Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Sent to:
Search glass icon
  • Login
  • Textbooks
  • Ask our Educators
  • Study Tools
    Study Groups Bootcamps Quizzes AI Tutor iOS Student App Android Student App StudyParty
  • For Educators
    Become an educator Educator app for iPad Our educators
  • For Schools

Problem

Patients undergo dialysis treatment to remove ure…

02:37

Question

Answered step-by-step

Problem 34 Hard Difficulty

It $ R $ denotes the reaction of the body to some stimulus of strength $ x. $ the sensitivity $ S $ is defined to be the rate of change of the reaction with respect to $ x. $ A particular example is that when the brightness $ x $ of a light source is increased, the eye reacts by decreasing the area $ R $ of the pupil. The experimental formula
$ R = \frac {40 + 24x^{0.4}}{1 + 4x^{0.4}} $
has been used to model the dependence of $ R $ on $ x $ when $ R $ is measured in square millimeters and $ x $ is measured in appropriate units of brightness,
(a) Find the sensitivity.
(b) Illustrate part (a) by graphing both $ R $ and $ S $ as functions of $ x. $ Comment on the values of $ R $ and $ S $ at low levels of brightness. Is this what you would expect?


Video Answer

Solved by verified expert

preview
Numerade Logo

This problem has been solved!

Try Numerade free for 7 days

Heather Zimmers
Numerade Educator

Like

Report

Textbook Answer

Official textbook answer

Video by Heather Zimmers

Numerade Educator

This textbook answer is only visible when subscribed! Please subscribe to view the answer

More Answers

01:12

Amrita Bhasin

Related Courses

Calculus 1 / AB

Calculus: Early Transcendentals

Chapter 3

Differentiation Rules

Section 7

Rates of Change in the Natural and Social Sciences

Related Topics

Derivatives

Differentiation

Discussion

You must be signed in to discuss.
Top Calculus 1 / AB Educators
Anna Marie Vagnozzi

Campbell University

Caleb Elmore

Baylor University

Kristen Karbon

University of Michigan - Ann Arbor

Joseph Lentino

Boston College

Calculus 1 / AB Courses

Lectures

Video Thumbnail

04:40

Derivatives - Intro

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

Video Thumbnail

44:57

Differentiation Rules - Overview

In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.

Join Course
Recommended Videos

02:21

Sensitivity of the eye to …

04:29

If $R$ denotes the reactio…

00:53

Pupil Size When the bright…

03:51

Pupil size When the bright…

05:34

Pupil Size When the bright…

04:01

Pupil Size When the bright…

02:53

Area of the Pupil The rela…

01:51

The relationship between t…

Watch More Solved Questions in Chapter 3

Problem 1
Problem 2
Problem 3
Problem 4
Problem 5
Problem 6
Problem 7
Problem 8
Problem 9
Problem 10
Problem 11
Problem 12
Problem 13
Problem 14
Problem 15
Problem 16
Problem 17
Problem 18
Problem 19
Problem 20
Problem 21
Problem 22
Problem 23
Problem 24
Problem 25
Problem 26
Problem 27
Problem 28
Problem 29
Problem 30
Problem 31
Problem 32
Problem 33
Problem 34
Problem 35
Problem 36
Problem 37
Problem 38
Problem 39

Video Transcript

in this problem were given an equation which represents the area of the pupil of the eye. And we're looking at the sensitivity to light as the brightness increases. And the way we find the sensitivity is by taking the derivative of the area of the Radius. And so we're going to use the quotient rule to get this derivative since we have a quotient. So here we have the bottom times, the derivative of the top, minus the top times, the derivative of the bottom, over the bottom squared. And now we can simplify that. So what we do is we distribute the 9.6 x to the negative 0.6 power, and we distribute the 1.6 X to the negative 0.6 power and and the next step, we noticed that we can cancel these opposite terms, and then we can add these, like, terms together. And so, for the derivative, we end up with negative 54.4 over X to the 0.6 power times 1.1 plus for two times X to the 0.4 squared. That's a mouthful. Okay, so now what we want to do is use a graphing calculator and look at the graph of R and look at the graph of s together. So I take her calculator and we type both functions in, and then we need to change the window dimensions so that we get a good view. And so I decided to let my ex values go up to 20 and my wife values go from negative 20 to 20. But you could mess around with those numbers and find something that you like. Okay, looking at the graph now, the blue one represents our that was the area of the pupil, and the red one represents s That was the rate of change of the area of the pupil. And that's the sensitivity on one thing we can notice for small values of X that would be values where the amount of light is low as exchanges. There's a more rapid change in the area of the pupil. You can see that the area function is decreasing more rapidly at first, so there's a quick change in the area. The pupil, when you're going from dark to light, there is a much more slow change in the area of the pupil when you're going from something that's already bright to something that's a little bit brighter and similarly in the rate of change of the area, the rate of change happens much more quickly at first as we're going from dark to light and then the rate of change slows as we're going from something that's already light to a little bit more bright.

Get More Help with this Textbook
James Stewart

Calculus: Early Transcendentals

View More Answers From This Book

Find Another Textbook

Study Groups
Study with other students and unlock Numerade solutions for free.
Math (Geometry, Algebra I and II) with Nancy
Arrow icon
Participants icon
128
Hosted by: Ay?Enur Çal???R
Math (Algebra 2 & AP Calculus AB) with Yovanny
Arrow icon
Participants icon
62
Hosted by: Alonso M
See More

Related Topics

Derivatives

Differentiation

Top Calculus 1 / AB Educators
Anna Marie Vagnozzi

Campbell University

Caleb Elmore

Baylor University

Kristen Karbon

University of Michigan - Ann Arbor

Joseph Lentino

Boston College

Calculus 1 / AB Courses

Lectures

Video Thumbnail

04:40

Derivatives - Intro

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

Video Thumbnail

44:57

Differentiation Rules - Overview

In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.

Join Course
Recommended Videos

02:21

Sensitivity of the eye to brightness If $R$ denotes the reaction of the body to…

04:29

If $R$ denotes the reaction of the body to some stimulus of strength $x,$ the …

00:53

Pupil Size When the brightness x of a light source is increased, the eye reacts…

03:51

Pupil size When the brightness $x$ of a light source is increased, the eye reac…

05:34

Pupil Size When the brightness $x$ of a light source is increased, the eye rea…

04:01

Pupil Size When the brightness $x$ of a light source is increased, the eye reac…

02:53

Area of the Pupil The relationship between the area of the pupil of the eye and…

01:51

The relationship between the area of the pupil of the eye and the intensity of …

Add To Playlist

Hmmm, doesn't seem like you have any playlists. Please add your first playlist.

Create a New Playlist

`

Share Question

Copy Link

OR

Enter Friends' Emails

Report Question

Get 24/7 study help with our app

 

Available on iOS and Android

About
  • Our Story
  • Careers
  • Our Educators
  • Numerade Blog
Browse
  • Bootcamps
  • Books
  • Notes & Exams NEW
  • Topics
  • Test Prep
  • Ask Directory
  • Online Tutors
  • Tutors Near Me
Support
  • Help
  • Privacy Policy
  • Terms of Service
Get started