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Establish the following rules for working with di…

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Problem 40 Hard Difficulty

When blood flows along a blood vessel, the flux $ F $ (the volume of blood per unit time that flows past a given point) is proportional to the fourth power of the radius $ R $ of the blood vessel:
$ F = kR^4 $
(This is known as Poiseuille's Law; we will show why it is true in Section 8.4.) A partially clogged artery can be expanded by an operation called angioplasty, in which a balloon-tipped catheter in inflated inside the artery in order to widen it and restore the normal blood flow.
Show that the relative change in $ F $ is about four times the relative changes in $ R. $ How will a $ 5\% $ increase in the radius affect the flow of blood?


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Related Courses

Calculus 1 / AB

Calculus: Early Transcendentals

Chapter 3

Differentiation Rules

Section 10

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Derivatives

Differentiation

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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.

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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.

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Video Transcript

okay, we know that if at this k r to the fourth and we know that d over d R before kay are cute, Therefore Delta after is four k are few times Delta are Therefore don't ask over K art of the fourth is four k are few times Delta Are you okay? Are before Therefore we have delta half over half is four times delta are over our remember The relative error is four times the relative Aaron are that gives us f Therefore, we know that we would have a percentage error for our 5% there for bath would be five times for which would give us 20%.

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Top Calculus 1 / AB Educators
Grace He

Numerade Educator

Anna Marie Vagnozzi

Campbell University

Kayleah Tsai

Harvey Mudd College

Kristen Karbon

University of Michigan - Ann Arbor

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
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