Evaluate the limit $$\lim_{x \to 0} \frac{e^{2x} - 1}{3x}$$.

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

FRQ 10: Chemical Kinetics Analysis

In a chemical reaction, the concentration of reactant A, denoted by [A], and time t (in minutes) are

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

Inflating Balloon

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

Inflating Balloon Rates

A spherical balloon is being inflated such that its volume increases at a constant rate of $$\frac{d

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

Limit Evaluation Using L'Hôpital's Rule

Evaluate the limit $$\lim_{x \to \infty} \frac{5x^3 - 4x^2 + 1}{7x^3 + 2x - 6}$$.

Linear Approximation in Estimating Function Values

Let $$f(x)= \ln(x)$$. Analyze its linear approximation.

Linearization for Approximating Powers

Let $$f(x) = x^3$$. Use linear approximation to estimate $$f(4.98)$$.

Linearization of a Nonlinear Function

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

Maximizing the Area of an Enclosure with Limited Fencing

A farmer has 240 meters of fencing available to enclose a rectangular field that borders a river (th

Medicine Dosage: Instantaneous Rate of Change

The concentration of a medicine in the bloodstream is given by $$C(t) = 25e^{-0.2t}+5$$, where $$t$$

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 & Linearization in Engineering Design

A material's strength is modeled by the function $$S(x)= 50*x^2 - 3*x^3$$, where $$x$$ (in centimete

Optimization for Minimizing Time in Road Construction

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

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: Minimizing Material for a Box

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

Particle Motion Analysis

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

Projectile Motion Analysis

The height of a projectile is modeled by the function $$h(t) = -4.9t^2 + 20t + 2$$, where $$t$$ is i

Projectile Motion: Maximum Height

A ball is thrown upward, and its height in meters after $$t$$ seconds is modeled by $$h(t)=-5*t^2+20

Rate of Change in Pool Volume

The volume $$V(t)$$ (in gallons) of water in a swimming pool is given by $$V(t)=10t^2-40t+20$$, wher

Rates of Change in Economics: Marginal Cost

A company's cost function is given by $$C(q)= 0.5*q^2 + 40$$, where q (in units) is the quantity pro

Region Area and Volume by Rotation

Consider the region R bounded by the curves $$y=\sqrt{x}$$ and $$y=\frac{x}{2}$$ from $$x=0$$ to $$x

Related Rates in a Conical Tank

Water is draining from a conical tank. The volume of water is given by $$V = \frac{1}{3}\pi r^2 h$$,

Rocket Thrust: Analyzing Exponential Acceleration

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

Temperature Change in Cooling Coffee

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

Vehicle Position and Acceleration

A vehicle's position along a straight road is modeled by $$s(t)=4\sqrt{t+1}$$ (in kilometers), where

Water Tank Volume Change

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

Absolute Value Piecewise Function Discontinuity Analysis

Consider the function $$ f(x) = \begin{cases} 1, & x > 2, \\ -1, & x < 2, \\ 0, & x = 2. \end{cases

Analyzing a Supply and Demand Model Using Derivatives

A product's price as a function of the number of units produced is given by $$P(q)= 50 - 3*q + 0.5*q

Analyzing Increasing/Decreasing Behavior of a Cubic Polynomial

Consider the function $$f(x)= x^3 - 6*x^2 + 9*x + 2$$. Analyze the function's behavior in terms of i

Bacterial Culture Growth: Identifying Critical Points from Data

A microbiologist records the population of a bacterial culture (in millions) at different times (in

Comprehensive Analysis of a Rational Function

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

Concavity and Inflection Points in a Quartic Function

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

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

Differentiability and Critical Points with an Absolute Value Function

Consider the function defined by $$ f(x)= \begin{cases} x^2, & \text{if } x \ge 0, \\ -x^2, & \

Discontinuity in a Rational Function Involving Square Roots

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

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

Estimating Total Revenue via Riemann Sums

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

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

Floodgate Operation Analysis

A dam uses a floodgate to control water flow. The inflow is given by $$Q_{in}(t)=60-4*t$$ m³/min and

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 $$x \ge 0$$.

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 $$[-1, 3]$$.

Investment with Continuous Compounding and Variable Rates

An investment grows continuously with a variable rate given by $$r(t)= 0.05+0.01e^{-0.5*t}$$. Its va

Mean Value Theorem Applied to Car Position Data

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

Monotonicity and Inverse Function Analysis

Consider the function $$f(x)= x + e^{-x}$$ defined for all real numbers. Investigate its monotonicit

Optimization of a Rectangular Enclosure

A rectangular pen is to be constructed along the side of a barn so that only three sides require fen

Relative Extrema of a Rational Function

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

Urban Water Supply Management

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

Accumulated Rainfall Estimation

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

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:

Antiderivatives of Trigonometric Functions

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

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

Average Temperature Calculation over 12 Hours

In a city, the temperature over a 12-hour period is modeled by $$T(t) = -2*t + 20$$ (in $$^\circ C$$

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

Bacterial Growth Modeling with Antibiotic Administration

A bacterial culture is subject to both growth and treatment simultaneously. The bacterial growth rat

Chemical Accumulation in a Reactor

A chemical reactor has a net accumulation rate given by $$R(t)=5*\cos(t) + 2$$ (in kg/hour), where $

Chemical Reactor Conversion Process

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

Computing Accumulated Volume from a Filling Rate Function

A small pond is being filled at a rate given by $$r(t)=2*t + 3$$ (in $$m^3/hr$$), where $$t$$ is in

Cost Accumulation from Marginal Cost Function

A company's marginal cost (in dollars per unit) is given by $$MC(x)=0.2*x+50$$, where $$x$$ represen

Cumulative Solar Energy Collection

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

Economic Analysis: Consumer Surplus

In a competitive market, the demand function is given by $$D(p)=100-2*p$$ and the supply function is

Electric Charge Accumulation

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

Evaluating an Integral with a Piecewise Function

Consider the function defined by $$f(x)=\begin{cases} x^2 & \text{if } x<2,\\ 4*x-4 & \text{if } x

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

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

FRQ13: Inverse Analysis of an Investment Growth Function

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

FRQ15: Inverse Analysis of a Quadratic Accumulation Function

Consider the function $$ Q(x)=\int_{0}^{x} (4*t+1)\,dt $$. Answer the following parts.

Integration of Exponential Functions with Shifts

Evaluate the integral $$\int_{0}^{2} e^{2*(x-1)}\,dx$$ using an appropriate substitution.

Medication Infusion in Bloodstream

A patient receives medication through an IV at a rate $$I(t)= 5\sqrt{t+1}$$ mg/min, while the body m

Net Change in Population Growth

A town's population grows at a rate given by $$P'(t)=0.1*t+50$$ (in individuals per year) where $$t$

Piecewise-Defined Function and Discontinuities

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

Sand Pile Dynamics

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

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

Trapezoidal Approximation for a Changing Rate

The following table represents the flow rate (in L/min) of water entering a tank at various times:

Trapezoidal Rule Application with Population Growth

A biologist recorded the instantaneous growth rate (in thousands per year) of a species over several

Trigonometric Integral with U-Substitution

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

Bank Account with Continuous Interest and Withdrawals

A bank account accrues interest continuously at an annual rate of $$6\%$$, while money is withdrawn

Bernoulli Differential Equation Challenge

Consider the nonlinear differential equation $$\frac{dy}{dt} - y = -y^3$$ with the initial condition

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

Cooling of Electronic Components

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

Direction Fields and Integrating Factor

Consider the differential equation $$\frac{dy}{dx}=\frac{2*x}{1+y^2}$$ with the initial condition $$

Epidemic Spread Modeling

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

Falling Object with Air Resistance

A falling object with mass $$m=70\,kg$$ is subject to gravity and a resistive force proportional to

Fishery Harvesting Model

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

Implicit Solution for $$\frac{dy}{dx}=\frac{x+2}{y+1}$$

Solve the differential equation $$\frac{dy}{dx} = \frac{x+2}{y+1}$$ with the initial condition $$y(0

Implicit Solution of a Differential Equation

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

Integrating Factor Method

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

Investment Growth with Continuous Deposits

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

Linear Differential Equation and Integrating Factor

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

Logistic Model with Harvesting

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

Mixing Problem with Constant Flow

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

Mixing Problem with Time-Dependent Inflow Concentration

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

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

Mixing Tank Problem

A tank initially contains $$100$$ liters of pure water. A salt solution with a concentration of $$0.

Modeling Cooling with Newton's Law

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

Newton's Law of Cooling with Variable Ambient Temperature

An object cools according to Newton's Law of Cooling, but the ambient temperature is not constant. I

Non-Separable to Linear DE

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

Nonlinear Differential Equation

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

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

Salt Mixing in a Tank

A tank initially contains 100 L of water with 5 kg of salt dissolved. Brine with a concentration of

Slope Field Sketching for $$\sin(x)$$ Model

Given the slope field for the differential equation $$\frac{dy}{dx} = \sin(x)$$, sketch a solution c

Traffic Flow Dynamics

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

Water Level in a Reservoir

A reservoir's water volume $$V$$ (in million m³) is measured at various times $$t$$ (in days) as sho

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 Curves: Exponential vs. Linear

Consider the functions $$f(x)=e^{-x}$$ and $$g(x)=1-x$$. A table of approximate values is provided b

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 $$

Area Between Transcendental Functions

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

Area Between Two Curves

Consider the curves $$f(x)= 5*x - x^2$$ and $$g(x)= x$$. Determine the area of the region enclosed b

Average Concentration in Medical Dosage

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

Average Density of a Rod

A rod of length $$10$$ cm has a linear density given by $$\rho(x)= 4 + x$$ (in g/cm) for $$0 \le x \

Average Force and Work Done on a Spring

A spring is compressed according to Hooke's Law, where the force required to compress the spring is

Average Force Calculation

An object experiences a variable force given by $$F(x)=3*x^2-2*x+1$$ (in Newtons) for $$0 \le x \le

Average Temperature Analysis

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

Average Value of a Polynomial Function

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

Average Velocity from Position Data

The position of a vehicle moving along a straight road is given in the table below. Use these data t

Bloodstream Drug Concentration

A drug enters the bloodstream at a rate given by $$R(t)= 5*e^{-0.5*t}$$ mg/min for $$t \ge 0$$. Simu

Boat Navigation Across a River with Current

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

Determining a Function from Its Derivative

A function $$F(x)$$ has a derivative given by $$F'(x)= 2*x - 4$$. Given that $$F(1)=3$$, determine $

Distance Traveled Analysis from a Velocity Graph

An object’s motion is recorded, and its velocity is shown on the graph provided for $$t \in [0,10]$$

Economic Profit Analysis via Area Between Curves

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

Electrical Charge Calculation

The current in an electrical circuit is given by $$I(t)=6*e^{-0.5*t} - 3*e^{-t}$$ (in amperes) for $

Filling a Container: Volume and Rate of Change

Water is being poured into a container such that the height of the water is given by $$h(t)=2*\sqrt{

Manufacturing Profit with Variable Rates

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

Motion along a Straight Path

A particle moving along the x-axis has its acceleration given by $$a(t)=6-4*t$$ (in m/s²) for $$t \g

Optimization of Average Production Rate

A manufacturing process has a production rate modeled by the function $$P(t)=50e^{-0.1*t}+20$$ (unit

Particle Motion and Integrated Functions

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

Piecewise Function Analysis

Consider a piecewise function defined by: $$ f(x)=\begin{cases} 3 & \text{for } 0 \le x < 2, \\ -x+5

Population Accumulation through Integration

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

Rebounding Ball

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

Resource Consumption in an Ecosystem

The rate of consumption of a resource in an ecosystem is given by $$C(t)=50*\ln(1+t)$$ (in units per

River Current Analysis

The velocity of a river is given by $$v(x)=2+x-0.1*x^2$$ (in m/s) for 0 ≤ x ≤ 10, where x measures t

Tank Draining with Variable Flow Rates

A water tank is undergoing simultaneous inflow and outflow. The inflow rate is given by $$I(t)=10+2\

Volume of a Rotated Region by the Disc Method

Consider the region bounded by the curve $$f(x)=\sqrt{x}$$ and the line $$y=0$$ for $$0 \le x \le 4$

Volume of Solid of Revolution: Bottle Design

A region under the curve $$f(x)=\sqrt{x}$$ on the interval 0 ≤ x ≤ 9 is rotated about the x-axis to

Water Flow in a River: Average Velocity and Flow Rate

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

Work in Pumping Water

A water tank is shaped as an inverted right circular cone with a height of $$10$$ meters and a top r

Work in Pumping Water from a Conical Tank

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