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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

If 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 = 40 + 24 x 0.4 1.4 x 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?

(a)

To determine

To find: The sensitivity of the body.

Explanation

Given:

The experimental formula is shown below.

R=40+24x0.41+4x0.4 (1)

Calculation:

Calculate the sensitivity S.

S=dRdx

Sensitivity is defined to be the rate of change of the reaction with respect to x.

Substitute 40+24x0.41+4x0.4 for R in the above equation.

S=ddx(40+24x0.41+4x0.4)

Apply the quotient rule below.

(uv)'=u'vv'uv2

Substitute (40+24x0.4) for u and (1+4x0.4) for v in the above equation.

S=(40+24x0.4)'(1+4x0.4)(1+4x0.4)'(40+24x0.4)(1+4x0.4)2=9

(b)

To determine

To illustrate: The part (a) by graphing both R and S as functions of x and comment on the values of R and S at low levels of brightness.

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