AP Calculus AB FRQ Room

Analysis of One-Sided Limits and Jump Discontinuity

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

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

Application of the Intermediate Value Theorem

Let the function $$f(x)= x^3 - 4*x - 1$$ be continuous on the interval $$[0, 3]$$. Answer the follow

Application of the Squeeze Theorem

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

Application of the Squeeze Theorem in Trigonometric Limits

Consider the function $$f(x) = x^2 * \sin(1/x)$$ for $$x \neq 0$$ with $$f(0)=0$$. Answer the follow

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

Asymptotic Analysis of a Radical Rational Function

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

Combined Limit Analysis of a Piecewise Function

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

Continuity and Asymptotic Behavior of a Rational Exponential Function

Consider the function $$q(x)= \frac{e^{2*x} - 4}{e^{x} - 2}$$. Notice that the function is not defin

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)=

Direct Evaluation of Polynomial Limits

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

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 a Limit with Radical Expressions

Evaluate the limit $$\lim_{x \to 9} \frac{\sqrt{x} - 3}{x - 9}$$. Answer the following:

Factoring a Cubic Expression for Limit Evaluation

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

Factorization and Limit Evaluation

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

Implicit Differentiation in an Exponential Equation

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

Intermediate Value Theorem in Equation Solving

A continuous function defined on [0, 10] is given by $$f(x)= \frac{x}{10} - \sin(x)$$.

Inverse Function and Limit Behavior Analysis

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

Limit Involving Radical Expressions

For the function $$f(x)=\frac{\sqrt{x+9}-3}{x}$$, evaluate the limit as x approaches 0.

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

Long-Term Behavior of Particle Motion: Horizontal Asymptotes

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

One-Sided Limits and an Absolute Value Function

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

Particle Motion with Squeeze Theorem Application

A particle moves along a line with velocity given by $$v(t)= t^2 \sin(1/t)$$ for $$t>0$$ and is defi

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

Rational Function Limits and Removable Discontinuities

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

Real-World Analysis of Vehicle Deceleration Using Data

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

Real-World Application: Temperature Sensor Calibration

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

Removable Discontinuity in a Cubic Function

Consider the function $$f(x)=\begin{cases} \frac{x^3-27}{x-3} & x\neq3 \\ 10 & x=3 \end{cases}$$. An

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 Application

Let $$f(x)=x^2\sin(1/x)$$ for \(x\neq 0\) and define \(f(0)=0\). Use the Squeeze Theorem to complete

Squeeze Theorem Application with Trigonometric Functions

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

Squeeze Theorem with an Oscillatory Function

Consider the function $$f(x) = x \cdot \sin\left(\frac{1}{x}\right)$$ for $$x \neq 0$$ and define $$

Vertical Asymptote and End Behavior

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

Analyzing a Projectile's Motion

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

Comparative Analysis of Secant and Tangent Slopes

A function $$f(x)$$ is represented by the data in the following table: | x | f(x) | |---|------| |

Derivative Applications in Motion Along a Curve

A particle moves such that its horizontal position is given by $$x(t)= t^2 + 2*t$$ and its vertical

Derivative from First Principles

Derive the derivative of the polynomial function $$f(x)=x^3+2*x$$ using the limit definition of the

Derivative from First Principles: The Function $$f(x)=\sqrt{x}$$

Consider the function $$f(x) = \sqrt{x}$$. Use the definition of the derivative to find an expressio

Derivatives in Economics: Cost Functions

A company's production cost is modeled by $$C(q)=500+20*q-0.5*q^2$$, where $$q$$ represents the quan

Deriving the Derivative from First Principles for a Reciprocal Square Root Function

Let $$f(x)=\frac{1}{\sqrt{x}}$$ for $$x > 0$$. Using the definition of the derivative, show that $$f

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

Differentiation Using the Product Rule

Consider the function \(p(x)= (2*x+3)*(x^2-1)\). Answer the following parts.

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

Exponential Growth Rate

Consider the function $$f(x)= 3*e^{2*x}$$ which models a quantity growing exponentially over time.

Inverse Function Analysis: Cubic Transformation

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

Inverse Function Analysis: Quadratic Function

Consider the function $$f(x)=x^2$$ restricted to $$x\geq0$$.

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

Linking Derivative to Kinematics: the Position Function

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

Optimization of Production Cost

A manufacturer’s cost function is given by $$C(x)=x^3-15x^2+60x+200$$, where x represents the produc

Optimizing Car Speed: Rate of Change Analysis

A car’s speed in km/h is modeled by the function $$s(t)=50+2*t^2-0.1*t^3$$ for $$0 \leq t \leq 10$$

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.

Polynomial Rate of Change Analysis

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

Population Growth Rate

Suppose the population of a species is modeled by $$P(t)= 1000*e^{0.07*t}$$, where $$t$$ is measured

Product and Quotient Rule Combination

Given $$u(x)=3*x^2+2$$ and $$v(x)=x-4$$, consider the function $$F(x)=\frac{u(x)*v(x)}{x+1}$$. Answe

Product Rule Application in Economics

A company's cost function for producing $$x$$ units is given by $$C(x)= (3*x+2)*(x^2+5)$$ (cost in d

Projectile Motion Analysis

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

Rate of Change for an Exponential Function

An amount of money grows according to the model $$A(t)=1000*e^{0.05*t}$$, where $$t$$ is measured in

Real-World Application: Temperature Change in a Chemical Reaction

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

Related Rates: Conical Tank Draining

A conical water tank drains so that its volume is given by $$V=\frac{1}{3}\pi r^2h$$. The radius r o

River Pollution Dynamics

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

Secant Slope from Tabulated Data

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

Slope of a Tangent Line from Experimental Data

Experimental data recording the distance traveled by an object over time is provided in the table be

Water Treatment Plant's Chemical Dosing

A water treatment plant adds a chemical at a rate of $$f(t)=5+0.2*t$$ (liters/min) while the chemica

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 in Population Modeling

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

Composite Differentiation with Nested Functions

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

Composite Function and Multiple Rates

An object's distance is modeled by the function $$s(t)= \sqrt{1+ [h(t)]^2}$$, where $$h(t)= \ln(5*t+

Composite Function Differentiation with Logarithms

A function is given by $$h(x)=\ln((5*x+1)^2)$$. Use the chain rule to differentiate $$h(x)$$.

Composite Function in Biomedical Model

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

Composite Function with Nested Chain Rule

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

Composite Function: Engineering Stress-Strain Model

In an engineering context, the stress σ as a function of strain ε is given by $$\sigma(\epsilon) = \

Composite Temperature Model

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

Composite, Implicit, and Inverse Combined Challenge

Consider a dynamic system defined by the equation $$\sin(y)+\sqrt{x+y}=x$$, which implicitly defines

Differentiation of an Inverse Trigonometric Composite Function

Consider the function $$y = \arctan(\sqrt{3x})$$.

Differentiation of Inverse Trigonometric Functions in Physics

In an optics experiment, the angle of refraction \(\theta\) is given by $$\theta= \arcsin\left(\frac

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 Biochemical Reaction

Consider a biochemical reaction modeled by the equation $$x*e^{y} + y*e^{x} = 10$$, where $$x$$ and

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 an Ellipse

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

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 with Exponentials and Logarithms

Consider the curve defined implicitly by $$x*e^(y) + \ln(y)= e$$. It is given that the point $$(1, 1

Implicit Differentiation with Logarithmic and Radical Components

Consider the equation $$\ln(x+y)=\sqrt{x*y}$$.

Implicit Differentiation with Mixed Trigonometric and Polynomial Terms

Consider the equation $$x*\cos(y) + y^2 = x^2$$, which mixes trigonometric and polynomial expression

Implicit Differentiation with Trigonometric and Logarithmic Terms

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

Inverse Function Derivative and Recovery

Let $$f(x)=x^3+x$$, which is one-to-one on a suitable interval. 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 an Exponential Function

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

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 Function Differentiation with an Exponential-Linear Function

Let $$f(x)=e^{2*x}+x$$ and assume it is invertible. Answer the following:

Inverse Trigonometric Differentiation

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

Inverse Trigonometric Differentiation

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

Inverse Trigonometric Function Differentiation

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

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,

Related Rates of a Shadow

A 2 m tall lamp post casts a shadow from a person who is 1.8 m tall. The person is moving away from

Analysis of Particle Motion

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

Analyzing Experimental Motion Data

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

Analyzing Position Data with Table Values

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

Balloon Inflation Analysis

A spherical balloon inflates such that its volume increases at a constant rate of 10 cubic inches pe

Biochemical Reaction Rate Analysis

A biochemical reaction proceeds with a rate modeled by $$R(t)=50t(1-t)^2$$ for $$0\le t\le1$$ (where

Chemical Reaction Rate Analysis

The concentration of a reactant in a chemical reaction is modeled by $$C(t)=\frac{10}{1+e^{0.5t}}$$,

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

Depth of a Well: Related Rates Problem

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

Differentiability of a Piecewise Function

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

Economic Cost Analysis Using Derivatives

A company’s cost function for producing $$x$$ units is given by $$C(x)=0.05*x^3 - 2*x^2 + 40*x + 100

Evaluating Indeterminate Limits via L'Hospital's Rule

Scientists are studying the limit of the function $$L(x)=\frac{5x^3-4x^2+1}{7x^3+2x-6}$$ as $$x \to

Expanding 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: Related Rates Problem

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

Falling Object Analysis

An object is dropped from a height and its position is modeled by $$s(t)=100-4.9t^2$$ (in meters), w

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 6: Particle Motion Analysis on a Straight Line

A particle moving along a straight line has its velocity described by $$v(t) = 3*t^2 - 4*t + 2$$, wh

FRQ 10: Chemical Kinetics Analysis

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

FRQ 13: Cost Function Linearization

A company’s cost function is given by $$C(x) = 5*x^3 - 60*x^2 + 200*x + 1000$$, where x represents t

FRQ 15: Evaluating Limits with L’Hôpital’s Rule

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

Implicit Differentiation and Related Rates in Conic Sections

A point moves along the ellipse defined by $$\frac{x^2}{9} + \frac{y^2}{16} = 1$$. At a certain inst

Inflating Spherical Balloon

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 $$\frac{dV}{dt}=50$$

Linearization and Differentials Approximation

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

Motion along a Straight Line: Changing Direction

A runner's position is modeled by $$s(t)= t^4 - 8*t^2 + 16$$, where $$s(t)$$ is in meters and $$t$$

Population Growth Rate Analysis

A town's population is modeled by the exponential function $$P(t) = 500e^{0.03t}$$, where $$t$$ is i

Projectile Motion: Maximum Height

A ball is thrown upward and its height is modeled by $$h(t)=-5t^2+20t+2$$ (in meters). Analyze its m

Rate of Change in a Freefall Problem

An object is dropped from a height. Its height (in meters) after t seconds is modeled by $$h(t)= 100

Rate of Change in Pool Volume

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

Related Rates: Expanding Circular Ripple

A circular ripple on a calm water surface is expanding such that its area is increasing at a rate of

Related Rates: Shadow Length

A 1.8-meter tall person is walking away from a 4.5-meter tall streetlight at a constant speed of 1.2

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

Vehicle Deceleration Analysis

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

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

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

Analysis of a Trigonometric Piecewise Function

Consider the function $$ f(x) = \begin{cases} \frac{\sin(x)}{x}, & x \neq 0, \\ 2, & x = 0. \end{ca

Application of Rolle's Theorem

Let $$f(x)$$ be a function that is continuous on $$[0,5]$$ and differentiable on $$(0,5)$$ with $$f(

Area and Volume: Polynomial Boundaries

Let $$f(x)= x^2$$ and $$g(x)= 4 - x^2$$. Consider the region bounded by these two curves.

Area Growth of an Expanding Square

A square has a side length given by $$s(t)= t + 2$$ (in seconds), so its area is $$A(t)= (t+2)^2$$.

Average Value of a Function and Mean Value Theorem for Integrals

Consider the function $$f(x)= e^{-x}$$ on the interval $$[0, 2]$$. Answer the following:

Behavior Analysis of a Logarithmic Function

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

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

Car Speed Analysis via MVT

A car's position is given by $$f(t) = t^3 - 3*t^2 + 2*t$$ (in meters) for $$t$$ in seconds on the cl

Floodgate Operation Analysis

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

FRQ 1: Car's Motion and the Mean Value Theorem

A car’s position along a straight road is modeled by $$s(t) = t^3 - 6*t^2 + 9*t + 5$$ (in meters) fo

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 8: Mean Value Theorem and Non-Differentiability

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

FRQ 15: Population Growth and the Mean Value Theorem

A town’s population (in thousands) is modeled by $$P(t)= t^3 - 3*t^2 + 2*t + 50$$, where $$t$$ repre

FRQ 16: Finding Relative Extrema for a Logarithmic Function

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

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

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

Graphical Analysis and Derivatives

A function \( f(x) \) is represented by the graph provided below. Answer the following based on the

Graphical Analysis Using First and Second Derivatives

The graph provided represents the function $$f(x)= x^3 - 3*x^2 + 2*x$$. Analyze this function using

Implicit Differentiation and Tangent Lines

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

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 $$[-1, 3]$$.

Increasing/Decreasing Behavior in a Financial Model

A financial analyst models the performance of an investment with the function $$f(x)= \ln(x) - \frac

Inverse Analysis of a Composite Trigonometric-Linear Function

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

Inverse Analysis of an Exponential Function

Consider the function $$f(x)=2*e^(x)+3$$. Analyze its inverse function as instructed in the followin

Liquid Cooling System Flow Analysis

A specialized liquid cooling system operates with non-linear flow rates. The inflow rate is given by

Mean Value Theorem Applied to Exponential Functions

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

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

Monotonicity and Inverse Function Analysis

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

Optimization of an Open-Top Box

A company is designing an open-top box with a square base. The volume of the box is modeled by the f

Piecewise Function with Trigonometric and Constant Segments

Consider the function $$ f(x) = \begin{cases} \cos(x), & x < \frac{\pi}{2}, \\ 0, & x = \frac{\pi}{

Predicting Fuel Efficiency in Transportation

A vehicle’s performance was studied by recording the miles traveled and the corresponding fuel consu

Relationship Between Integration and Differentiation

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

Reservoir Evaporation and Rainfall

A reservoir gains water through rainfall and loses water by evaporation. Rainfall occurs at a rate g

Reservoir Sediment Accumulation

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

Solving a Log-Exponential Equation

Solve the equation $$\ln(x)+x=0$$ for $$x>0$$. Answer the following:

Traffic Flow Modeling

A highway segment experiences varying traffic flows. Cars enter at a rate $$I(t)=50+10*\sin(\frac{\p

Analyzing Bacterial Growth via Riemann Sums

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

Analyzing Tabular Data via Integration Methods

A vehicle's speed in km/h is recorded over 4 hours, as shown in the table below.

Approximating Area Under a Curve with Riemann Sums

Consider a function $$f(x)$$ whose values are tabulated below for different values of $$x$$. Use the

Area Between Curves

Consider the curves defined by $$f(x)=x^2$$ and $$g(x)=2*x$$. The region enclosed by these curves is

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

Definite Integral and the Fundamental Theorem of Calculus

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

Economic Accumulation of Revenue

The marginal revenue (MR) for a company is given by $$MR(x)=50*e^{-0.1*x}$$ (in dollars per item), w

Environmental Modeling: Pollution Accumulation

The pollutant enters a lake at a rate given by $$P(t)=5*e^{-0.3*t}$$ (in kg per day) for $$t$$ in da

Estimating the Area Under a Curve Using Riemann Sums

A function $$f(x)$$ is defined on the interval $$[0,6]$$ and its values are listed in the table belo

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

Exact Area Under a Transformed Function Using U-Substitution

Evaluate the area under the curve described by the integral $$\int_{1}^{5} 2*(x-1)^{3}\,dx$$ using u

FRQ1: Analysis of an Accumulation Function and its Inverse

Consider the function $$ F(x)=\int_{1}^{x} (2*t+3)\,dt $$ for $$ x \ge 1 $$. Answer the following pa

Graphical Analysis of an Accumulation Function

Let $$f(t)$$ represent the rate of water flow (in $$m^3/hr$$) into a reservoir, and suppose the grap

Marginal Cost and Total Cost

In a production process, the marginal cost (in dollars per unit) for producing x units is given by $

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

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 $$

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

Population Growth: Accumulation through Integration

A certain population grows at a rate modeled by $$R(t)= 0.5*t^2 - 3*t + 10$$ (individuals per year),

Rainfall Accumulation Analysis

The rainfall intensity at a location is modeled by the function $$i(t) = 0.5*t$$ (inches per hour) f

Rainfall and Evaporation in a Greenhouse

In a greenhouse, rainfall is modeled by $$R(t)= 8\cos(t)+10$$ mm/hr, while evaporation occurs at a c

Riemann Sum Approximation of f(x) = 4 - x^2

Consider the function $$f(x)=4-x^2$$ on the interval $$[0,2]$$. Use Riemann sums to approximate the

Ski Lift Passengers: Boarding and Alighting Rates

On a ski lift, passengers board at a rate $$B(t)= 3e^{-t/2}$$ persons/min and alight at a constant r

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 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 Flow in a Tank

Water flows into a tank at a rate given by $$R(t)=3*t+2$$ (in liters per minute) for $$0 \le t \le 6

Work Done by a Variable Force

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

Balloon Inflation with Leak

A balloon is being inflated at a rate of $$5$$ liters/min, but it is also leaking air at a rate prop

Bernoulli Differential Equation

Solve the Bernoulli differential equation $$\frac{dy}{dx}-\frac{1}{x}y=-x*y^2$$ for $$x>0$$ with the

Bernoulli Differential Equation Challenge

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

Charging a Capacitor in an RC Circuit

In an RC circuit, the charge $$Q$$ on a capacitor satisfies the differential equation $$\frac{dQ}{dt

Chemical Reaction in a Vessel

A 50 L reaction vessel initially contains a solution of reactant A at a concentration of 3 mol/L (i.

Chemical Reaction Rate

The concentration $$y$$ (in moles per liter) of a reactant in a chemical reaction is modeled by the

Chemical Reaction Rate with Second-Order Decay

A chemical reaction follows the rate law $$\frac{d[A]}{dt}=-k[A]^2$$, where $$[A](t)$$ (in M) is the

CO2 Absorption in a Lake

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

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

Exponential Growth: Separable Equation

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

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 and Slope Analysis

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

Investment Growth with Withdrawals

An investment account grows at a rate proportional to its current balance, but a constant amount is

Logistic Population Growth

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

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 in a Tank

A tank initially contains 200 L of water with 10 kg of dissolved salt. Brine with a salt concentrati

Mixing Problem with Changing Volume

A tank initially contains 100 L of water with 5 kg of salt. Brine enters the tank at 3 L/min with a

Mixing Problem with Constant Flow

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

Mixing Tank Problem

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

Mixing with Variable Inflow Rate

A 50-liter tank initially contains water with 1 kg of dissolved salt. Water containing 0.2 kg of sal

Modeling Continuous Compound Interest

An account accrues interest continuously according to the differential equation $$\frac{dA}{dt}=rA$$

Nonlinear Differential Equation

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

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 Model with Harvesting

A fish population in a lake is modeled by the differential equation $$\frac{dP}{dt}=0.3*P\left(1-\fr

Radioactive Decay

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

Radioactive Decay

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

Radioactive Decay Model

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

Radioactive Material with Continuous Input

A radioactive substance decays at a rate proportional to its amount while being produced continuousl

Separable Differential Equation: $$dy/dx = x*y$$

Consider the differential equation $$dy/dx = x*y$$ with the initial condition $$y(0)=2$$. Solve the

Separable Differential Equation: y and x

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

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

Solving a Differential Equation Using the SIPPY Method

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

Tumor Growth with Allee Effect

The growth of a tumor is modeled by the differential equation $$\frac{dP}{dt}=rP\left(1-\frac{P}{K}\

Volume by Revolution of a Differential Equation Derived Region

The function $$y(x) = e^{-x} + x$$, which is a solution to a differential equation, and the line $$y

Area Between Cost Functions in a Business Analysis

A company analyzes its cost structure using two functions: the fixed-plus-variable cost function $$C

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

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

Arithmetic Savings Account

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

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 of a Logarithmic Function

Let $$f(x)=\ln(x+2)$$ represent a measured quantity over the interval $$[0,6]$$.

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 Temperature Analysis

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

Average Value Calculation for a Polynomial Function

Consider the function $$f(x)=2*x^2-3*x+1$$ defined on the interval $$[0,5]$$. Compute the average va

Economics: Consumer Surplus Calculation

Given the demand function $$d(p)=100-2p$$ and the supply function $$s(p)=20+3p$$, determine the cons

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

Particle Motion Analysis using Velocity Data

A particle moves along a straight line and its velocity (in m/s) is recorded at various times as sho

Population Growth Rate Analysis

Suppose the instantaneous growth rate of a population is given by $$r(t)=0.04 - 0.002*t$$ for $$t \i

Sales Increase in a Store

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

Tank Draining with Variable Flow Rates

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

Total Distance Traveled from a Velocity Profile

A particle’s velocity over the interval $$[0, 6]$$ seconds is given in the table below. Note that th

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

A region in the xy-plane is bounded by $$y=x$$ and $$y=x^2$$ for $$0 \le x \le 1$$. This region is r

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 by Washer Method

Consider the region R bounded by $$y= x$$ and $$y= x^2$$ on the interval $$x \in [0,1]$$. This regio

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: Curve Raised to a Power

Consider the region under the curve $$f(x)=x^{1/3}$$ for $$x\in [0,8]$$. This region is revolved abo

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

Work Done by a Variable Force

A variable force is applied along a straight line such that $$F(x)=6-0.5*x$$ (in Newtons). The force

Work in Pumping Water

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