AP Calculus AB FRQ Room

Absolute Value Function and Discontinuity

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

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 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 Limits from Experimental Data (Table)

The table below shows measured values of a function $$f(x)$$ near $$x = 1$$. | x | f(x) | |-----

Analyzing Multiple Discontinuities in a Rational Function

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

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:

Estimating Limits from a Data Table

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

Graph Analysis of a Discontinuous Function

A function f has been graphed below and exhibits a discontinuity at x = 1. Use the graph to answer t

Graph Transformations and Continuity

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

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

Horizontal Asymptote and End Behavior

Consider the rational function $$f(x)=\frac{3*x^2 + x - 5}{6*x^2 - 4*x + 7}$$. Answer the following

Implicit Differentiation Involving Logarithms

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

Intermediate Value Theorem and Root Existence

Consider the function $$f(x)= x^3 - 6*x + 1$$ on the interval [1, 3].

Intermediate Value Theorem Application

Suppose a continuous function $$f(x)$$ is defined on the interval $$[1,5]$$, with $$f(1)=-3$$ and $$

Intermediate Value Theorem Application

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

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 Involving a Removable Discontinuity

Consider the function $$g(x)= \frac{(x+3)(x-2)}{x-2}$$ defined for $$x \neq 2$$. Answer the followin

One-Sided Limits and Discontinuity Analysis

Consider the function $$f(x)= \begin{cases} \frac{x^2 - 4}{x - 2}, & x \neq 2 \\ 5, & x = 2 \end{cas

Oscillatory Function and the Squeeze Theorem

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

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 $$t > 1$$.

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

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

Removable Discontinuity in a Rational Function

Consider the function $$f(x)=\begin{cases} \frac{x^2-16}{x-4} & x\neq4 \\ 3*x+1 & x=4 \end{cases}$$.

Removing Discontinuities

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

Squeeze Theorem for an Exponential Damped Function

A physical process is modeled by the function $$h(x)= x*e^{-1/(x*x)}$$ for $$x \neq 0$$ and is defin

Squeeze Theorem for an Oscillatory Function

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

Table Analysis for Estimating a Limit

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

Trigonometric Limit Evaluation

Examine the function $$ f(x)= \frac{\sin(3*x)}{x} $$ for $$x\ne0$$.

Vertical Asymptotes and Horizontal Limits

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

Analyzing a Function with a Removable Discontinuity

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

Analyzing a Projectile's Motion

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

Analyzing the Derivative of a Trigonometric Function

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

Approximating Derivatives Using Secant Lines

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

Bacterial Culture Growth with Washout

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

Car Fuel Consumption vs. Refuel

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

Chain Rule Application

Consider the composite function $$f(x)=\sqrt{1+4*x^2}$$, which may describe a physical dimension.

Concavity and the Second Derivative

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

Derivative of an Exponential Decay Function

Consider the function $$f(t)=e^{-0.5*t}$$, which may represent the decay of a substance over time. A

Derivative of the Square Root Function via Limit Definition

Let $$g(x)=\sqrt{x}$$. Using the limit definition of the derivative, answer the following parts.

Derivatives and Optimization in a Real-World Scenario

A company’s profit is modeled by $$P(x)=-2*x^2+40*x-150$$, where $$x$$ represents the number of item

DIY Rainwater Harvesting System

A household's rainwater harvesting system collects rain at a rate of $$f(t)=12-0.5*t$$ (liters/min)

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, \\

Finding the Derivative Using First Principles

Consider the function $$f(x)= 5*x^3 - 4*x + 7$$. Use the definition of the derivative to find the de

Finding the Derivative using the Limit Definition

Let $$h(x)= 5*x^2 + 3*x - 7$$. Use the limit definition of the derivative to determine $$h'(x)$$.

Graph Interpretation of the Derivative

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

Graph vs. Derivative Graph

A graph of a function $$f(x)$$ and a separate graph of its derivative $$f'(x)$$ are provided in the

Implicit Differentiation in Motion

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

Inverse Function Analysis: Hyperbolic-Type Function

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

Inverse Function Analysis: Rational Decay Function

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

Inverse Function Analysis: Rational Function 2

Consider the function $$f(x)=\frac{x+4}{x+2}$$ defined for $$x\neq -2$$, with the additional restric

Marginal Cost Analysis

A company's total cost function is given by $$C(x)=5*x^2+20*x+100$$, where $$x$$ represents the numb

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 $$0 \leq t \leq 10$$

Particle Motion on a Straight Road

A particle moves along a straight road. Its position at time $$t$$ seconds is given by $$s(t) = t^3

Polynomial Rate of Change Analysis

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

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.

Sand Pile Growth with Erosion Dynamics

A sand pile is growing as sand is added at a rate of $$f(t)=8+0.3*t$$ (kg/min) and simultaneously lo

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 Approximation Convergence and the Derivative

Consider the natural logarithm function $$f(x)= \ln(x)$$. Investigate its rate of change using the d

Secant Line Slope Approximations in a Laboratory Experiment

In a chemistry lab, the concentration of a solution is modeled by $$C(t)=10*\ln(t+1)$$, where $$t$$

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

Tangent Line and Differentiability

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

Tangent Line to a Parabola

Consider the function $$f(x)=x^2 - 4*x + 3$$. A graph of this quadratic function is provided. Answer

Using the Difference Quotient with a Polynomial Function

Let $$g(x)=2*x^2 - 5*x + 3$$. Answer the following questions:

Advanced Composite Function Differentiation in Biological Growth

A biologist models bacterial growth by the function $$P(t)= e^{\sqrt{t+1}}$$, where $$t$$ is time in

Advanced Implicit and Inverse Function Differentiation on Polar Curves

Consider the curve defined implicitly by $$x^2+y^2= \sin(x*y)$$. Although not a typical polar curve,

Analyzing a Function and Its Inverse

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

Combining Chain Rule, Implicit, and Inverse Differentiation

Consider the equation $$\sqrt{x+y}+\ln(y)=x^2$$, where $$y$$ is defined implicitly as a function of

Combining Composite and Implicit Differentiation

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

Composite and Rational Function Differentiation

Let $$P(x)=\frac{x^2}{\sqrt{1+x^2}}$$.

Composite Differentiation of an Inverse Trigonometric Function

Let $$H(x)= \arctan(\sqrt{x+3})$$.

Composite Function Chain Reaction

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

Composite Function with Nested Chain Rule

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

Composite Functions in Population Dynamics

The population of a species is modeled by the composite function $$P(t) = f(g(t))$$, where $$g(t) =

Differentiating an Inverse Trigonometric Function

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

Differentiation of a Composite Rational Function

Let $$f(x)=\frac{(2*x+1)^3}{\sqrt{5*x-2}}$$. Use the chain rule and the quotient (or product) rule t

Differentiation of Nested Composite Logarithmic-Trigonometric Function

Consider the function $$f(x)=\ln(\sin(3x^2+2))$$.

Implicit Curve Analysis: Horizontal Tangents

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

Implicit Differentiation in an Exponential Context

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

Implicit Differentiation Involving Trigonometric Functions

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

Implicit Differentiation of a Logarithmic Equation

Given the equation $$\ln(x) + \ln(y) = \ln(10)$$, answer the following parts.

Implicit Differentiation of a Logarithmic-Exponential Equation

Consider the equation $$\ln(x+y) + e^{x*y} = 7$$, which implicitly defines $$y$$ as a function of $$

Implicit Differentiation with Exponential-Trigonometric Functions

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

Inverse Function Derivative

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

Inverse Function Derivative with Composite Functions

Consider the function $$f(x)=x^3+2*x+1$$, which is one-to-one on its domain. Given that $$f(1)=4$$,

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 in Temperature Conversion

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

Inverse Trigonometric and Logarithmic Function Composition

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

Manufacturing Optimization via Implicit Differentiation

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

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,

Optimization in an Implicitly Defined Function

The curve defined by $$x^2y + \sin(y) = 10$$ implicitly defines $$y$$ as a function of $$x$$ near $$

Related Rates via Chain Rule

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

Second Derivative via Implicit Differentiation

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

Analysis of Wheel Rotation

Consider a wheel whose angular position is given by $$\theta(t) = 2t^2 + 3t$$, in radians, where $$t

Analyzing a Nonlinear Rate of Revenue Change

A company's revenue in thousands of dollars is modeled by the function $$R(x)=100\ln(x+1) + 0.5x$$,

Analyzing Experimental Motion Data

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

Balloon Inflation Analysis

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

Complex Piecewise Function Analysis

Consider the function $$f(x)=\begin{cases}\frac{\sin(x)}{x} & x<\pi \\ 2 & x=\pi \\ 1+\cos(x-\pi) &

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

Differentiability of a Piecewise Function

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

Error Estimation in Pendulum Period

The period of a simple pendulum is given by $$T=2\pi\sqrt{\frac{L}{g}}$$, where $$L$$ is the length

Estimating Function Change Using Differentials

Let $$f(x)=x^{1/3}$$. Use differentials to approximate the change in $$f(x)$$ when $$x$$ increases f

Falling Object's Velocity Analysis

A rock is thrown upward from the top of a building with a velocity function $$v(t)= 20 - 9.8*t$$ (in

FRQ 1: Vessel Cross‐Section Analysis

A designer is analyzing the cross‐section of a vessel whose shape is given by the ellipse $$\frac{x^

FRQ 2: Balloon Inflation Analysis

A spherical balloon is being inflated. Its volume is given by $$V = \frac{4}{3}\pi r^3$$, and the ra

FRQ 3: Ladder Sliding Problem

A 13­m ladder leans against a vertical wall. Its position satisfies the equation $$x^2 + y^2 = 169$$

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

Implicit Differentiation in Related Rates

A 5-foot ladder leans against a wall such that its bottom slides away from the wall. The relationshi

Inflating Balloon

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

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

Interpretation of the Derivative from Graph Data

The graph provided represents the position function $$s(t)$$ of a particle moving along a straight l

Linear Approximation in Estimating Function Values

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

Linearization and Differentials

Given the function $$f(x)=x^4$$, use linear approximation to estimate the value of $$(3.98)^4$$.

Linearization for Approximating Powers

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

Marginal Profit Analysis

A company's profit in thousands of dollars is given by $$P(x)= -0.5*x^2+20*x-50$$, where $$x$$ (in h

Medicine Dosage: Instantaneous Rate of Change

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

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

Particle Motion Analysis

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

Population Change Rate

The population of a town is modeled by $$P(t)= 50*e^{0.3*t}$$, where $$t$$ is in years and $$P(t)$$

Projectile Motion Analysis

A projectile is launched vertically, and its height (in meters) as a function of time is given by $$

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

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

Route Optimization for a Rescue Boat

A rescue boat must travel from a point on the shore to an accident site located 2 km along the shore

Seasonal Water Reservoir

A reservoir's water volume (in million m³) changes with the seasons according to $$V(t)=5+2\sin\left

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 in a Cooling Process

A cup of coffee cools according to the function $$T(t)= 80 + 20e^{-0.3t}$$, where t is measured in m

Transcendental Function Temperature Change

A cooling object has its temperature modeled by $$T(t)= 100 + 50e^{-0.2*t}$$, where t is measured in

Using L'Hospital's Rule to Evaluate a Limit

Consider the limit $$L=\lim_{x\to\infty}\frac{5x^3-4x^2+1}{7x^3+2x-6}$$. Answer the following:

Water Tank Dynamics

A water tank is subjected to an inflow and an outflow. The inflow rate is given by $$f(t)=10+2*t$$ m

Water Tank Volume Change

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

Analyzing Concavity and Inflection Points

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

Analyzing Continuity and Discontinuity in a Function with a Square Root

Consider the function $$ f(x) = \begin{cases} \frac{\sqrt{x+4}-2}{x}, & x < 0, \\ 1 + \sqrt{1+x}, &

Analyzing the Function $$f(x)= x*\ln(x) - x$$

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

Approximating Displacement from Velocity Data

A vehicle's velocity (in $$m/s$$) over time (in seconds) was recorded during a test run. The table b

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

Concavity Analysis of a Cubic Function

Consider the function $$f(x)= x^3 - 6*x^2 + 9*x + 2$$. Use the second derivative to investigate the

Concavity and Inflection Points in a Quartic Function

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

Continuity Analysis of a Rational Piecewise Function

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

Continuous Compound Interest

An investment account is governed by the formula $$A(t)= A_0 * e^{r*t}$$, where $$r$$ is the continu

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, & \

Evaluating Pollution Concentration Changes

A study recorded the concentration of a pollutant (in ppm) in a river over time (in hours). Use the

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 13: Water Tank Volume Analysis

The volume of water in a tank is given by $$V(t)= t^3 - 12*t^2 + 36*t + 100$$ (in liters), where $$t

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 Using First and Second Derivatives

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

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 Function with Square Root and Linear Term

Consider the function $$f(x)=\sqrt{3*x+1}+x$$. Answer the following questions regarding its inverse.

Inverse Analysis of an Exponential Function

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

Inverse Analysis: Logarithmic Ratio Function in Financial Context

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

Optimization in a Physical Context with the Mean Value Theorem

A car's velocity is modeled by $$v(t) = t^2 - 4*t + 5$$ (in m/s) for time $$t$$ in seconds on the in

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

Optimizing an Open-Top Box from a Metal Sheet

A rectangular sheet of metal with dimensions 24 cm by 18 cm is used to create an open-top box by cut

Rational Function Optimization

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

Relative Extrema in an Economic Demand Model

An economic study recorded the quantity demanded of a product at different price points. Use the tab

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 Total Change in a Population Model

A population grows at a rate given by $$r(t)=0.2*t^2 - t + 5$$ (in thousands per year), where t is i

Analyzing Tabular Data via Integration Methods

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

Application of the Fundamental Theorem in a Discounted Cash Flow Model

A continuous cash flow is given by $$C(t)=500(1+0.05*t)$$ dollars per year. Using a continuous disco

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

Approximating the Area with Riemann Sums

Consider the linear function $$f(x) = 2*x + 1$$ on the interval $$[1,5]$$. Use Riemann sums to appro

Car Fuel Consumption Analysis

A car engine’s fuel dynamics are modeled such that fuel is consumed at a rate of $$f(t)=0.1t^2$$ L/m

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 $

Computing a Definite Integral Using the Fundamental Theorem of Calculus

Let the function be defined as $$f(x) = 2*x$$. Use the Fundamental Theorem of Calculus to evaluate t

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

Convergence of Riemann Sum Estimations

Consider the function $$f(x)=\sqrt{x}$$ on the interval $$[1,4]$$. Answer the following questions re

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

Definite Integral Approximation Using Riemann Sums

Consider the function $$f(x)= x^2 + 3$$ defined on the interval $$[2,6]$$. A table of sample values

Displacement from a Velocity Function

A particle moves along a straight line with velocity function $$v(t)=3*t^2 - 4*t + 2$$ (in m/s). Det

Electric Charge Accumulation

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

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 a Radical Integral via U-Substitution

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

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

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 Trigonometric Function

Consider the function $$f(x)=|\sin(x)|$$. Evaluate the definite integral $$\int_0^{2\pi} |\sin(x)|\,

Marginal Cost and Total Cost

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

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

Particle Motion on a Road with Varying Speed

A particle moves along a straight road with velocity $$v(t)=4-0.5*t^2$$ (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). T

Piecewise-Defined Function and Discontinuities

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

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

Rainwater Collection in a Reservoir

Rainwater falls into a reservoir at a rate given by $$R(t)= 12e^{-0.5t}$$ L/min while evaporation re

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

Riemann Sum Approximation from a Table

The table below gives values of a function $$f(x)$$ at selected points: | x | 0 | 2 | 4 | 6 | 8 | |

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

Tabular Riemann Sums for Electricity Consumption

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

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 $$

Trapezoidal Approximation for a Changing Rate

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

Trigonometric Integration via U-Substitution

Evaluate the integral $$I=\int_{0}^{\frac{\pi}{4}} \tan(x)*\sec^2(x)\,dx.$$ Answer the following par

Using Integration to Determine Average Value

A function given by $$f(x)= \ln(1+x)$$ is defined on the interval $$[0,3]$$. Use integration to dete

Volume of a Solid by Washer Method

A region is bounded by the curves $$y=\sqrt{x}$$ and $$y=\frac{x}{2}$$. This region, between the cur

Water Accumulation in a Tank

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

A Separable Differential Equation: Growth Model

Consider the differential equation $$\frac{dy}{dx}=3*x*y^2$$ that models a growth process. Use separ

Bacterial Growth under Logistic Model

A bacterial culture grows according to the logistic differential equation $$\frac{dB}{dt}=rB\left(1-

Bacterial Population with Time-Dependent Growth Rate

A bacterial population grows according to the differential equation $$\frac{dP}{dt}=\frac{k}{1+t^2}P

Bernoulli Differential Equation via Substitution

Consider the differential equation $$\frac{dy}{dx}=y+x*y^2$$. Recognize that this is a Bernoulli equ

Charging of an RC Circuit

An RC circuit is being charged with a battery of voltage $$12\,V$$. The voltage across the capacitor

Chemical Reaction Rate

In a chemical reaction, the concentration $$C$$ (in mol/L) of a reactant is recorded over time as sh

Environmental Contaminant Dissipation in a Lake

A lake has a pollutant concentration $$C(t)$$ (in mg/L) that evolves according to $$\frac{dC}{dt}=-0

Evaporation of a Liquid

A liquid evaporates from an open container and its volume $$V$$ (in liters) changes over time (in ho

First Order Linear Differential Equation

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

Fishery Harvesting Model

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

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 Tangent Lines of an Ellipse

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

Integrating Factor Method

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

Investigating a Piecewise Function's Discontinuity

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

Logistic Growth Model Analysis

A population $$y(t)$$ grows according to the logistic differential equation $$\frac{dy}{dt} = k * y

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 of organisms is modeled by the logistic differential equation $$\frac{dP}{dt}=rP\left(1

Logistic Population Model Analysis

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

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 the differential equation $$\frac{dN}{dt}=-\lambda N$$,

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: y and x

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

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 a Linear Differential Equation

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

Tank Draining Differential Equation

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

Vehicle Deceleration

A vehicle undergoing braking has its speed $$v$$ (in m/s) recorded over time (in seconds) as shown.

Average Reaction Rate Determination

A chemical reaction’s rate is modeled by the function $$r(t)=k*e^{-t}$$, where $$t$$ is in seconds a

Average Temperature Analysis

A researcher models the temperature during a day using the function $$T(t)=10+15*\sin\left(\frac{\pi

Average Value of a Trigonometric Function

Consider the function $$f(x)=\sin(x)+1$$ defined on the interval $$[0,\pi]$$. This function models a

Boat Navigation Across a River with Current

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

Calculation of Consumer Surplus

The demand function for a product is given by $$p(x)=20-0.5*x$$, where $$p$$ is the price (in dollar

Car Braking Analysis

A car decelerates with acceleration given by $$a(t)=-4e^{-t/2}$$ (in m/s²) and has an initial veloci

Designing an Open-Top Box

An open-top box with a square base is to be constructed with a fixed volume of $$5000\,cm^3$$. Let t

Economic Profit Analysis via Area Between Curves

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

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{

Net Change and Total Distance in Particle Motion

A particle has acceleration $$a(t)=12-8*t$$ (in $$m/s^2$$) for $$t \ge 0$$, with initial velocity $$

Optimization of Average Production Rate

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

Particle Motion and Integrated Functions

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

Population Accumulation through Integration

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

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(

Tank Filling Process Analysis

Water flows into a tank at a rate modeled by $$R(t)=5+0.5*t$$ (in liters per minute) for $$0 \le t \

Traveling Particle with Piecewise Motion

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

Volume by Cylindrical Shells

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

Volume by the Cylindrical Shells Method

A region bounded by $$y=\ln(x)$$, $$y=0$$, and the vertical line $$x=e$$ is rotated about the y-axis

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 Using the Disc Method

Consider the region in the xy-plane bounded by $$y = \sqrt{x}$$ and $$y=0$$ for $$0 \le x \le 9$$. T

Water Tank Filling with Graduated Inflow

A water tank is filled daily by adding a certain amount of water that increases by a fixed amount ea

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