AP Calculus AB FRQ Room

Absolute Value Function Limits

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

Algebraic Manipulation and Limit Evaluation

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

Analyzing Process Data for Continuity

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

Area and Volume Setup with Bounded Regions

Consider the region R bounded by the curves $$y = x^2$$, $$y = 4$$, and $$x = 0$$. Though integratio

Capstone Problem: Continuity and Discontinuity in a Compound Piecewise Function

Consider the function $$f(x)=\begin{cases} \frac{x^2-1}{x-1} & x<2 \\ \frac{x^2-4}{x-2} & x\ge2 \end

Complex Rational Function with Removable and Essential Discontinuities

Consider the function $$f(x)=\begin{cases} \frac{x^3-8}{x^2-4} & x\neq -2,2 \\ 4 & x=2 \end{cases}$$

Composite Function and Continuity Analysis

Define \(f(x)=\sqrt{1-\ln(x)}\) for \(x>0\) and consider the composite function \(g(x)=f(e^x)\). Ans

Compound Interest and Geometric Series

A bank account accrues interest compounded annually at an annual rate of 10%. The balance after $$n$

Discontinuities in a Rational-Exponential Function

Consider the function $$ f(x) = \begin{cases} \frac{e^{x} - 1}{x}, & x \neq 0 \\ k, & x = 0. \en

Ensuring Continuity for a Piecewise-Defined Function

Consider the piecewise function $$p(x)= \begin{cases} ax + 3 & \text{if } x < 2, \\ x^2 + b & \text{

Evaluating a Limit with Radical Expressions

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

Inverse Function Analysis and Derivative

Let $$f(x)= x^3+2$$, defined for all real numbers.

Jump Discontinuity in a Piecewise Function

Consider the function $$f(x)=\begin{cases} x+2, & x < 1 \\ 3, & x = 1 \\ 2*x, & x > 1 \end{cases}$$.

Jump Discontinuity in a Piecewise Function

Consider the function $$g(x)=\begin{cases} \frac{x^2-4}{x-2} & x<2\\ 5 & x=2\\ x+3 & x>2 \end{cases}

Limit Analysis in Population Modeling

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

Modeling Temperature Change: A Real-World Limit Problem

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

One-Sided Limits and Discontinuity Analysis

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

One-Sided Limits and Vertical Asymptotes

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

One-Sided Limits of a Piecewise Function

Let $$f(x)$$ be defined by $$f(x) = \begin{cases} x + 2 & \text{if } x < 3, \\ 2x - 3 & \text{if }

Oscillatory Behavior and Non-Existence of Limit

Let \(f(x)=\sin(1/x)\) for \(x\neq0\). Answer the following:

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

Piecewise-Defined Function Continuity Analysis

Let $$f(x)$$ be defined as follows: For $$x < 2$$: $$f(x)= 3x - 1$$. For $$2 \leq x \leq 5$$: $$f(x

Removable Discontinuity and Direct Limit Evaluation

Consider the function $$f(x)=\frac{(x-2)(x+3)}{x-2}$$ defined for $$x\neq2$$. The function is not de

Removal of Discontinuity by Redefinition

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

Squeeze Theorem Application with Trigonometric Functions

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

Squeeze Theorem for an 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 with Trigonometric Function

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

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 Computation

Consider the function $$f(x)= \frac{\sin(5*(x-\pi/4))}{x-\pi/4}$$.

Air Quality and Pollution Removal

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

Analysis of Savings Account Growth

A savings account has a balance given by $$S(t)= 1000*(1.005)^t$$, where $$t$$ is the number of mont

Analyzing a Function's Derivative from its Graph

A graph of a smooth function is provided. Answer the following questions:

Approximating Derivative using Secant Lines

Consider the function $$f(x)= \ln(x)$$. A student records the following data: | x | f(x) | |---

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

Derivative of a Logarithmic Function

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

Derivative of a Trigonometric Function

Let \(f(x)=\sin(2*x)\). Answer the following parts.

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.

Derivative using the Limit Definition for a Linear Function

For the linear function $$f(x)= 5*x - 3$$, perform an analysis of its derivative using the limit def

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

Derivatives of Trigonometric Functions

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

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 Second Derivative

Given $$f(x)= x^4 - 4*x^2 + 7$$, compute its first and second derivatives.

Identifying Horizontal Tangents

A continuous function $$f(x)$$ has a derivative $$f'(x)$$ such that $$f'(4)=0$$ and $$f'(x)$$ change

Inverse Function Analysis: Exponential Transformation

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

Inverse Function Analysis: Hyperbolic-Type Function

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

Inverse Function Analysis: Logarithmic Transformation

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

Inverse Function Analysis: Rational Function

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

Logarithmic Differentiation of a Composite Function

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

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

Mountain Stream Flow Adjustment

A mountain stream receives additional water from snowmelt at a rate of $$f(t)=4*t$$ (cubic feet/seco

Polynomial Rate of Change Analysis

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

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 with Exponential Function

Consider the function $$f(x)= x*e^{x}$$ which exhibits both polynomial and exponential behavior.

Proof of Scaling in Derivatives

Let $$f(x)$$ be a differentiable function and let $$k$$ be a constant. Consider $$g(x)= k*f(x)$$. Us

Quotient Rule Challenge

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

Secant and Tangent Lines

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

Secant and Tangent Lines to a Curve

Consider the function $$f(x)=x^2 - 4*x + 5$$. Answer the following questions:

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 $$t$$

Using the Quotient Rule for a Rational Function

Let $$f(x) = \frac{3*x+5}{x-2}$$. Differentiate $$f(x)$$ using the quotient rule.

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

Chain Rule in a Light Intensity Model

The intensity of light is modeled by $$I(r) = \frac{1}{r^2}$$, where r is the distance (in meters) f

Combining Composite and Implicit Differentiation

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

Composite Function and Inverse Analysis via Graph

Consider the function $$f(x)= \sqrt{4*x-1}$$, defined for $$x \geq \frac{1}{4}$$. Analyze the functi

Composite Function Kinematics

A particle moves along a straight line with its position given by $$s(t) = (2*t+3)^4$$. Analyze the

Designing a Tapered Tower

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

Differentiating an Inverse Trigonometric Function

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

Differentiation of a Log-Exponential-Trigonometric Composite

Consider the function $$f(x)= \ln\left(e^(\cos(x)) + x^2\right)$$. Solve the following:

Estimating Derivatives Using a Table

An experiment measures a one-to-one function $$f$$ and its inverse $$g$$, yielding the following dat

Implicit Differentiation and Rate Change in Biology

In an ecosystem, the relationship between two population parameters is given by $$e^y+ x*y= 10$$, wh

Implicit Differentiation in a Circle

Consider the circle $$x^2 + y^2 = 25$$. Answer the following parts.

Implicit Differentiation in a Hyperbola

Consider the hyperbola defined by $$x*y=10$$. Answer the following parts.

Implicit Differentiation of a Trigonometric Composite Function

Consider the curve defined implicitly by $$\sin(y) + y^2 = x$$.

Implicit Differentiation on a Circle

Consider the circle defined by $$x^2+y^2=25$$.

Implicit Differentiation with Trigonometric Components

Consider the equation $$\sin(x) + \cos(y) = x*y$$, which implicitly defines $$y$$ as a function of $

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 Derivative

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

Inverse Function Derivative for a Log-Linear Function

Let $$f(x)= x+ \ln(x)$$ for $$x > 0$$ and let g be the inverse of f. Solve the following parts:

Inverse Function Differentiation

Let $$f(x)=x^3+x$$ and assume it is invertible. Answer the following:

Inverse Function Differentiation for a Log Function

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

Inverse Function Differentiation in a Biological Growth Model

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

Inverse Function Differentiation Involving a Polynomial

Let $$f(x)= x^3 + 2*x + 1$$. Analyze its invertibility and the derivative of its inverse function.

Inverse Function Differentiation: Composite Inversion

Let $$f(x) = \frac{x}{1-x}$$ for x < 1, and let g denote its inverse function. Answer the following

Pendulum Angular Displacement Analysis

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

Temperature Change Model Using Composite Functions

The temperature of an object is modeled by the function $$T(t)=e^{-\sqrt{t+2}}$$, where $$t$$ is tim

Analysis of a Piecewise Function with Discontinuities

Consider the function $$f(x)=\begin{cases} \frac{x^2-4}{x-2} & \text{if } x<2 \\ x+1 & \text{if } x\

Analyzing Speed Changes in a Particle’s Motion

A particle moves along a straight line with a velocity function given by $$v(t) = (t-2)^2(t+1)$$ for

Balloon Inflation Related Rates

A spherical balloon is being inflated, and its volume is increasing at a constant rate of $$12$$ cub

Chemical Reaction Rate Analysis

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

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

Cooling Hot Beverage

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

Determining the Tangent Line

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

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

Economics and Marginal Analysis: Revenue and Cost Differentiation

A company’s revenue and cost functions are given by $$R(x)=100*x - 0.5*x^2$$ and $$C(x)=20*x + 50$$,

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

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

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

Free Fall Motion Analysis

An object in free fall near Earth's surface has its position modeled by $$s(t)=-4.9t^2+20t+1$$ (in m

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 3: Ladder Sliding Problem

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

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 18: Chemical Reaction Concentration Changes

During a chemical reaction, the concentrations of reactants A and B are related by $$[A]^2 + 3*[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 Spherical Balloon

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

Interpretation of the Derivative from Graph Data

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

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 of Natural Logarithm

Estimate $$\ln(1.05)$$ using linear approximation for the function $$f(x)=\ln(x)$$ at $$a=1$$.

Linearization and Differentials Approximation

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

Linearization for Function Estimation

Use linear approximation to estimate the value of $$\ln(4.1)$$. Let the function be $$f(x)=\ln(x)$$

Local Linearization Approximation

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

Modeling a Bouncing Ball with a Geometric Sequence

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

Optimization for Minimizing Time in Road Construction

A new road connecting two towns involves two segments with different construction speeds. The travel

Optimization of Production Costs

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

Optimizing Road Construction Costs

An engineer is designing a road that connects a point on a highway to a town located off the highway

Particle Motion Analysis

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

Particle Motion Analysis

A particle moves along a straight line with displacement given by $$s(t)=t^3-6t^2+9t+2$$ for $$0\le

Projectile Motion and Maximum Height

A basketball is thrown such that its height (in meters) is modeled by $$h(t)= -4.9*t^2 + 14*t + 1.5$

Projectile Motion with Velocity Components

A projectile is launched from the ground with a constant horizontal velocity of 15 m/s and a vertica

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

Rates of Change in Economics: Marginal Cost

A company's cost function is given by $$C(q)= 0.5*q^2 + 40$$, where q (in units) is the quantity pro

Related Rates: Expanding Circular Ripple

A ripple in a still pond expands in the shape of a circle. The area of the ripple is given by $$A=\p

Revenue and Cost Analysis

A company’s revenue is modeled by $$R(t)=200e^{0.05t}$$ and its cost by $$C(t)=10t^3-30t^2+50t+200$$

Revenue Sensitivity to Advertising

A firm's revenue (in thousands of dollars) is given by $$R(t)=50\sqrt{t+1}$$, where $$t$$ represents

Rocket Thrust: Analyzing Exponential Acceleration

A rocket’s velocity is modeled by $$v(t) = 100(1 - e^{-0.05t})$$, where $$t$$ is in seconds and $$v(

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

Swimming Pool with Leak

A swimming pool is simultaneously being filled and leaking. The filling rate is constant at $$R_{in}

Tangent Line and Linearization Approximation

Let $$f(x)=\sqrt{x}$$. Use linearization at $$x=16$$ to approximate $$\sqrt{15.7}$$. Answer the foll

Temperature Change in Cooling Coffee

A cup of coffee cools according to the model $$T(t) = 90e^{-0.05t} + 20$$, where $$t$$ is the time i

Temperature Cooling in a Cup of Coffee

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

Temperature Rate Change in Cooling Coffee

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

The Sliding Ladder

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

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

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

Area Growth of an Expanding Square

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

Behavior Analysis of a Logarithmic Function

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

Concavity Analysis of a Trigonometric Function

For the function $$f(x)= \sin(x) - \frac{1}{2}\cos(x)$$ defined on the interval $$[0,2\pi]$$, analyz

Continuous Compound Interest

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

Cost Function and the Mean Value Theorem in Economics

An economic model gives the cost function as $$C(x)= 100 + 20*x - 0.5*x^2$$, where x represents the

Determining Intervals of Increase and Decrease with a Rational Function

Consider the function $$f(x) = \frac{x^2}{x+2}$$ defined on the interval $$[0, 4]$$. Answer the foll

Discontinuity and Derivative in a Hybrid Piecewise Function

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

Drag Force and Rate of Change from Experimental Data

Drag force acting on an object was measured at various velocities. The table below presents the expe

Evaluating Rate of Change in Electric Current Data

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

FRQ 4: Intervals of Increase and Decrease Analysis

Examine the function $$f(x) = 2*x^3 - 9*x^2 + 12*x + 5$$.

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

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

FRQ 11: Particle Motion with Non-Constant Acceleration

A particle moves along a straight line with acceleration given by $$a(t)= 12*t - 6$$ (in m/s²). If 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 and Derivatives

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

Inflection Points in a Population Growth Model

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

Inverse Analysis of a Function with an Absolute Value Term

Consider the function $$f(x)=x+|x-2|$$ with the domain restricted to $$x\ge 2$$. Analyze the inverse

Inverse Analysis of a Piecewise Function

Consider the piecewise function $$f(x)=\begin{cases}2*x+1 & x\le 0,\\ 3*x+1 & x>0\end{cases}$$. Ans

Mean Value Theorem Analysis

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

Mean Value Theorem for a Cubic Function

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

Mean Value Theorem for a Piecewise Function

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

Modeling Disease Spread with an Exponential Model

In an epidemic, the number of infected individuals is modeled by $$I(t)= I_0 * e^{r*t}$$, where $$t$

Motion Analysis via Derivatives

A particle moves along a straight line with its position described by $$s(t)= t^3 - 6*t^2 + 9*t + 5$

Oil Spill Cleanup

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

Quartic Polynomial Concavity Analysis

Consider the quartic function $$f(x)= x^4 - 6*x^3 + 11*x^2 - 6*x$$, defined on the interval $$[0,4]$

Sign Analysis of f'(x)

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

Slope Analysis for Parametric Equations

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

Verifying the Mean Value Theorem for a Polynomial Function

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

Accumulated Altitude Change: Hiking Profile

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

Accumulation and Flow Rate in a Tank

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

Antiderivative of a Transcendental Function

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

Antiderivatives and Initial Value Problems

Given that $$f'(x)=\frac{2}{\sqrt{x}}$$ for $$x>0$$ and $$f(4)=3$$, find the function $$f(x)$$.

Antiderivatives with Initial Conditions: Temperature

The rate of temperature change in a chemical reaction is given by $$T'(t)=-0.2*t+3$$ (in °C/min), wi

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

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

Area Under a Piecewise Function

Consider the piecewise function $$ f(x)= \begin{cases} x & \text{for } 0\le x<3,\\ 9-x & \text{for

Average Temperature Calculation over 12 Hours

In a city, the temperature over a 12-hour period is modeled by $$T(t) = -2*t + 20$$ (in $$^\circ C$$

Chemical Production via Integration

The production rate of a chemical in a reactor is given by $$r(t)=5*(t-2)^3$$ (in kg/hr) for $$t\ge2

Coffee Brewing Dynamics

An advanced coffee machine drips water into the brewing chamber at a rate of $$W(t)=10+t$$ mL/s, whi

Comparing Riemann Sum Methods for $$\int_1^e \ln(x)\,dx$$

Consider the function $$f(x)= \ln(x)$$ on the interval $$[1,e]$$. A table of approximate values is p

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

Cost Accumulation from Marginal Cost Function

A company's marginal cost (in dollars per unit) is given by $$MC(x)=0.2*x+50$$, where $$x$$ represen

Economic Cost Function Analysis

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

Electric Charge Accumulation

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

Estimating Area Under a Curve Using Riemann Sums

Consider the function whose values are given in the table below. Use the table to estimate the area

Estimating Work Done Using Riemann Sums

In physics, the work done by a variable force is given by $$W=\int F(x)\,dx$$. A force sensor record

Evaluating a Definite Integral Using U-Substitution

Compute the integral $$\int_{0}^{3} (2*t+1)^5\,dt$$ using u-substitution.

FRQ14: Inverse Analysis of a Logarithmic Accumulation Function

Let $$ L(x)=\int_{1}^{x} \frac{1}{t}\,dt $$ for x > 0. Answer the following parts.

General Antiderivatives and the Constant of Integration

Given the function $$f(x)= 4*x^3$$, address the following questions about antiderivatives.

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,

Mixed Method Approximation of an Integral

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

Net Displacement and Total Distance Calculation

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

Oxygen Levels in a Bioreactor

In a bioreactor, oxygen is introduced at a rate $$O_{in}(t)= 7 - 0.5t$$ mg/min and is consumed at a

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

The Accumulation Function for a Linear Rate Model

Consider the accumulation function defined by $$A(t)= \int_0^t (3*t' + 2)\,dt'$$, where $$t'$$ is a

Volume of a Melting Ice Sculpture

An ice sculpture is melting, losing volume at a rate of $$M(t)= 2t+5$$ cubic meters per hour, while

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

Bank Account with Continuous Interest and Withdrawals

A bank account accrues interest continuously at an annual rate of $$6\%$$, while money is withdrawn

Bernoulli Differential Equation via Substitution

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

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

Direction Fields and Integrating Factor

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

Drug Infusion and Elimination

The concentration of a drug in a patient's bloodstream is modeled by the differential equation $$\fr

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

Exponential Growth and Doubling Time

A bacterial culture grows according to the differential equation $$\frac{dy}{dt} = k * y$$ where $$y

Implicit Differentiation of a Transcendental Equation

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

Logistic Population Growth

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

Logistic Population Model Analysis

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

Mixing a Salt Solution

A mixing tank experiment records the salt concentration $$C$$ (in g/L) at various times $$t$$ (in mi

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 in a Tank

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

Mixing Problem with Evaporation and Drainage

A tank initially contains 200 L of water with 20 kg of pollutant. Water enters the tank at 2 L/min w

Modeling Orbital Decay

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

Population with Constant Harvesting

A fish population in a lake grows according to the differential equation $$\frac{dy}{dt} = r*y - H$$

Radioactive Decay Model

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

Radioactive Isotope in Medicine

A radioactive isotope used in medical imaging decays according to $$\frac{dA}{dt}=-kA$$, where $$A$$

Related Rates: Expanding Balloon

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

Reversible Chemical Reaction

In a reversible chemical reaction, the concentration $$C(t)$$ of a product is governed by the differ

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

A slope field for the differential equation $$\frac{dy}{dx}=y-x$$ is provided in the stimulus. Use t

Tumor Treatment with Chemotherapy

A patient's tumor cell population $$N(t)$$ is modeled by the differential equation $$\frac{dN}{dt}=r

Water Tank Flow Analysis

A water tank receives an inflow of water at a rate $$Q_{in}(t)=50+10*\sin(t)$$ (liters/min) and an o

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 Damped Oscillatory Motion

A mass-spring system exhibits a damped oscillation modeled by $$f(t)=e^{-0.3*t}*\sin(t)$$ (in meters

Area Between Curves in an Ecological Study

In an ecological study, the population densities of two species are modeled by the functions $$P_1(x

Area Between Two Curves from Tabulated Data

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

Average Concentration in Medical Dosage

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

Average of a Logarithmic Function

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

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 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 weather scientist models the temperature during a day by the function $$f(t)=5+2*t-0.1*t^2$$ where

Designing a Water Slide

A water slide is designed along the curve $$y=-0.1*x^2+2*x+3$$ (in meters) over the interval $$[0,10

Distance Traveled Analysis from a Velocity Graph

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

Electric Charge Accumulation

The current flowing into a capacitor is defined by $$I(t)=\frac{10}{1+e^{-2*(t-3)}}$$ (in amperes) f

Environmental Impact: Average Pollutant Concentration

The pollutant concentration in a river is modeled by $$h(x)=0.01*x^3-0.5*x^2+5*x$$ (in mg/L) over a

Funnel Design: Volume by Cross Sections

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

Implicit Differentiation in an Economic Equilibrium Model

In an economics model, the relationship between price $$p$$ and quantity $$q$$ is given implicitly b

Implicit Differentiation in an Electrical Circuit

In an electrical circuit, the voltage $$V$$ and current $$I$$ are related by the equation $$V^2 + (3

Medication Dosage Increase

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

Modeling Bacterial Growth

A bacterial culture grows at a rate modeled by $$g(t)=a*e^{0.3*t}$$, where $$t$$ is time in hours an

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 $$

Projectile Motion: Position, Velocity, and Maximum Height

A projectile is launched vertically upward with an initial velocity of $$20\,m/s$$ from a height of

Projectile Motion: Time of Maximum Height

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

Retirement Savings Auto-Increase

A person contributes to a retirement fund such that the monthly contributions form an arithmetic seq

River Current Analysis

The velocity of a river is given by $$v(x)=2+x-0.1*x^2$$ (in m/s) for 0 ≤ x ≤ 10, where x measures t

Temperature Average Calculation

A scientist records the temperature in a lab using a continuous function $$T(t)=3*t^2 - 4*t + 5$$, w

Total Distance from a Runner's Variable Velocity

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

Traveling Particle with Piecewise Motion

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

Volume of a Rotated Region by the Disc Method

Consider the region bounded by the curve $$f(x)=\sqrt{x}$$ and the line $$y=0$$ for $$0 \le x \le 4$

Volume of a Solid of Revolution: Disc Method

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

Volume with Semicircular Cross‐Sections

A region in the first quadrant is bounded by the curve $$y=x^2$$ and the x-axis for $$0 \le x \le 3$

Water Flow in a River: Average Velocity and Flow Rate

A river has a width of 10 meters. The velocity of the water at a distance $$x$$ (in meters) from one

Water Reservoir Inflow‐Outflow Analysis

A water reservoir receives water through an inflow pipe and loses water through an outflow valve. Th

Work Done by a Variable Force

A force acting along a straight line is given by $$F(x)=3*x^2+2$$ (in Newtons) when an object is dis

Work in Pumping Water

A water pump is used to empty a reservoir. The force required to pump water out at a depth $$y$$ (in

Work in Pumping Water from a Conical Tank

A water tank is in the shape of an inverted right circular cone with height $$10\,m$$ and top radius

Work in Spring Stretching

A spring obeys Hooke's law, where the force required to stretch the spring a distance $$x$$ from its

Work to Pump Water from a Cylindrical Tank

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