AP Calculus AB FRQ Room

Analysis of a Rational Function with Exponential and Logarithmic Components

Consider the function $$g(x)=\frac{e^{x}-1-\ln(1+x)}{x}$$ for $$x \neq 0$$. Evaluate the limit as $$

Analyzing Continuity in a Piecewise Function

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

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

Analyzing Multiple Discontinuities in a Rational Function

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

Analyzing Process Data for Continuity

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

Antiderivatives and Discontinuities in Acceleration to Velocity Transition

A particle's acceleration is given by $$a(t)= \frac{4*t-8}{t-2}$$ for \(t \neq 2\) and is defined to

Asymptotic Behavior of a Logarithmic Function

Consider the function $$w(x)=\frac{\ln(x+e)}{x}$$ for $$x>0$$. Analyze its behavior as $$x \to \inft

Continuity Analysis with a Piecewise-defined Function

A particle’s displacement is described by the piecewise function $$s(t)= \begin{cases} t^2+1, & t <

Continuity of Composite Functions

Let $$f(x)=x+2$$ for all x, and let $$g(x)=\begin{cases} \sqrt{x}, & x \geq 0 \\ \text{undefined},

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

Consider the function $$f(x)= \begin{cases} 2*x + k, & x < 1 \\ x^2, & x \geq 1 \end{cases}$$, where

Direct Substitution in a Polynomial Function

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

Estimating Limits from a Data Table

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

Evaluating Limits Involving Square Roots

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

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

Exponential Limit Parameter Determination

Consider the function $$f(x)=\frac{e^{3*x} - e^{k*x}}{x}$$ for $$x \neq 0$$, and define $$f(0)=L$$,

Factorization and Limit Evaluation

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

Graph-Based Analysis of Discontinuities

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

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

Identifying Discontinuities in a Rational Function

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

Intermediate Value Theorem in Context

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

Intermediate Value Theorem in Temperature Modeling

A continuous function $$ f(x) $$ describes the temperature (in °C) throughout a day, with $$f(0)=15$

Investigation of Continuity in a Piecewise Log-Exponential Function

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

Limits of a Nested Logarithmic Function

Given the function $$t(x)=\ln\left(\frac{e^{x}+1}{e^{x}-1}\right)$$, study its behavior as $$x \to 0

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 }

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 Removable Discontinuity

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

Particle Motion with Vertical Asymptote in Velocity

A particle moves along a number line with velocity function $$v(t)= \frac{3*t}{t-1}$$ for $$t > 1$$.

Rational Functions with Removable Discontinuities

Examine the function $$f(x)= \frac{x^2 - 5x + 6}{x - 2}$$. (a) Factor the numerator and simplify th

Real-World Analysis of Vehicle Deceleration Using Data

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

Real-World Application: Temperature Sensor Calibration

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

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

Related Rates: Shadow Length of a Moving Object

A 1.8 m tall person is walking away from a 3 m tall lamppost at a speed of 1.2 m/s. Let $$x$$ be the

Removing a Removable Discontinuity

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

Removing Discontinuities

Consider the function $$f(x)=\frac{x^2-9}{x-3}$$.

Squeeze Theorem with Bounded Function

Suppose that for all x in some interval around 0, the function $$f(x)$$ satisfies $$-x^2 \le f(x) \l

Water Tank Inflow-Outflow Analysis

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

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 Temperature Change via Derivatives

The temperature in a chemical reactor is modeled by $$T(x)=x^3 - 6*x^2 + 9*x$$, where $$T(x)$$ is in

Analyzing Differentiability of an Absolute Value Function

Consider the function $$f(x)= |x-2|$$.

Approximating the Instantaneous Rate of Change Using Secant Lines

A function $$f(t)$$ models the position of an object. The following table shows selected values of $

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

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

Cost Optimization and Marginal Analysis

A manufacturer’s cost function is given by $$C(q)=500+4*q+0.02*q^2$$, where $$q$$ is the quantity pr

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 from Definition for a Piecewise Function

Consider the piecewise function $$ f(x)= \begin{cases} e^{x} & x < 0 \\ \ln(x+1) + 1 & x \ge

Derivatives on an Ellipse

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

Differentiation of a Log-Linear Function

Consider the function $$f(x)= 3 + 2*\ln(x)$$ which might model a process with a logarithmic trend.

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, \\

Instantaneous and Average Velocity

A particle's position is given by $$s(t)=t^3-6*t^2+9*t+2$$, where $$s(t)$$ is in meters and $$t$$ is

Instantaneous Rate of Temperature Change in a Coffee Cup

The temperature of a cup of coffee is recorded at several time intervals as shown in the table below

Inverse Function Analysis: Cosine and Linear Combination

Consider the function $$f(x)=\cos(x)+x$$ defined on the interval $$[0,\frac{\pi}{2}]$$.

Inverse Function Analysis: Hyperbolic-Type Function

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

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

Logarithmic Differentiation of a Composite Function

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

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.

Product Rule Application

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

Quotient Rule Challenge

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

Sand Pile Accumulation

A sand pile is being formed by a conveyor belt that drops sand at a rate of $$f(t)=5+0.5*t$$ (kg/min

Secant and Tangent Lines Analysis

Consider the function $$g(t)=t^3-6*t^2+9*t+2$$ modeling the height (in meters) of a ball at time $$t

Secant and Tangent Lines for a Cubic Function

Consider the function $$f(x)= x^3 - 4*x$$.

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

Chain Rule and Implicit Differentiation in Radial Motion

Let $$r$$ be a function of time $$t$$ defined implicitly by the equation $$r^2 + (\ln(r))^2 = t$$, w

Chain Rule in Population Modeling

A biologist models the population of a species with the function $$P(t)= f(g(t))$$, where $$g(t)=25*

Chain Rule with Logarithmic and Radical Functions

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

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

Chain Rule with Trigonometric Function

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

Combining Composite and Implicit Differentiation

Consider the equation $$e^{x*y}+x^2-y^2=7$$.

Composite Function with Logarithm and Trigonometry

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

Composite Inverse Trigonometric Function Evaluation

Let $$f(x)= \tan(2*x)$$, defined on a restricted domain where it is invertible. Analyze this functio

Composite Log-Exponential Function Analysis

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

Composite Temperature Model

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

Composite Trigonometric Function Analysis in Pendulum Motion

A pendulum's angular displacement is modeled by the function $$\theta(t)= \sin(\sqrt{2*t+1})$$.

Concavity Analysis of an Implicit Curve

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

Differentiation Involving Exponentials and Inverse Trigonometry

Consider the function $$M(x)=e^{\arctan(x)}\cdot\cos(x)$$.

Differentiation Under Implicit Constraints in Physics

A particle moves along a path defined by the equation $$\sin(x*y)=x-y$$. This equation implicitly de

Expanding Spherical Balloon

A spherical balloon has its volume given by $$V=\frac{4}{3}\pi r^3$$. The radius of the balloon incr

Finding Second Derivative via Implicit Differentiation

Given the curve defined by $$x^2+y^2+ x*y=7$$, answer the following:

Implicit Differentiation for an Ellipse

Consider the ellipse defined by the equation $$\frac{x^2}{9} + \frac{y^2}{4} = 1$$. This equation re

Implicit Differentiation in a Financial Model

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

Implicit Differentiation in a 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 Economic Model

In an economic model, the relationship between the quantity supplied $$x$$ and the market price $$y$

Implicit Differentiation Involving Exponential Functions

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

Implicit Differentiation Involving Trigonometric Functions

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

Implicit Differentiation of an Exponential-Product Equation

Consider the curve defined implicitly by $$e^(x*y) + x - y = 0.$$ Solve 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 $$e^{x*y}+\sin(y)= x$$, which relates \(x\) and \(y\). This equation may repre

Implicit Differentiation with Exponential-Trigonometric Functions

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

Implicit Differentiation with Product Rule

Consider the equation $$x*y+e^{y}=x^2$$. Answer the following:

Implicit Differentiation with Product Rule

Consider the implicit equation $$x*y+y^2=6$$ which defines $$y$$ as a function of $$x$$. Use implici

Inverse Analysis in Exponential Decay

A radioactive substance decays according to $$N(t)= N_0*e^(-0.5*t)$$, where N(t) is the quantity at

Inverse Function Differentiation for an Exponential Function

Let $$f(x)= e^{2*x} + 1$$. This function involves an exponential model shifted upward.

Inverse Function Differentiation in a Biological Growth Model

In a bacterial growth experiment, the population $$P$$ (in colony-forming units) at time $$t$$ (in h

Inverse Function Differentiation in an Exponential Model

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

Inverse Trigonometric Differentiation in a Geometry Problem

Consider a scenario in which an angle $$\theta$$ in a geometric configuration is given by $$\theta=\

Accelerating Car Motion Analysis

A car's velocity is modeled by $$v(t)=4t^2-16t+12$$ in m/s for $$t\ge0$$. Analyze the car's motion.

Analyzing Rate of Change in a Compound Interest Model

The amount in a bank account is modeled by $$A(t)= P e^{rt}$$, where $$P = 1000$$, r = 0.05 (per yea

Chemistry Reaction Rate

The concentration of a chemical in a reaction is given by $$C(t)= \frac{100}{1+5*e^{-0.3*t}}$$ (in m

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

Differentiability of a Piecewise Function

Consider the piecewise function $$ f(x)=\begin{cases} x^2, & x \leq 2 \\ 4x-4, & x>2 \end{cases} $$

Drainage Analysis in a Conical Tank

Water is draining from a conical tank at a constant rate of 3 cubic meters per minute. The tank has

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 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

Error Approximation in Engineering using Differentials

The cross-sectional area of a circular pipe is given by $$A=\pi r^2$$. If the radius is measured as

Estimating Small Changes using Differentials

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

Evaluating an Indeterminate Limit using L'Hospital's Rule

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

Expanding Circular Oil Spill

An oil spill on water forms a circular shape. The area of the spill is increasing at a rate of $$200

Exponential Decay in Radioactive Material

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

Friction and Motion: Finding Instantaneous Rates

A block slides down an inclined plane. The height of the plane at a horizontal distance $$x$$ is giv

FRQ 5: Coffee Cooling Experiment

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

FRQ 11: Shadow Length Change

A 2‑m tall person walks away from a 10‑m tall lamp post. Let x be the distance from the lamp post to

Graphing a Function via its Derivative

Consider the function $$f(x) = x^{1/3}$$ defined for all real numbers.

Hybrid Exponential-Logarithmic Convergence

Consider the function $$f(x)=e^{-x}\ln(1+2x)$$, which combines exponential decay with logarithmic gr

Inflating Balloon Rates

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

Inflation of a Balloon: Surface Area Rate of Change

A spherical balloon is being inflated so that its volume increases at a rate of $$\frac{dV}{dt}=50$$

Kinematics on a Straight Line

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

Linear Approximation in Estimating Function Values

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

Linearization of a Machine Component's Length

A machine component's length is modeled by $$L(x)=x^4$$, where x is a machine setting in inches. Use

Logarithmic Profit Optimization

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

Open-top Box Optimization

A manufacturer wants to design an open‐top rectangular box with a square base that has a fixed volum

Optimization of Production Costs

A manufacturing company’s cost function for producing $$x$$ units per hour is given by $$C(x)=\frac{

Optimizing Crop Yield

The yield per acre of a crop is modeled by the function $$Y(p) = 100\,p\,e^{-0.1p}$$, where $$p$$ is

Population Growth with Changing Rates

A population is modeled by the piecewise function $$P(t)=\begin{cases}50e^{0.1t}&t<10\\500e^{0.05t}&

Projectile Motion Analysis

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

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: Inflating Spherical Balloon

A spherical balloon is being inflated such that its volume increases at a constant rate of $$20$$ cu

Revenue Function and Marginal Revenue Analysis

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

Shadow Length Problem

A person 1.80 m tall walks away from a 3.0 m tall lamppost at a rate of 1.2 m/s. Let $$x$$ be the di

Temperature Change Analysis

The temperature of a chemical solution is recorded over time. Use the table below, where $$T(t)$$ (i

Temperature Change in a Cooling Process

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

Vehicle Deceleration Analysis

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

Water Tank Volume Change

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

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

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:

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}]$

Bacterial Culture Growth: Identifying Critical Points from Data

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

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

Designing an Optimal Can

A closed cylindrical can is to have a volume of $$600$$ cubic centimeters. The surface area of the c

Determining Absolute and Relative Extrema

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

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, & \

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

Finding Local Extrema Using the First Derivative Test

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

FRQ 3: Relative Extrema for a Cubic Function

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

Implicit Differentiation and Tangent Lines

Consider the curve defined implicitly by the equation $$x^2 + x*y + y^2= 7$$.

Inverse Analysis of a Cooling Temperature Function

A cooling process is described by the function $$f(t)=20+80*e^{-0.05*t}$$, where t is the time in mi

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 of a Rational Function

Consider the function $$f(x)=\frac{2*x-1}{x+3}$$. Perform the following analysis regarding its inver

Investigating a Piecewise Function with a Vertical Asymptote

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

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

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

Logarithmic Transformation of Data

A scientist models an exponential relationship between variables by the equation $$y= A*e^{k*x}$$. T

Minimizing Average Cost in Production

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

Optimal Production Level: Relative Extrema from Data

A manufacturer recorded profit (in thousands of dollars) at different levels of unit production. Use

Optimizing a Box with a Square Base

A company is designing an open-top box with a square base of side length $$x$$ and height $$h$$. The

Population Growth Analysis via the Mean Value Theorem

A country's population data over a period of years is given in the table below. Use the data to anal

Profit Function Concavity Analysis

A company’s profit is modeled by $$P(x) = -2*x^3 + 18*x^2 - 48*x + 10$$, where $$x$$ is measured in

Related Rates in an Evaporating Reservoir

A reservoir’s volume decreases due to evaporation according to $$V(t)= V_0*e^{-a*t}$$, where $$t$$ i

Revenue Optimization in Economics

A company's revenue is modeled by the function $$R(x)= x*e^{-0.1*x}$$, where $$x$$ (in thousands) re

Trigonometric Function Behavior

Consider the function $$f(x)= \sin(x) + \cos(2*x)$$ defined on the interval $$[0,2\pi]$$. Analyze it

Antiderivatives and the Constant of Integration in Modelling

A moving car has its velocity modeled by $$v(t)= 5 - 2*t$$ (in m/s). Answer the following parts to o

Application of the Fundamental Theorem of Calculus

A particle moves along a straight line with an instantaneous velocity given by $$v(t)=3*t^2+2*t$$ (i

Area Between Curves

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

Area Under a Curve Using Riemann Sums

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

Area Under a Polynomial Curve

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

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 $

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

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

Cumulative Solar Energy Collection

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

Evaluating an Integral with a Trigonometric Function

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

Exact Area Under a Parabolic Curve

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

FRQ2: Inverse Analysis of an Antiderivative Function

Consider the function $$ G(x)=\int_{0}^{x} (t^2+1)\,dt $$ for all real x. Answer the following parts

FRQ6: Inverse Analysis of a Displacement Function

Let $$ S(t)=\int_{0}^{t} (6-2*u)\,du $$ for t in [0, 3], representing displacement in meters. Answer

FRQ16: Inverse Analysis of an Integral Function via U-Substitution

Let $$ U(x)=\int_{0}^{x} 2*(t-3)^2\,dt $$ for x ≥ 3. Answer the following parts.

FRQ19: Inverse Analysis with a Fractional Integrand

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

Fuel Consumption Analysis

A truck's fuel consumption rate (in L/hr) is recorded at various times during a 12-hour drive. Use t

Integration of a Piecewise Function

A function $$f(x)$$ is defined piecewise as follows: $$f(x)= \begin{cases} x^2, & x < 3, \\ 2*x+1,

Integration of Exponential Functions with Shifts

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

Integration to Determine Work Done by a Variable Force

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

Optimizing Fencing Cost for a Garden Adjacent to a River

A farmer plans to fence a rectangular garden adjacent to a river, so that no fence is required along

Rainfall Accumulation via Integration

A region experiences rain where the rate of rainfall (in inches per hour) is given by $$r(t)=0.5+0.2

Related Rates: Expanding Balloon

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

Reservoir Accumulation Problem

A reservoir is filled at a rate given by $$R(t)=\frac{8}{1+e^{-0.5*t}}$$ cubic meters per minute, wh

Roller Coaster Work Calculation

An amusement park engineer recorded the force applied by a roller coaster engine (in Newtons) at var

Seismic Data Analysis: Ground Acceleration

A seismograph records ground acceleration (in m/s²) during an earthquake. Use the data in the table

Temperature Change in a Chemical Reaction

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

Total Distance Traveled from Velocity Data

A particle moves along a line with velocity given by $$v(t)=t^3 - 6*t^2 + 9*t$$ (in m/s) for t in [0

Total Water Volume from a Flow Rate Function

A river’s flow rate (in cubic meters per second) is modeled by the function $$Q(t)=4+2*t$$, where $$

Volume of a Solid: Exponential Rotation

Consider the region bounded by the curve $$y=e^{-x}$$, the x-axis, and the vertical lines $$x=0$$ an

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

Chemical Reactor Mixing

In a continuously stirred chemical reactor, a reactant is added at a rate that results in an inflow

Implicit Differential Equation and Asymptotic Analysis

Consider the differential equation $$\frac{dy}{dx}= \frac{y(1-y)}{x}$$ for $$x > 0$$ with the initia

Implicit Differentiation Involving a Logarithmic Function

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

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

Logistic Population Model Analysis

A population $$P$$ grows according to the logistic equation $$\frac{dP}{dt}=0.4P\left(1-\frac{P}{100

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

Nonlinear Differential Equation with Powers

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

Pollutant Concentration in a Lake

A lake receives a constant pollutant input so that the concentration $$C(t)$$ (in mg/L) satisfies th

Population Dynamics with Harvesting

A fish population is governed by the differential equation $$\frac{dP}{dt} = 0.4*P\left(1-\frac{P}{1

RC Circuit Charging

In an RC circuit, the charge $$Q(t)$$ on the capacitor satisfies the differential equation $$\frac{d

Related Rates: Expanding Balloon

A spherical balloon is inflated such that its radius increases at a constant rate of $$\frac{dr}{dt}

Separable Differential Equation

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

Separable Differential Equation with Trigonometric Component

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

Sketching Solution Curves on a Slope Field

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

Slope Field Analysis for $$\frac{dy}{dx}=\frac{y}{x}$$

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

Slope Field Analysis for a Linear Differential Equation

Consider the linear differential equation $$\frac{dy}{dx}=\frac{1}{2}*x-y$$ with the initial conditi

Slope Field Exploration

Consider the differential equation $$\frac{dy}{dx} = \sin(x)$$. The provided slope field (see stimul

Traffic Flow Dynamics

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

Amusement Park Ride Waiting Time

At an amusement park, the waiting times (in minutes) for a popular ride form an arithmetic sequence.

Analysis of a Rational Function's Average Value

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

Analysis of an Inverse Function

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

Area Between a Function and Its Tangent

A function $$f(x)$$ and its tangent line at $$x=a$$, given by $$L(x)=m*x+b$$, are considered on the

Area Between an Exponential Function and a Linear Function

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

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 Growth Rate in a Biological Process

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

Average Speed from a Velocity Function

A car’s velocity is given by $$v(t)=t^2-4*t+5$$ (in m/s) for $$0 \le t \le 5$$. Assume that $$v(t)$$

Average Speed from Variable Acceleration

A car accelerates along a straight road with acceleration given by $$a(t)=2*t-1$$ (in m/s²) for $$t\

Average Temperature Analysis

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

Average Temperature Over a Day

In a city, the temperature (in $$^\circ C$$) is modeled by $$T(t)=10+5*\cos\left(\frac{\pi*t}{12}\ri

Average Value of a Deposition Rate Function

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

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

Center of Mass of a Lamina with Variable Density

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

Combining Position, Area, and Average Concepts in River Navigation

A boat navigates a river where its speed relative to the water is given by $$v(t)=4+\sin(t)$$ km/h.

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: Area Between Quadratic Cost Functions

Two cost functions for production are given by $$C_1(x)=0.5*x^2+3*x+10$$ and $$C_2(x)=0.3*x^2+4*x+5$

Economic Analysis of Consumer Surplus

A market demand function is given by $$P(x)=50 - 10*\ln(x+1)$$, where $$x$$ represents quantity dema

Exponential Decay Function Analysis

A lab experiment models the decay of a chemical concentration with the function $$f(t)=8*e^{-0.5*t}$

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{

Graduated Rent Increase

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

Kinematics with Variable Acceleration

A particle is moving along a straight path with an acceleration given by $$a(t)=10-6*t$$ (in m/s²) f

Medication Dosage Increase

A patient receives a daily medication dose that increases by a fixed amount each day. The first day'

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

Pollutant Accumulation in a River

Along a 20 km stretch of a river, a pollutant enters the water at a rate described by $$p(x)=0.5*x+2

Population Growth with Variable Growth Rate

A city's population changes with time according to a non-constant growth rate given in thousands per

Position Analysis of a Particle with Piecewise Acceleration

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

Position and Velocity Relationship in Car Motion

A car's position along a highway is modeled by $$s(t)=t^3-6*t^2+9*t+2$$ (in kilometers) with time $$

Sales Increase in a Store

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

Traveling Particle with Piecewise Motion

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

Voltage and Energy Dissipation Analysis

The voltage across an electrical component is modeled by $$V(t)=12*e^{-0.1*t}*\ln(t+2)$$ (in volts)

Volume by Cylindrical Shells

Consider the region bounded by $$y=x$$, $$y=4$$, and $$x=0$$. This region is revolved about the $$y$

Volume by the Washer Method: Solid of Revolution

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

Volume of a Solid 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 Semicircular Cross Sections

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

Volume of a Solid with Square Cross Sections

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

Washer Method with Logarithmic and Exponential Curves

Consider the region bounded by the curves $$f(x)=\ln(x+1)$$ and $$g(x)=e^{-x}$$ on the interval $$[0

Water Tank Volume and Average Cross-Sectional Area

A water tank has a shape where the horizontal cross-sectional area at a depth $$x$$ (in feet) from t