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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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Calculus 1 / AB
Chapter 3
Differentiation Rules
Section 10
Linear Approximation and Differentials
Derivatives
Differentiation
Campbell University
Harvey Mudd College
University of Michigan - Ann Arbor
Lectures
04:40
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.
44:57
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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