2.6 Instantaneous Rates of Change

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We have established a graphical interpretation of the derivative as the slope of a tangent line. Now, we want to understand this notion in connection to other subjects and applications.

s

m

s

m

tt

yyM 2

12

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Consider dimensions: Suppose that function y=f(t) gives a distance in meters, and its argument, t, denotes time in seconds. Chose first a linear dependence f(t), say, f(t)=2t+1. This is a line, and its slope is

The slope M is just the velocity of the object. More generally, the slope is the rate of change of the function with respect to the argument. The derivative defines the slope in this and more complex cases, but since the slope may change, we need to define the derivative as instantaneous rate of change.

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dt

dsv

Velocity: If s denotes distance and t denotes time, then

Example:Describe the motion of a particle moving in the vertical direction according to the equation

Solution: We find the velocity first:

Next, construct a time-table, i.e., estimate values of s and v at sample points... At what instants the particle moves up? Down?At what instants it turns around (solve v=0)?

24 tts

tdt

dsv 24

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dt

dva

Acceleration: If v denotes velocity and t denotes time, then

Example (cntd):We have found Acceleration now is

meaning that the velocity decreases by 2 m/s every second.

.24 tv

,22

s

m

dt

dva

Note:Speed is the absolute value of the velocity.

Consequence: while velocity increases, given a positive acceleration, the speed may decrease.

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Other rates of change:

Current in a circuit:

Voltage across an inductor:

Current to a capacitor: (as a der. of q=Cv)

dt

diLv

dt

dqi

dt

dvCi

Example:Find the current to a capacitor of 0.002 F after 2 seconds if the voltage is given by

Solution:

At t=2,

.2010 2 ttv

)1(2.02021001.0)2010(

01.02

ttdt

ttd

dt

dvCi

A. 2.0)12(2.0 i

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Example: Consider a blood vessel in the shape of a cylindrical tube of radius R and length l. Because of the friction at the walls, the velocity of the blood is greatest in the center of the cylinder. The relationship between the velocity and the radius is given by the law of laminar flow (Poiseuille, 1840):

where, is the viscosity, and P is the pressure difference. Find the instantaneous rate of change of the velocity with

respect to r (velocity gradient) at =0.02, R=0.01cm, r=0.005cm, l=1cm, and P=4000dynes/[cmcm].

),(4

)( 22 rRl

Prv

J. Stewart, Calculus

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HomeworkSection 2.6: 7,9,15,17,19,23,25,27.