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 reation with respect to x.     A particular example is that...

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 reation 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= (39+16x^5)/(1+0.5x^5)

can be used to model the dependance of R on x when R is measured in square millimeters and x is measured in appropriate units of brightness. Find the sensitivity corresponding to x=2

Asked on by veteran25

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to find the derivative of the function `R(x)`  with respect to x, using quotient rule, such that:

`R'(x) = ((39+16x^5)'*(1+0.5x^5) - (39+16x^5)*(1+0.5x^5)')/((1+0.5x^5)^2)`

`R'(x) = (80x^4*(1+0.5x^5) - 2.5x^4*(39+16x^5))/((1+0.5x^5)^2)`

`R'(x) = (80x^4 + 40x^9 - 97.5x^4 - 40x^9)/((1+0.5x^5)^2)`

`R'(x) = (-17.5x^4)/((1+0.5x^5)^2)`

You need to evaluate the sensitivity at  x=2, hence you need to substitute 2 for x in `R'(x)`  such that:

`R'(2) = (-17.5*2^4)/((1+0.5*2^5)^2)`

`R'(2) = -280/289 = -0.968`

Hence, evaluating the sensitivity at x=2 under given conditions yields `R'(2) = -0.968` .

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