AP Calculus AB FRQ Room

Absolute Value Function and Discontinuity

Consider the function $$f(x)=\begin{cases} \frac{|x-5|}{x-5} & x\neq5 \\ 0 & x=5 \end{cases}$$. Answ

Absolute Value Function Limits

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

Analysis of a Removable Discontinuity in a Log-Exponential Function

Consider the function $$p(x)= \frac{e^{x}-e}{\ln(x)-\ln(1)}$$ for $$x \neq 1$$. The function is unde

Analyzing End Behavior and Asymptotes

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

Analyzing Process Data for Continuity

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

Applying the Squeeze Theorem with Trigonometric Function

Consider the function $$ f(x)= x^2 \sin(1/x) $$ for $$x\ne0$$, with $$f(0)=0$$. Use the Squeeze Theo

Complex Rational Limit and Removable Discontinuity

Consider the function $$f(x)=\frac{x^3 - 27}{x^2 - 9}$$ for x \(\neq\) 3. Answer the following parts

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 in a Piecewise Function with Square Root and Rational Expression

Consider the function $$f(x)=\begin{cases} \sqrt{x+6}-2 & x<-2 \\ \frac{(x+2)^2}{x+2} & x>-2 \\ 0 &

Continuity of a Radical Function

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

Direct Substitution in a Polynomial Function

Consider the polynomial function $$f(x)=2*x^2 - 3*x + 5$$. Answer the following parts concerning lim

Economic Limit and Continuity Analysis

A company's profit (in thousands of dollars) from producing x items is modeled by the function $$P(x

Graph Analysis of Discontinuities

A graph of a function f(x) shows a jump discontinuity at x = 1 and a removable discontinuity (a hole

Horizontal Asymptote of a Rational Function

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

Implicit Differentiation with Rational Exponents

Consider the curve defined by $$x^{2/3} + y^{2/3} = 4$$. Answer the following:

Intermediate Value Theorem Application

Consider the continuous function $$f(x)= x^3 - 4*x + 1$$. Answer the following parts.

Inverse Function and Limit Behavior Analysis

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

Investigating Discontinuities in a Rational Function

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

Limit Analysis in Population Modeling

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

Limits Involving Radical Functions

Evaluate the limit $$\lim_{x \to 4} \frac{\sqrt{x} - 2}{x - 4}$$.

Limits Involving Radicals and Algebra

Consider the function $$f(x)= \sqrt{x^2 + x} - x$$. Answer the following parts.

Limits Near Vertical Asymptotes

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

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 Population Growth with a Limit

A population P(t) is modeled by the function $$P(t) = \frac{5000}{1 + 40e^{-0.5*t}}$$ for t ≥ 0. Ans

Modeling Temperature Change: A Real-World Limit Problem

A scientist records the temperature (in °C) of a chemical reaction during a 24-hour period using the

One-Sided Limits and Vertical Asymptotes

Consider the function $$ f(x)= \frac{1}{x-4} $$.

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 Continuity and IVT

Let $$f(x)=\begin{cases} x^2, & x \leq 1 \\ a*x+b, & x > 1 \end{cases}$$. Determine constants a and

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 Evaluation

Consider the function $$f(x) = \frac{(x + 3) * (x - 2)}{x + 3}$$ for $$x \neq -3$$. Answer the follo

Return on Investment and Asymptotic Behavior

An investor’s portfolio is modeled by the function $$P(t)= \frac{0.02t^2 + 3t + 100}{t + 5}$$, where

Squeeze Theorem with an Oscillatory Term

Consider the function $$f(x) = x^2 \cdot \cos\left(\frac{1}{x^2}\right)$$ for $$x \neq 0$$, and defi

Analyzing a Function with a Removable Discontinuity

Consider the function $$f(x)= \frac{x^2 - 4}{x-2}$$, defined for $$x \neq 2$$.

Analyzing Concavity and Inflection Points Using Derivatives

Let $$f(x)=x^4 - 4*x^3 + 6*x^2$$. Answer the following questions:

Analyzing Rate of Change in Economics

The cost function for producing $$x$$ units of product (in dollars) is given by $$C(x)= 0.5*x^2 - 8*

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

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

Difference Quotient for a Cubic Function

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

Differentiation of a Composite Motion Function

A particle’s position is given by $$s(t) = t^2 * \ln(t)$$ for $$t > 0$$. Use differentiation to anal

Differentiation of Exponential Functions

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

Economic Marginal Revenue

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

Evaluating Derivative of a Composite Function using the Definition

Consider the function $$h(x)=\sqrt{4+x}$$. Answer the following questions:

Evaluating Limits and Discontinuities in a Piecewise Function

Consider the function given by $$ f(x)=\begin{cases} \frac{x^2-9}{x-3} & \text{if } x\neq 3, \\

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

Highway Traffic Flow Analysis

Vehicles enter a highway ramp at a rate given by $$f(t)=60+4*t$$ (vehicles/min) and exit the highway

Implicit Differentiation in Motion

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

Implicit Differentiation of a Circle

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

Instantaneous Rate of Change of Temperature

The temperature in a room is modeled by $$T(t)=20+5*\sin\left(\frac{\pi*t}{12}\right)$$, where $$t$$

Inverse Function Analysis: Exponential Function

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

Inverse Function Analysis: Rational Function

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

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 Profit Calculation

A company’s profit (in thousands of dollars) is given by $$P(x)= -2*x^2 + 50*x - 100$$, where $$x$$

Medication Infusion with Clearance

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

Optimization in Revenue Models

A company's revenue function is given by $$R(x)= x*(50 - 2*x)$$, where $$x$$ represents the number o

Projectile Motion Analysis

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

Quotient Rule Application

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

Rainfall-Runoff Model

A reservoir receives water from rainfall at a rate modeled by $$R_{in}(t)=10*\sin\left(\frac{\pi*t}{

Rates of Change from Experimental Data

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

River Crossover: Inflow vs. Damming

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

Secant and Tangent Lines

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

Secant Slopes Limit Interpretation

For a function $$f(x)$$, the secant slopes over the interval from $$x$$ to $$x+h$$ are given by the

Temperature Change Analysis

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

Advanced Composite Function Differentiation with Multiple Layers

Consider the function $$f(x)= \ln\left(\sqrt{1+e^{3*x}}\right)$$.

Analyzing Composite Functions Involving Inverse Trigonometry

Let $$y=\sqrt{\arccos\left(\frac{1}{1+x^2}\right)}$$. Answer the following:

Chain Rule in an Economic Model

In a manufacturing process, the cost function is given by $$C(q)= (\ln(1+3*q))^2$$, where $$q$$ is t

Chain Rule in Angular Motion

An object rotates such that its angular position is given by $$\theta(t)= \arctan(3*t^2)$$, where $$

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 Logarithms

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

Chemical Reaction Rate: Exponential and Logarithmic Model

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

Composite and Inverse Differentiation in Production Analysis

A factory’s production output is modeled by the composite function $$Q(x)= f(g(x))$$, where $$g(x)=

Composite Function from an Implicit Equation

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

Derivative of an Inverse Function: Quadratic Case

Let $$f(x)=x^2+2$$ for $$x \ge 0$$ and let $$g = f^{-1}$$ be its inverse function.

Differentiating an Inverse Trigonometric Function

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

Differentiation of Inverse Trigonometric Function via Implicit Differentiation

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

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 and Inverse Function Differentiation Combined

Suppose that $$x$$ and $$y$$ are related by the equation $$x^2+y^2-\sin(x*y)=4$$. Answer the followi

Implicit Differentiation in a Cubic Relationship

Examine the curve defined by $$x^3 + y^3 = 6*x*y$$, which represents a complex relationship between

Implicit Differentiation in Elliptical Orbits

Consider an elliptical orbit described by the equation $$\frac{x^2}{16}+\frac{y^2}{9}=1$$, where the

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 of an Ellipse

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

Implicit Differentiation of an Exponential-Product Equation

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

Implicit Differentiation of Quadratic Curve

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

Implicit Differentiation with Mixed Functions

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

Implicit Differentiation with Trigonometric and Logarithmic Terms

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

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 Derivative

Suppose that $$f$$ is a differentiable and one-to-one function. Given that $$f(4)=10$$ and $$f'(4)=2

Inverse Function Derivative for a Logarithmic Function

Let $$f(x)=\ln(x+1)-\sqrt{x}$$, which is one-to-one on its domain.

Inverse Function Differentiation in Temperature Conversion

In a temperature conversion model, the function $$f(T)=\frac{9}{5}*T+32$$ converts Celsius temperatu

Inverse Function Differentiation in Temperature Conversion

Consider the function $$f(x)= \frac{1}{1+e^{-0.5*x}}$$, which converts a temperature reading in Cels

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

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:

Multiple Applications: Chain Rule, Implicit, and Inverse Differentiation

Consider the function \(f(x)= e^{x^2}\) and note that it has an inverse function \(g\). In addition,

Nested Trigonometric Function Analysis

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

Rate of Change in a Circle's Shadow

The equation of a circle is given by $$x^2 + y^2 = 36$$. A point \((x,y)\) on the circle corresponds

Related Rates in a Circular Colony

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

Second Derivative via Implicit Differentiation

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

Tangent Lines on an Ellipse

Consider the ellipse given by $$\frac{x^2}{9} + \frac{y^2}{4} = 1$$. Use the graph provided to aid i

Temperature Reaction Rate Analysis

A chemical reaction experiment measures the temperature T (in °C) at various times t (in minutes) as

Airplane Altitude Change

An airplane's altitude (in meters) as a function of time is modeled by $$A(t)= 500*t - 4.9*t^2 + 100

Chemical Reaction Rate Analysis

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

Coffee Cooling Analysis Revisited

The temperature of a cup of coffee is modeled by $$T(t)=70+50e^{-0.05t}$$ (in °F), where $$t$$ is ti

Cost Estimation using Linearization

The cost (in dollars) to manufacture $$x$$ items is given by $$C(x) = 0.005x^3 - 0.2x^2 + 50x + 200$

Deceleration with Air Resistance

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

Determining the Tangent Line

Consider the function $$f(x)=\ln(x)+ x$$. The graph of the function is provided for reference.

Dynamics of a Car: Stopping Distance and Deceleration

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

Economic Cost 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 $

Economic Inflation Rate

The cost of a commodity is modeled by $$C(t)=100e^{0.03*t}$$ dollars, where t is in years.

Function with Vertical Asymptote

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

L’Hôpital’s Rule in Limit Evaluation

Evaluate the limit $$\lim_{x\to\infty}\frac{5*x^3 - 4*x + 1}{2*x^3 + 7*x^2 - x + 6}$$ using L’Hôpita

L'Hôpital’s Rule in Limits with Contextual Application

Consider the function $$f(x)= \frac{e^{2*x} - 1}{5*e^{2*x} - 5}$$, which models a growth phenomenon.

Linear Approximation for Function Values

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

Linear Approximations: Estimating Function Values

Let $$f(x)=x^4$$. Use linear approximation to estimate $$f(3.98)$$. Answer the following:

Linearization for Approximating Powers

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

Local Linearization Approximation

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

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

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

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

Radioactive Decay: Rate of Change and Half-life

A radioactive substance decays according to the formula $$N(t)=N_0e^{-kt}$$, where $$N(t)$$ is the a

Ramped Conveyor Belt

Boxes on a conveyor belt move along a ramp with position given by $$s(t)=2*t^2+3*t$$ meters. Their s

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 Rate and Temperature

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

Related Rates: The Expanding Ripple

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

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

Temperature Cooling in a Cup of Coffee

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

Temperature Rate Change in Cooling Coffee

A cup of coffee cools following the model $$x(t)=70+50e^{-0.1t}$$, where x is in degrees Fahrenheit

The Sliding Ladder

A 10 m long ladder is leaning against a vertical wall. The bottom of the ladder slides away from the

Train Motion Analysis

A train’s acceleration is given by $$a(t)=\sin(t)+0.5$$ (m/s²) for $$0 \le t \le \pi$$ seconds. The

Absolute Extrema via the Candidate's Test

Consider the function $$f(x)= \sqrt{x} - x$$ on the closed interval $$[0,4]$$. Use the candidate's t

Analyzing Concavity and Inflection Points

Consider the function $$f(x) = x^4 - 4*x^3 + 6*x^2$$. 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:

Application of the Mean Value Theorem on a Piecewise Function

Consider the function $$ f(x) = \begin{cases} x^2, & x < 2, \\ 4x - 4, & x \ge 2. \end{cases} $$ A

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

Chemical Mixing in a Tank

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

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 Minimization in Transportation

A transportation company recorded shipping costs (in thousands of dollars) for different numbers of

Cost Optimization Using Derivatives

A company’s cost function for producing a certain product is modeled by $$C(x)= 2*x^3 - 9*x^2 + 12*x

Evaluating Rate of Change in Electric Current Data

An electrical engineer recorded the current (in amperes) in a circuit over time. The table below sho

FRQ 5: Concavity and Points of Inflection for a Cubic Function

For the function $$f(x) = x^3 - 6*x^2 + 9*x + 2$$, analyze its concavity.

FRQ 7: Maximizing Revenue in Production

A company’s revenue function is modeled by $$R(x)= -2*x^2 + 40*x$$ (in thousands of dollars), where

FRQ 16: Finding Relative Extrema for a Logarithmic Function

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

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 Linear Function

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

Inverse Analysis of a Quadratic Function (Restricted Domain)

Consider the function $$f(x)=x^2-4*x+7$$ defined on the restricted domain $$[2, \infty)$$. Analyze t

Inverse Analysis: Logarithmic Ratio Function in Financial Context

Consider the function $$f(x)=\ln\left(\frac{x+4}{x+1}\right)$$ with domain $$x > -1$$. This function

Jump Discontinuity in a Piecewise Linear Function

Consider the piecewise function $$ f(x) = \begin{cases} 2x + 1, & x < 3, \\ 2x - 4, & x \ge 3. \end

Local Linear Approximation of a Trigonometric Function

Consider the function $$f(x)= \cos(x)$$ and its behavior near $$x=0$$.

Mean Value Theorem Analysis

A particle moves along a straight line and its position is given by $$s(t)= t^3 - 3*t^2 + 2*t$$ for

Mean Value Theorem for a Cubic Function

Consider the function $$f(x)= x^3 - 2*x^2 + x$$ on the closed interval $$[0,2]$$. In this problem, y

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

Oil Spill Cleanup

In an oil spill scenario, oil continues to enter an affected area while cleanup efforts remove oil.

Optimization of a Rectangle Inscribed in a Semicircle

A rectangle is inscribed in a semicircle of radius 5 (with the base along the diameter). The top cor

Pharmaceutical Drug Delivery

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

Pharmacokinetics: Drug Concentration Decay

A drug in the bloodstream decays according to $$D(t)= D_0*e^{-k*t}$$, where $$t$$ is in hours. Answe

Piecewise Function and the Mean Value Theorem

Consider the piecewise function $$f(x)= \begin{cases} x^2 & \text{if } x \le 1, \\ 2*x - 1 & \text{

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

Relationship Between Integration and Differentiation

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

Sand Pile Dynamics

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

Sign Analysis of f'(x)

The first derivative $$f'(x)$$ of a function is known to have the following behavior on $$[-2,2]$$:

Slope Analysis for Parametric Equations

A curve is defined parametrically by $$x(t)= t^2$$ and $$y(t)= t^3 - 3*t$$ for $$t$$ in the interval

Temperature Analysis Over a Day

The temperature $$f(x)$$ (in $$^\circ C$$) at time $$x$$ (in hours) during the day is modeled by $$f

Temperature Regulation in a Greenhouse

A greenhouse is regulated by an inflow of warm air and an outflow of cooler air. The inflow temperat

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

Accumulated Altitude Change: Hiking Profile

During a hike, a climber's rate of change of altitude (in m/hr) is recorded as shown in the table be

Accumulated Bacteria Growth

A laboratory observes a bacterial colony whose rate of growth (in bacteria per hour) is modeled by t

Accumulated Change Function Evaluation

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

Accumulation and Flow Rate in a Tank

Water flows into a tank at a rate given by $$R(t)=3*t^2-2*t+1$$ (in m³/hr) for $$0\le t\le2$$. The t

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].

Comparing Riemann Sum Methods for a Complex Function

Consider the function $$f(x)=\frac{1}{1+x^2}$$ on the interval [0,1]. Answer the following:

Cooling of a Liquid Mixture

In a tank, the cooling rate is given by $$C(t)=20e^{-0.3t}$$ J/min while an external heater adds a c

Evaluating a Radical Integral via U-Substitution

Evaluate the integral $$\int_{1}^{9}\sqrt{2*x+1}\,dx$$ using U-substitution. Answer the following pa

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

FRQ19: Inverse Analysis with a Fractional Integrand

Let $$ M(x)=\int_{2}^{x} \frac{t}{t+2}\,dt $$. Answer the following parts.

Mixed Method Approximation of an Integral

A function $$f(t)$$ that represents a biological rate is recorded over time. Use the table below to

Modeling Accumulated Revenue over Time

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

Net Surplus Calculation

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

Particle Motion with Changing Direction

A particle moves along a line with its velocity given by $$v(t)=t*(t-4)$$ (in m/s) for $$0\le t\le6$

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 $$

Pollutant Concentration in a River

Pollutants are introduced into a river at a rate $$D(t)= 8\sin(t)+10$$ kg/hr while a treatment plant

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 for Sin(x)

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

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 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

Volume of a Solid with Square Cross Sections

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

Water Accumulation in a Tank

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

Analyzing Slope Fields for $$dy/dx=x\sin(y)$$

Consider the differential equation $$\frac{dy}{dx}=x\sin(y)$$. A corresponding slope field is provid

Bacterial Growth with Constant Removal

A bacterial colony (in thousands) grows according to the differential equation $$\frac{dP}{dt}=0.4P-

Bacterial Nutrient Depletion

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

Chemical Reaction Rate and Concentration Change

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

CO2 Absorption in a Lake

A lake absorbs CO2 from the atmosphere. The concentration $$C(t)$$ of dissolved CO2 (in mol/m³) in t

Combined Cooling and Slope Field Problem

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

Comparative Population Decline

A population declines according to two models. Model 1 follows simple exponential decay: $$\frac{dN}

Cooling of a Hot Beverage

According to Newton's Law of Cooling, the temperature $$T(t)$$ of a hot beverage satisfies $$\frac{d

Differential Equation with Substitution using u = y/x

Consider the differential equation $$\frac{dy}{dx}= \frac{y}{x}+\sqrt{\frac{y}{x}}$$. Use the substi

Implicit Differentiation of a Transcendental Equation

Consider the equation $$e^{x*y} + y^3= x$$. Answer the following:

Integrating Factor Method

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

Inverse Function Analysis of a Differential Equation Solution

Consider the function $$f(x)=\sqrt{4*x+9}$$, which arises as a solution to a differential equation i

Investigating a Piecewise Function's Discontinuity

Consider the function $$ f(x)=\begin{cases} \frac{x^2-9}{x-3}, & x\neq 3\\ 5, & x=3 \end{cases} $$

Linear Differential Equation and Integrating Factor

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

Logistic Growth Model

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

Logistic Population Growth Model

A fish population in a lake is modeled by the logistic differential equation $$\frac{dP}{dt} = 0.3\,

Mixing of a Pollutant in a Lake

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

Mixing Problem in a Tank

A tank initially contains 100 liters of brine with 10 kg of dissolved salt. Brine with a concentrati

Mixing Problem with Time-Dependent Inflow Rate

A tank initially holds 200 L of water with 10 kg of salt. Brine containing 0.2 kg/L of salt flows in

Mixing Problem with Variable Volume

A tank initially contains 200 liters of solution with 10 kg of solute. A solution with concentration

Newton's Law of Cooling with Temperature Data

A thermometer records the temperature of an object cooling in a room. The object's temperature $$T(t

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

Oil Spill Cleanup Dynamics

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

Radioactive Decay

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

Radioactive Decay and Half-Life

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

Salt Tank Mixing Problem

A tank initially contains 100 liters of pure water. A salt solution with concentration 0.5 kg/L is p

Sand Erosion in a Beach Model

During a storm, a beach loses sand. Let $$S(t)$$ (in tons) be the amount of sand on a beach at time

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)

Soot Particle Deposition

In an environmental study, the thickness $$P$$ (in micrometers) of soot deposited on a surface is me

Tank Draining Differential Equation

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

Area Between Two Curves from Tabulated Data

Consider two functions, $$f(x)$$ and $$g(x)$$, whose values are recorded in the table below over the

Average and Instantaneous Rates in a Cooling Process

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

Average Drug Concentration in the Bloodstream

The concentration of a drug in the bloodstream is modeled by $$C(t)=\frac{20*t}{1+t^2}$$ (in mg/L) f

Average Temperature of a Cooling Liquid

The temperature of a cooling liquid is modeled by $$T(t)=50*e^{-0.1*t}+20$$ (in $$^\circ C$$) for $$

Average Value of a Function in a Production Process

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

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

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

Determining Field Area from Intersection of Curves

A farmer's field is bounded by the curves $$y=0.5*x^2$$ and $$y=4*x$$. Find the area of the field wh

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{

Funnel Design: Volume by Cross Sections

A funnel is designed by rotating the region bounded by the curve $$y=4-x^2$$ for -2 ≤ x ≤ 2 about th

Investment Compound Interest

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

Loaf Volume Calculation: Rotated Region

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

Motion Experiment with Sinusoidal Acceleration

A particle has an acceleration given by $$a(t)=2\sin(t)$$ (in m/s²) for 0 ≤ t ≤ 2π. The initial cond

Net Change in Concentration of a Chemical Reaction

In a chemical reaction, the rate of production of a substance is given by $$r(t)$$ (in mol/min). The

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 $$t=0$$,

Population Growth and Average Rate

A town's population is modeled by the function $$P(t)=1000*e^{0.03*t}$$, where $$t$$ is measured in

Position Analysis of a Particle with Piecewise Acceleration

A particle moving along a straight line experiences a piecewise constant acceleration given by $$a(

Projectile Motion: Time of Maximum Height

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

Surface Area of a Rotated Curve

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

Temperature Increase in a Chemical Reaction

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

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 Disc Method for a Rotated Region

Consider a function $$f(x)$$ that represents the radius (in cm) of a region rotated about the x-axis

Volume Calculation via Cross-Sectional Areas

A solid has cross-sectional areas perpendicular to the x-axis that are circles with radius given by

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

Volume of a Solid with Square Cross Sections

A solid is formed over the region under the line $$f(x)=4-x$$ from $$x=0$$ to $$x=4$$ in the x-y pla

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

Water Pumping from a Parabolic Tank

A water tank has ends shaped by the region bounded by $$y=x^2$$ and $$y=4$$, and the tank has a unif

Work Done by a Variable Force

A variable force $$F(x)$$ (in Newtons) acts on an object as it moves along a straight line from $$x=