AP Calculus AB FRQ Room

Absolute Value Function Limits

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

Analysis of a Vertical Asymptote

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

Analysis of Three Functions

The table below lists the values of three functions f, g, and h at selected x-values. Use the table

Analysis of Vertical Asymptotes

Examine the function $$h(x)= \frac{x^2-9}{x^2-4*x+3}$$. Answer the following:

Analyzing a Piecewise Function for Continuity

Consider the piecewise function $$ f(x)=\begin{cases} 2x+1, & x<2 \\ x^2-1, & x\geq2 \end{cases}$$.

Application of the Intermediate Value Theorem in a Logistic Model

Let $$ f(x)=\frac{1}{1+e^{-x}} $$, a logistic function that is continuous for all x. Analyze its beh

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

Asymptotic Analysis of a Radical Rational Function

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

Complex Rational Limit and Removable Discontinuity

Consider the function $$f(x)=\frac{x^3 - 27}{x^2 - 9}$$ for x \(\neq\) 3. Answer the following parts

Determining Asymptotes and Holes in a Rational Function

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

Economic Limit and Continuity Analysis

A company's profit (in thousands of dollars) from producing x items is modeled by the function $$P(x

Estimating Derivatives Using Limit Definitions from Data

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

Evaluating a Limit with Radical Expressions

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

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

Graphical Estimation of a Limit

The following graph shows the function $$f(x)$$. Use the graph to answer the subsequent questions re

Identifying Discontinuities in a Rational Function

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

Intermediate Value Theorem Application

Let $$p(x)=x^3-4x-5$$. Use the Intermediate Value Theorem (IVT) to show that the equation \(p(x)=0\)

Intermediate Value Theorem in Equation Solving

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

Intermediate Value Theorem with an Exponential-Logarithmic Function

Consider the function $$u(x)=e^{x}-\ln(x+2)$$, defined for $$x > -2$$. Since $$u(x)$$ is continuous

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 and Integration in Non-Polynomial Particle Motion

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

Limits Involving Absolute Value Functions

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

Limits Involving Composition and Square Roots

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

Limits Involving Radicals and Algebra

Consider the function $$f(x)= \sqrt{x^2 + x} - x$$. Answer the following parts.

Limits Near Vertical Asymptotes

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

One-Sided Limits and an Absolute Value Function

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

One-Sided Limits and Continuity of a Piecewise Function

Consider the piecewise function $$w(x)= \begin{cases} \frac{e^{x}-1}{x} & \text{if } x<0, \\ \frac{\

Optimization and Continuity in a Manufacturing Process

A company designs a cylindrical can (without a top) for which the cost function in dollars is given

Oscillatory Behavior and Non-Existence of Limit

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

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 with Two Critical Points

Consider the function $$f(x)=\begin{cases} \frac{x^2+x-6}{x^2-9} & x\neq -3,3 \\ \frac{5}{6} & x=-3

Rational Functions with Removable Discontinuities

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

Related Rates: Shadow Length of a Moving Object

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

Robotic Arm and Limit Behavior

A robotic arm moving along a linear axis has a velocity function given by $$v(t)= \frac{t^3-8}{t-2}$

Squeeze Theorem Application

Consider the function $$f(x)=x^2\sin(\frac{1}{x})$$ for $$x\neq0$$ and $$f(0)=0$$. Answer the follow

Trigonometric Function Discontinuity

Consider the function $$f(x)=\begin{cases} \frac{\sin(2*x)}{x} & x\neq0 \\ 4 & x=0 \end{cases}$$. An

Vertical Asymptote Analysis

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

Analyzing the Derivative of a Trigonometric Function

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

Application of Derivative in Calculating Slope of a Curve

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

Approximating Derivatives Using Secant Lines

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

Chain Rule Application

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

Derivative from First Principles

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

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

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

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

Difference Quotient for a Cubic Function

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

Differentiability of a Piecewise Function

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

Differentiability of an Absolute Value Function

Consider the function $$f(x)=|x-3|$$, representing the error margin (in centimeters) in a calibratio

Differentiating an Absolute Value Function

Consider the function $$f(x)= |3*x - 6|$$.

Estimating Instantaneous Slope of a Logarithmic Function

Consider the function \(f(x)=\ln(x)\). Without directly using the derivative rules, estimate the ins

Exponential Growth Rate

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

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 vs. Derivative Graph

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

Graphical Analysis of Secant and Tangent Slopes

A function $$f(x)$$ is represented by the red curve in the graph below. Answer the following questio

Inverse Function Analysis: Cubic with Linear Term

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

Inverse Function Analysis: Exponential Transformation

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

Radioactive Decay Analysis

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

Related Rates: Expanding Ripple Circle

Water droplets create circular ripples on a surface. The area of a ripple is given by $$A = \pi * r^

Related Rates: Shadow Length Change

A person 1.8 m tall is walking away from a streetlight that is 5 m high. Let x represent the distanc

Sand Pile Accumulation

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

Secant and Tangent Lines for a Cubic Function

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

Secant and Tangent Lines for a Trigonometric Function

Let $$f(x)=\sin(x)+x^2$$. Use the definition of the derivative to find $$f'(x)$$ and evaluate it at

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 Approximation

Suppose a continuous function $$f(x)$$ is differentiable with $$f(2)=8$$ and $$f'(2)=5$$. Use this i

Tangent Line Equation for an Exponential Function

Consider the function $$f(x)= e^{x}$$ and its graph.

Analyzing a Function and Its Inverse

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

Chain and Product Rules in a Rate of Reaction Process

In a chemical reaction, the rate is modeled by the function $$R(t)= \sqrt{t+3}*\cos(2*t)$$, where $$

Chain Rule Basics

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

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

Chain Rule in an Economic Model

In a manufacturing process, the cost function is given by $$C(q)= (\ln(1+3*q))^2$$, where $$q$$ is t

Chain Rule with Exponential and Trigonometric Functions

A particle's position is given by $$s(t)=e^{3*t}\sin(2*t)$$. Use appropriate differentiation techniq

Chain Rule with Multiple Nested Functions in a Physics Model

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

Chain Rule with Trigonometric Function

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

Combining Composite and Implicit Differentiation

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

Composite Function 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 via Chain Rule in a Financial Context

A company’s profit (in dollars) based on production level (in thousands of units) is modeled by the

Composite Functions in Population Dynamics

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

Composite Log-Exponential Function Analysis

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

Differentiation of Inverse Function with Polynomial Functions

Let \(f(x)= x^3+2*x+1\) be a one-to-one function. Its inverse is denoted by \(f^{-1}\).

Implicit Curve Analysis: Horizontal Tangents

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

Implicit Differentiation in a Circle

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

Implicit Differentiation in a Financial Model

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

Implicit Differentiation in an Elliptical Orbit

Consider the ellipse defined by $$4*x^2 + 9*y^2 = 36$$, which can model the orbit of a satellite.

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 on a Circle

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

Implicit Differentiation with Exponential-Trigonometric Functions

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

Implicit Differentiation with Exponential-Trigonometric Functions

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

Implicit Differentiation with Product Rule

Consider the equation $$x*e^{y} + y*\ln(x)=5$$. Answer the following:

Implicit Trigonometric Equation Analysis

Consider the equation $$x \sin(y) + \cos(y) = x$$. Answer the following parts.

Inverse Derivative of a Sum of Exponentials and Linear Terms

Let $$f(x)= e^(x)+ x$$ and let g be its inverse function satisfying $$g(f(x))= x$$. Answer the follo

Inverse Function Differentiation in a Biological Growth Model

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

Inverse Function Differentiation in an Exponential Context

Let $$f(x)= e^(3*x) - 2$$ and let $$g(x)$$ be the inverse function of f. Answer the following:

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 Function Differentiation with Exponentials and Trigonometry

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

Inverse Function Differentiation with Trigonometric Component

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

Inverse Trigonometric Differentiation

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

Inverse Trigonometric Differentiation in Engineering Mechanics

In an engineering application, the angle of elevation $$\theta$$ is given by the function $$\theta=

Inverse Trigonometric Function Differentiation

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

Particle Motion: Logarithmic Position Function

The position of a particle moving along a line is given by $$s(t)= \ln(3*t+2)$$, where s is in meter

Pendulum Angular Displacement Analysis

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

Population Dynamics via Composite Functions

A biological population is modeled by $$P(t)= \ln\left(20*e^(0.1*t^2)+ 5\right)$$, where t is measur

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

Related Rates via Chain Rule

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

Temperature Reaction Model Using Composite Functions

A chemical reactant’s concentration is given by $$C(t)= e^{-0.3*t^2}$$ and the reaction rate dependi

Analysis of a Composite Function involving Logarithm

The revenue function is given by $$R(x)= x\ln(100/x)$$ for x > 0, where x is the number of units sol

Analyzing Rate of Change in a Compound Interest Model

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

Cooling Coffee Temperature Analysis

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

Designing a Flower Bed: Optimal Shape

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

Differentiability of a Piecewise Function

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

Drug Concentration in the Blood

A drug’s concentration in the bloodstream is modeled by $$C(t)= \frac{5}{1+e^{0.2(t-30)}}$$, where $

Economic Cost Function Linearization

A company's production cost is modeled by $$C(x)= 0.02*x^3 - 1.5*x^2 + 40*x + 200$$ dollars, where $

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 Balloon: Related Rates with a Sphere

A spherical balloon is being inflated so that its volume increases at a constant rate of $$dV/dt = 1

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

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

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 ln(1.05)

Let $$f(x)=\ln(x)$$. Use linearization at x = 1 to approximate the value of $$\ln(1.05)$$.

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 in Medicine Dosage

A drug’s concentration in the bloodstream is modeled by $$C(t)=\frac{5}{1+e^{-t}}$$, where $$t$$ is

Linearization of a Nonlinear Function

Suppose $$f(x)=\ln(x)$$. Use linearization about $$x=4$$ to approximate $$\ln(4.1)$$. Answer the fol

Local Linearization Approximation

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

Logarithmic Profit Optimization

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

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

Motion Along a Curved Path

An object moves along the curve given by $$y=\ln(x)$$ for $$x\geq 1$$. Suppose the x-component of th

Optimization in a Manufacturing Process

A company designs an open-top container whose volume is given by $$V = x^2 y$$, where x is the side

Optimization of Production Costs

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

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

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: Evaluating Maximum Height

A projectile is launched vertically with its height given by $$h(t)= -4.9*t^2 + 19.6*t + 3$$, where

Radioactive Decay: Rate of Change and Half-life

A radioactive substance decays according to the formula $$N(t)=N_0e^{-kt}$$, where $$N(t)$$ is the a

Rate of Change in a Population Model

A population model is given by $$P(t)=30e^{0.02t}$$, where $$P(t)$$ is the population in thousands a

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 in a Conical Tank

Water is draining from a conical tank. The volume of water is given by $$V = \frac{1}{3}\pi r^2 h$$,

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

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

Vehicle Deceleration Analysis

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

Analysis of a Parametric Curve

Consider the parametric equations $$x(t)= t^2 - 3*t$$ and $$y(t)= 2*t^3 - 9*t^2 + 12*t$$. Analyze th

Analysis of an Exponential-Logarithmic Function

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

Analyzing a Piecewise Function and Differentiability

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

Analyzing Concavity and Inflection Points

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

Analyzing Differentiability of a Piecewise Function

Consider the piecewise function $$ f(x)= \begin{cases} x^2, & \text{if } x \le 1, \\ 2*x - 1, &

Application of Rolle's Theorem for a Quadratic Function

Let $$f(x)= x^2 - 4$$ be defined on the interval $$[-2,2]$$. In this problem, you will verify the co

Application of Rolle's Theorem to a Trigonometric Function

Consider the function $$f(x) = \sin(x)$$ defined on the interval $$[0, \pi]$$. Answer the following:

Applying the Mean Value Theorem and Analyzing Discontinuities

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

Area and Volume: Polynomial Boundaries

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

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

Comprehensive Analysis of a Rational Function

Given the rational function $$f(x)= \frac{x^2-4}{x^2+1}$$, perform a comprehensive analysis includin

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

Cost Minimization in Transportation

A transportation company recorded shipping costs (in thousands of dollars) for different numbers of

Determining Absolute and Relative Extrema

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

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

FRQ 4: Intervals of Increase and Decrease Analysis

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

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

Implicit Differentiation and Tangent Lines

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

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 a Trigonometric Function on a Restricted Domain

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

Investigating Limits and Discontinuities in a Rational Function with Complex Denominator

Consider the function $$ f(x) = \begin{cases} \frac{x^2-9}{x-3}, & x < 3, \\ \frac{x^2-9}{x-3} + 4,

Liquid Cooling System Flow Analysis

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

Logarithmic Transformation of Data

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

Mean Value Theorem for a Logarithmic Function

Consider the function $$f(x)= \ln(x)$$ defined on the interval $$[1, e^2]$$. Use the Mean Value Theo

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

Pharmacokinetics: Drug Concentration Decay

A drug in the bloodstream decays according to $$D(t)= D_0*e^{-k*t}$$, where $$t$$ is in hours. Answe

Transcendental Function Analysis

Consider the function $$f(x)= \frac{e^x}{x+1}$$ defined for $$x > -1$$ and specifically on the inter

Water Cooling Tower Efficiency

In a water cooling tower, water is pumped in at a rate $$R_{in}(t)=10+0.5*t^2$$ L/min and discharged

Water Droplet Free Fall Analysis

A water droplet is released from a ceiling, and its height (in meters) above the ground is modeled b

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 Function and Its Derivative

Define the function $$F(x)= \int_0^x \Big(e^{t} - 1\Big)\,dt$$. Answer the following parts related t

Antiderivatives and the Constant of Integration in Modelling

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

Application of the Fundamental Theorem of Calculus

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

Area Between Curves

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

Area Between Curves

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

Area Under a Parabola

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

Composite Functions and Accumulation

Let the accumulation function be defined by $$F(x)=\int_{2}^{x} \sqrt{t+1}\,dt.$$ Answer the followi

Cost Accumulation in a Production Process

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

Economic 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

Economic Cost Function Analysis

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

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 Displacement with a Midpoint Riemann Sum

A vehicle’s velocity is modeled by the function $$v(t) = -t^{2} + 4*t$$ (in meters per second) over

Evaluating an Integral with a Piecewise Function

Consider the function defined by $$f(x)=\begin{cases} x^2 & \text{if } x<2,\\ 4*x-4 & \text{if } x

Evaluating the Accumulated Drug Concentration

In a pharmacokinetics study, a drug is infused into a patient's bloodstream at a rate given by $$R(t

Evaluating the Definite Integral of a Power Function

Let $$f(t)= 3*t^{1/3}$$. Evaluate the definite integral $$\int_{27}^{64} 3*t^{1/3}\,dt$$. Answer th

Experimental Data Analysis using Trapezoidal Sums

A chemical reaction is monitored over time, and the reaction rate $$f(t)$$ (in moles per minute) is

Medication Concentration and Absorption Rate

A patient's blood concentration of a drug (in mg/L) is monitored over time before reaching its peak.

Midpoint Riemann Sum for Temperature Data

A weather station records temperature (in degrees Celsius) at hourly intervals. The data for a 4-hou

Modeling Savings with a Geometric Sequence

A person makes annual deposits into a savings account such that the first deposit is $100 and each s

Net Change Calculation

The net change in a quantity $$Q$$ is modeled by the rate function $$\frac{dQ}{dt}=e^{t}-1$$ for $$0

Population Growth in a Bacterial Culture

A bacterial culture grows at a rate given by $$r(t)=0.5*e^{-0.3*t}$$ (in thousands of bacteria per h

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

Applying the SIPPY Method to $$dy/dx = \frac{4x}{y}$$

Solve the differential equation $$\frac{dy}{dx}=\frac{4x}{y}$$ with the initial condition $$y(0)=5$$

Boat Crossing a River with Current

A boat is attempting to cross a river flowing at a constant speed of $$2$$ m/s. The boat is directed

Carbon Dating and Radioactive Decay

Carbon dating is based on the radioactive decay model given by $$\frac{dC}{dt}=-kC$$. Let the initia

Charging a Capacitor in an RC Circuit

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

Charging of a Capacitor

The voltage $$V$$ (in volts) across a capacitor being charged in an RC circuit is recorded over time

Chemical Reactor Temperature Profile

In a chemical reactor, the temperature $$T$$ (in °C) is recorded over time (in minutes) as shown. Th

Cooling of Electronic Components

After shutdown, the temperature $$T$$ (in °C) of an electronic component is recorded over time (in s

Direction Fields and Integrating Factor

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

Economic Decay Model

An asset depreciates in value according to the model $$\frac{dC}{dt}=-rC$$, where $$C$$ is the asset

Ecosystem Nutrient Cycle

In a forest ecosystem, nitrogen is deposited from the atmosphere at a rate of $$2$$ kg/ha/year while

Falling Object with Air Resistance

A falling object with mass $$m=70\,kg$$ is subject to gravity and a resistive force proportional to

Falling Object with Air Resistance

A falling object experiences air resistance proportional to its velocity. Its motion is modeled by t

Integrating Factor Method

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

Investment Account with Continuous Withdrawals

An investment account grows continuously at an annual rate of 5% and experiences continuous withdraw

Logistic Population Model Analysis

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

Mixing of a Pollutant in a Lake

A lake with a constant volume of $$10^6$$ m\(^3\) receives polluted water from a river at a rate of

Newton's Law of Cooling with Variable Ambient Temperature

An object cools according to Newton's Law of Cooling, but the ambient temperature is not constant. I

Population Dynamics with Harvesting

A wildlife population (in thousands) is monitored over time and is subject to harvesting. The popula

Population Dynamics with Harvesting

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

Radioactive Decay

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

Radioactive Decay Differential Equation

A radioactive substance decays according to the differential equation $$\frac{dN}{dt}= -\lambda N$$,

Radioactive Material with Constant Influx

A laboratory receives radioactive waste material at a constant rate of $$3$$ g/day. Simultaneously,

RC Circuit Charging

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

Salt Tank Mixing Problem

A tank initially contains 100 liters of pure water. A salt solution with concentration 0.5 kg/L is p

Sand Erosion in a Beach Model

During a storm, a beach loses sand. Let $$S(t)$$ (in tons) be the amount of sand on a beach at time

Separable Differential Equation

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

Separable Differential Equation with Trigonometric Factor

Consider the differential equation $$\frac{dy}{dx}=(2y+3)\cos(x)$$. Answer each part using separatio

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 by Substitution

Solve the differential equation $$\frac{dy}{dx}=\frac{2xy}{x^2-4}$$. (a) Separate the variables to

Tank Draining Differential Equation

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

Tumor Treatment with Chemotherapy

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

Water Level in a Reservoir

A reservoir's water volume $$V$$ (in million m³) is measured at various times $$t$$ (in days) as sho

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 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 a Parabola and a Line

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

Area Between Curves: Revenue and Cost Analysis

A company’s revenue and cost are modeled by the functions $$f(x)=10-x^2$$ and $$g(x)=2*x$$, where $$

Arithmetic Savings Account

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

Average Concentration in Medical Dosage

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

Average Force Calculation

An object experiences a variable force given by $$F(x)=3*x^2-2*x+1$$ (in Newtons) for $$0 \le x \le

Average 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 Over a Day

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

Cell Phone Battery Consumption

A cell phone’s battery life degrades over time such that the effective battery life each month forms

Consumer Surplus Calculation

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

Consumer Surplus Calculation

The demand and supply for a product are given by $$p_d(x)=20-0.5*x$$ and $$p_s(x)=10+0.2*x$$ respect

Displacement from a Velocity Graph

A moving object has its velocity given as a function of time. A velocity versus time graph is provid

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

Graduated Rent Increase

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

Kinematics with Variable Acceleration

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

Manufacturing Output Increase

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

Piecewise Function Analysis

Consider a piecewise function defined by: $$ f(x)=\begin{cases} 3 & \text{for } 0 \le x < 2, \\ -x+5

Reconstructing Position from Acceleration Data

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

Revenue Optimization via Integration

A company’s revenue is modeled by $$R(t)=1000-50*t+2*t^2$$ (in dollars per hour), where $$t$$ (in ho

Surface Area of a Rotated Curve

Consider the function $$f(x)=\sqrt{x}$$ on the interval $$[0,4]$$. This curve is rotated about the $

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\

Volume by the Disc Method for a Rotated Region

Consider a function $$f(x)$$ that represents the radius (in cm) of a region rotated about the x-axis

Volume of a Solid with Square Cross-Sections

A solid has a base in the xy-plane bounded by $$y=x$$ and $$y=x^2$$ for $$0 \le x \le 1$$. Every cro

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

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