AP Calculus AB FRQ Room

Advanced Analysis of a Piecewise Function

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

Analyzing a Discontinuous Function with a Sequence Component

The function is given by $$f(x) = \frac{\sin(\pi x)}{\pi (x - 1)}$$ for $$x \neq 1$$ (with f(1) unde

Analyzing a Piecewise Function’s Limits and Continuity

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

Analyzing Asymptotic Behavior in a Rational Function

Consider the function $$f(x)=\frac{2*x^2 - 3*x + 1}{x^2 + 4}$$. Answer the following:

Analyzing Multiple Discontinuities in a Rational Function

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

Application of the Squeeze Theorem

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

Asymptotic Analysis of a Radical Rational Function

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

Continuity Analysis of a Radical Function

Consider the function $$f(x) = \frac{\sqrt{x+4} - 2}{x}$$. (a) Evaluate $$\lim_{x \to 0} f(x)$$. (b

Continuity of a Composite Log-Exponential Function

Let $$f(x)=\begin{cases} \frac{\ln(1+e^{x})-\ln(2)}{x} & \text{if } x\neq 0, \\ C & \text{if } x=0.

Continuity of a Sine-over-x Function

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

Determining Asymptotes and Holes in a Rational Function

Consider the function $$f(x) = \frac{2x^2 - 3x + 1}{x^2 - 4}$$. This function may exhibit vertical a

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

Determining Parameters for Continuity in a Piecewise Function

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

End Behavior of Rational Functions

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

Estimating Derivatives Using Limit Definitions from Data

The position of an object (in meters) is recorded at various times (in seconds) in the table below.

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 Transformations and Continuity

Let $$f(x)=\sqrt{x}$$ and consider the function $$g(x)= f(x-2)+3= \sqrt{x-2}+3$$.

Graph-Based Analysis of Discontinuities

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

Graph-Based Analysis of Discontinuity

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

Investigating Discontinuities in a Rational Function

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

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 Near Vertical Asymptotes

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

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

Numerical Estimation of a Limit Using a Table

A student investigates the function \(f(x)=\frac{x^2-1}{x-1}\) for \(x\neq1\) by creating a table of

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 }

Oscillatory Function and the Squeeze Theorem

Consider the function $$f(x)= x*\sin(1/x)$$ for $$x \neq 0$$, with $$f(0)=0$$. Answer the following:

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

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 $$t > 1$$.

Redefining a Function for Continuity

A function is defined by $$f(x) = \frac{x^2 - 1}{x - 1}$$ for $$x \neq 1$$, while $$f(1)$$ is left u

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 $

Removing a Removable Discontinuity

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

Squeeze Theorem Application with Trigonometric Functions

Let $$f(x)= x^2 \sin(1/x)$$ for $$x\neq0$$, and define $$f(0)=0$$.

Squeeze Theorem for an Oscillatory Function

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

Squeeze Theorem with Trigonometric Function

Consider the function \(h(x)=x^2\cos(1/x)\) for \(x\neq0\) with \(h(0)=0\). Answer the following:

Vertical Asymptote Analysis

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

Vertical Asymptote Analysis

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

Air Quality and Pollution Removal

A city experiences pollutant inflow at a rate of $$f(t)=30+2*t$$ (micrograms/m³·hr) and pollutant re

Analyzing the Derivative of a Trigonometric Function

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

Application of Product Rule

Differentiate the function $$f(x)=(3x^2+2x)(x-4)$$ by two methods. Answer the following:

Approximating Derivatives Using Secant Lines

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

Behavior of $$f(x)= e^{x} - x$$

Consider the function $$f(x)= e^{x} - x$$, which combines exponential growth and a linear term.

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

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 Applications in Population Growth

A population of cells is modeled by $$P(t)=100*e^{0.2*t}$$, where $$t$$ is in hours. Answer the foll

Derivative of a Logarithmic Function

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

Derivatives of Trigonometric Functions

Let $$f(x)=\sin(x)+\cos(x)$$, where $$x$$ is measured in radians. This function may represent a comb

Difference Quotient for a Cubic Function

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

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

Differentiation of Exponential Functions

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

Graphical Interpretation of Rate of Change

Consider the graph of a function provided in the stimulus which shows a vehicle's displacement over

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

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

Inverse Function Analysis: Cubic Function

Consider the function $$f(x)=x^3+2*x+1$$ defined for all real numbers.

Inverse Function Analysis: Logarithmic Transformation

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

Inverse Function Analysis: Logarithmic-Hyperbolic Function

Consider the function $$f(x)=\ln\left(x+\sqrt{x^2+1}\right)$$ defined for all real x. (This function

Inverse Function Analysis: Trigonometric Function with Linear Term

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

Marginal Cost 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

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 +

Radioactive Decay Analysis

The amount of a radioactive substance is modeled by $$N(t)= 200*e^{-0.03*t}$$, where $$t$$ is in yea

Real-World Cooling Process

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

Riemann Sums and Derivative Estimation

A car’s position $$s(t)$$ in meters is recorded in the table below at various times $$t$$ in seconds

Secant and Tangent Lines

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

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

Secant Slope from Tabulated Data

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

Secant vs. Tangent Rate Comparison

For the function $$f(x)=x^2$$, we analyze the relationship between the secant and tangent approximat

Tangent Line and Differentiability

Let $$h(x)=\frac{1}{x+2}$$, modeling the concentration of a substance in a chemical solution over ti

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

Analyzing Composite Functions Involving Inverse Trigonometry

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

Chain Rule in an Implicitly Defined Function

Consider the equation $$\tan(x+y)=x^2-y^2$$. Answer the following:

Chain Rule with Multiple Nested Functions in a Physics Model

In a physics experiment, the displacement of a particle is modeled by the function $$s(t)=\sqrt{\cos

Composite Function and Tangent Line

Consider the function $$f(x)=\sqrt{3*x^2+2*x+1}$$. (Note: All derivatives are to be computed without

Composite Function Chain Reaction

A chemist models the concentration of a reacting solution at time $$t$$ (in seconds) with the compos

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 Finance

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

Composite Function with Inverse Trigonometric Outer Function

Consider the function $$H(x)=\arctan(\sqrt{x^2+1})$$. Answer the following parts.

Composite Log-Exponential Function Analysis

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

Designing a Tapered Tower

A tower has a circular cross-section where the relationship between the radius r (in meters) and the

Implicit Curve Analysis: Horizontal Tangents

Consider the curve defined implicitly by $$x^2+ e^(y)= 5$$. Answer the following:

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 in Circular Motion

A runner is moving along a circular track described by the equation $$x^2+y^2=16$$, where $$x$$ and

Implicit Differentiation in Logarithmic Functions

Consider the equation $$\ln(x)+\ln(y)=1$$. Answer the following:

Implicit Differentiation Involving Sine

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

Implicit Differentiation Involving Trigonometric Functions

For the relation $$\sin(x) + \cos(y) = 1$$, consider the curve defined implicitly.

Implicit Differentiation of a Circle

Consider the circle given by $$x^2+y^2=25$$. Answer the following:

Implicit Differentiation on a Circle

Consider the circle defined by $$x^2+y^2=25$$. Use implicit differentiation to find the slope of the

Implicit Differentiation with Exponential and Trigonometric Functions

Consider the equation $$e^{y}\cos(x)+ x*y=1$$. Answer the following:

Implicit Differentiation with Mixed Terms

Consider the equation $$x*y + y^2 = 10$$. Answer the following parts.

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 for a Log Function

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

Inverse Function Differentiation in 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 with Exponentials and Trigonometry

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

Inverse Function Differentiation with Logarithmic Function

Let $$f(x) = x + \ln(x)$$ and let g denote its inverse function. Answer the following parts.

Inverse Function Differentiation with Trigonometric Component

Let $$f(x) = \sin(x) + x$$ and let g denote its inverse function. Answer the following parts.

Inverse Trigonometric Differentiation

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

Manufacturing Optimization via Implicit Differentiation

A manufacturing cost relationship is given implicitly by $$x^2*y + x*y^2 = 1000$$, where $$x$$ repre

Multilayer Composite Function Differentiation

Let $$y=\cos(\sqrt{5*x+3})$$. Answer the following:

Pendulum Angular Displacement Analysis

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

Projectile Motion and Composite Function Analysis

A projectile is launched and its height $$h(t)$$ (in meters) is recorded at various times t (in seco

Second Derivative via Implicit Differentiation

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

Water Tank Optimization Using Composite Functions

A water tank has an inflow rate given by $$I(t)= 3+2\sin(0.1*t)$$ and an outflow rate given by $$O(t

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

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

Blood Drug Concentration

In a pharmacokinetic study, the concentration of a drug in the bloodstream is modeled by $$D(t)=\fra

Cooling Coffee: Exponential Decay Model

A cup of coffee cools according to $$T(t) = 70 + 50e^{-0.1t}$$, where $$T(t)$$ (in °F) is the temper

Cost Efficiency in Production

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

Designing a Flower Bed: Optimal Shape

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

Economic Inflation Rate

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

Expanding Circular Ripple in a Pond

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

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

Exponential Decay in Radioactive Material

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

FRQ 7: Conical Water Tank Filling

A conical water tank has a total height of 10 m and a top radius of 4 m. The water in the tank has a

FRQ 20: Market Demand Analysis

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

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 Rates

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

Modeling a Bouncing Ball with a Geometric Sequence

A ball is dropped from a height of 10 m. Each time it bounces, it reaches 70% of the height of the p

Optimization in Packaging

An open-top box with a square base is to be constructed so that its volume is fixed at $$1000\;cm^3$

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

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

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

Related Rates in Shadows: A Lamp and a Tree

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

Related Rates: Inflating Balloon

A spherical balloon is being inflated such that its volume increases at a rate of $$15\;cm^3/s$$. Th

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(

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

Water Flow Rate in a Tank

Water flows into a tank at a rate given by $$r(t)=\frac{2t+1}{t+4}$$ liters per minute, where $$t$$

Analysis of an Exponential-Logarithmic Function

Consider the function $$f(x)= e^{x} - 3*\ln(x)$$ defined for $$x>0$$. Answer the following:

Analyzing a Piecewise Function and Differentiability

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

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 Concavity and Inflection Points

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

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

Consider the functions $$f(x)= \sin(x)$$ and $$g(x)= \cos(x)$$ on the interval $$[0, \frac{\pi}{2}]$

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

Cooling of a Cup of Coffee

A cup of coffee cools according to the model $$T(t)= T_{room}+(T_{initial}-T_{room})e^{-kt}$$ with $

Cubic Polynomial Analysis

Consider the cubic function $$f(x)= x^3 - 6*x^2 + 9*x + 2$$ defined on the interval $$[0,4]$$. Analy

Determining Intervals of Concavity for a Logarithmic Function

Consider the function $$f(x)= \ln(x) - x$$ defined on the interval \([1, e]\). Answer the following:

Discontinuity and Derivative in a Hybrid Piecewise Function

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

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

FRQ 2: Daily Temperature Extremes and the Extreme Value Theorem

A function modeling the ambient temperature (in $$^\circ C$$) during the first 6 hours of a day is g

FRQ 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 18: Marginal Cost Analysis and Concavity

The cost per unit of producing $$x$$ units is given by $$C(x)= 100 + 20*x - 0.5*x^2$$ for $$0 \le x

Function Behavior Analysis Using Derivatives

Examine the function $$f(x) = \ln(x) + x$$, where $$x > 0$$.

Identification of Extrema and Critical Points

Let $$f(x)= x^3 - 6*x^2 + 9*x + 1$$ be defined on the interval $$[0,4]$$. Use your understanding of

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 $$[-1, 3]$$.

Inflection Points and Concavity in a Real-World Cost Function

A company's cost function is given by $$C(x) = 0.5*x^3 - 6*x^2 + 20*x + 100$$, where \( x \) represe

Inflection Points in a Population Growth Model

Population data from a species over several years is provided in the table below. Use this informati

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 Cubic Function

Consider the function $$f(x)=x^3+1$$. Answer all parts regarding its inverse below.

Inverse Analysis of a Trigonometric Function on a Restricted Domain

Consider the function $$f(x)=\sin(x)$$ with the restricted domain $$\left[-\frac{\pi}{2},\frac{\pi}{

Logistic Population Model Analysis

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

Maximizing Revenue

A company’s revenue (in hundreds of dollars) is modeled by the function $$R(x)= 80*x - 2*x^3$$, wher

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

Polynomial Rational Discontinuity Investigation

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

Rate of Change in an Oil Spill

An oil spill is spreading so that its area is increasing at a constant rate of $$100$$ square meters

Rational Function Optimization

Consider the rational function $$f(x)= \frac{x^2 + 1}{x - 1}$$ defined on the interval $$[2,6]$$. An

Reservoir Sediment Accumulation

A reservoir experiences sediment deposition from rivers and sediment removal via dredging. The sedim

Verifying the Mean Value Theorem for a Polynomial Function

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

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

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

Analyzing Bacterial Growth via Riemann Sums

A biologist measures the instantaneous growth rate of a bacterial population (in thousands of cells

Calculating Total Distance Traveled from a Changing Velocity Function

A particle moves with a velocity given by $$v(t)=t^2 - 4*t + 3$$ (in m/s) for $$0 \le t \le 5$$. Not

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 $

Consumer Surplus and Definite Integrals in Economics

The demand function for a product is given by $$p(q)= 100 - 2*q$$, where $$p$$ is the price in dolla

Cost Accumulation in a Production Process

A factory's marginal cost function is given by $$C'(x)=5*\sqrt{x}$$ dollars per item, where $$x$$ re

Economic Cost Function Analysis

A company's marginal cost (in dollars per unit) is recorded at various production levels. Use the da

Elevation Profile Analysis on a Hike

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

Estimating an Integral Using the Midpoint Rule

For the function $$f(x)=\ln(x)$$ defined on the interval [1, e], answer the following:

Evaluating an Integral with a Trigonometric Function

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

Exploring the Fundamental Theorem of Calculus

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

Finding the Area of a Parabolic Arch

An architect designs an arch described by the parabola $$y = 10 - \frac{x^{2}}{5}$$. The arch spans

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

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.

FRQ17: Inverse Analysis of a Biologically Modeled Accumulation Function

In a biological study, the net concentration of a chemical is modeled by $$ B(t)=\int_{0}^{t} (0.6*t

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

Implicit Differentiation of a Conic

Consider the relation $$x^2 + x*y + y^2 = 7.$$ Answer the following parts:

Improving Area Approximations with Increasing Subintervals

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

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

Modeling Water Volume in a Tank via Integration

A tank is being filled with water at a rate given by $$R(t)= \frac{50}{t+2}$$ cubic meters per minut

Net Change vs Total Accumulation in a Velocity Function

A particle moves with velocity $$v(t)=5-t^2$$ (in m/s) for t in [0,4]. Answer the following:

Population Accumulation in a Lake

A researcher is studying a fish population in a lake. The rate of change of the fish population is m

Population Change in a Wildlife Reserve

In a wildlife reserve, animals immigrate at a rate of $$I(t)= 10\cos(t) + 20$$ per month, while emig

Related Rates: Expanding Balloon

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

Tabular Riemann Sums for Electricity Consumption

A household's daily electricity consumption (in kWh) over 5 consecutive days is recorded in the tabl

Temperature Change in a Chemical Reaction

During an exothermic chemical reaction, the temperature (in °C) is recorded over a 15-minute period.

Temperature Change in a Reactor

In a chemical reactor, the internal heating is modeled by $$H(t)= 10+2\cos(t)$$ °C/min and cooling o

Temperature Cooling: An Initial Value Problem

An object cools according to the differential equation $$T'(t)=-0.2\,(T(t)-20)$$, where $$T(t)$$ is

Bacterial Growth with Constant Removal

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

Drug Concentration with Continuous Infusion

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

Environmental Pollution Model

Pollutant concentration in a lake is modeled by the differential equation $$\frac{dC}{dt}=\frac{R}{V

Heating and Cooling in an Electrical Component

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

Implicit Differentiation and Slope Analysis

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

Implicit Differentiation and Tangent Lines of an Ellipse

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

Implicit Differentiation Involving a Logarithmic Function

Consider the function defined implicitly by $$\ln(y) + x^2y = 7$$. Answer the following:

Investment Growth with Continuous Contributions

An investment account grows continuously with an annual interest rate of 5% while continuous deposit

Investment Growth with Continuous Deposits

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

Logistic Model with Harvesting

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

Logistic Population Growth

A population $$P(t)$$ grows according to the logistic model $$\frac{dP}{dt}=0.3P\Bigl(1-\frac{P}{100

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

Modeling Orbital Decay

A satellite’s altitude $$h(t)$$ decreases over time due to atmospheric drag, following $$\frac{dh}{d

Motion Along a Curve with Implicit Differentiation

A particle moves along the curve defined by $$x^2+ y^2- 2*x*y= 1$$. At a certain instant, its horizo

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 $$

Particle Motion with Variable Acceleration

A particle moves along a straight line with acceleration $$a(t)=3-2*t$$ (in m/s²). Its initial veloc

Population Growth with Harvesting

A fish population in a lake grows at a rate proportional to its current size, but fishermen harvest

Population Growth with Logistic Equation

A population grows according to the logistic differential equation $$\frac{dy}{dx} = 0.5*y\left(1-\f

Radioactive Decay

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

Separable Differential Equation

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

Separable Differential Equation with Initial Condition

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

Separable Differential Equation with Trigonometric Component

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

Separable Differential Equation: Growth Model

Consider the separable differential equation $$\frac{dy}{dx} = 3*x*y$$ with the initial condition $$

Slope Field and Solution Curve Analysis

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

Solving a Differential Equation Using the SIPPY Method

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

Tank Mixing and Salt Concentration

A tank initially contains 100 L of solution with 5 kg of dissolved salt. A salt solution with concen

Accumulated Nutrient Intake from a Drip

A medical nutrient drip administers a nutrient at a variable rate given by $$N(t)=-0.03*t^2+1.5*t+20

Accumulated Rainfall Calculation

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

Area Between a Parabola and a Line

Consider the functions $$f(x)=5*x-x^2$$ and $$g(x)=x$$. A set of experimental data provided the foll

Area Between an Exponential Function and a Linear Function

Consider the functions $$f(x)=e^{x}$$ and $$g(x)=x+1$$ on the interval $$[0,1]$$.

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

Arithmetic Savings Account

A person makes monthly deposits into a savings account such that the amount deposited each month for

Average Concentration in a Reaction

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

Average Concentration in Medical Dosage

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

Average 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

Average Temperature Analysis

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

Average Value of a Deposition Rate Function

During a sediment deposition experiment, the deposition rate (in mm/hr) was recorded over a 10-hour

Bacterial Colony Growth Analysis

A bacterial colony grows at a rate given by $$r(t)=20e^{0.1*t}$$ (in thousands per hour) over the ti

Consumer Surplus Calculation

The demand function for a certain product is given by $$D(p)=100-5*p$$ and the supply function by $$

Cost Analysis Through Area Between Curves

A company analyzes two different manufacturing cost models represented by the curves $$C_1(x)=50+3*x

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 $

Discontinuities in a Piecewise Function

Consider the function $$f(x)=\begin{cases} \frac{\sin(x)}{x} & \text{if } x \neq 0 \\ 2 & \text{if }

Distance Traveled Analysis from a Velocity Graph

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

Finding the Area Between Two Curves

Let the functions $$f(x)=5*x-x^2$$ and $$g(x)=x$$ be given. Find the area of the region bounded by t

Graduated Rent Increase

An apartment’s yearly rent increases by a fixed amount. The initial annual rent is $$1200$$ dollars

Hollow Rotated Solid

Consider the region in the first quadrant bounded by the curves $$y=x$$ and $$y=x^2$$. This region i

Manufacturing Output Increase

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

Motion Analysis Using Integration of a Sinusoidal Function

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

Radioactive Decay Accumulation

A radioactive substance decays at a rate given by $$r(t)= C*e^{-k*t}$$ grams per day, where $$C$$ an

Reconstructing Position from Acceleration Data

A particle traveling along a straight line has its acceleration given by the values in the table bel

Related Rates: Shadow Length Change

A 2-meter tall lamp post casts a shadow of a moving 1.7-meter tall person. Let $$x$$ be the distance

Total Distance from a Runner's Variable Velocity

A runner’s velocity (in m/s) is modeled by the function $$v(t)=t^2-10*t+16$$ for $$0 \le t \le 10$$

Volume by the Washer Method: Solid of Revolution

Consider the region bounded by the curves $$y=x$$ and $$y=x^2$$ for $$0 \le x \le 1$$. This region i

Volume of a 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 a Solid by the Washer Method

Consider the region in the first quadrant bounded by the line $$y=x$$, the line $$y=0$$, and the ver

Volume of a Solid of Revolution Rotated about a Line

Consider the region bounded by $$y=x^2$$ and $$y=x$$ for $$x\in [0,1]$$. This region is rotated abou

Volume of a Solid of Revolution: Disc Method

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

Volume of a Solid with a Hole Using the Washer Method

Consider the region bounded by $$y=x^2$$ and $$y=4$$. This region is revolved about the $$x$$-axis t

Work Calculation from an Exponential Force Function

An object is acted upon by a force modeled by $$F(x)=5*e^{-0.2*x}$$ (in newtons) along a displacemen

Work to Pump Water from a Cylindrical Tank

A cylindrical tank with a radius of 3 m and a height of 10 m is completely filled with water (densit