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Calculus help - Find the sensitivity?
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= (35+16x^4)/(1+0.5x^4)
can be 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.
Find the sensitivity corresponding to x=2
Sensitivity=?????
1 Answer
- Meng tianLv 71 decade agoFavorite Answer
In short, the sensitivity is the derivative of R.
R= (35 + 16x^4) / (1 + 0.5x^4)
Using the quotient rule, we have:
R'
= [(1 + 0.5x^4)(64x^3) - (35 + 16x^4)(2x^3)] / (1 + 0.5x^4)^2
= [64x^3 + 32x^7 - 70x^3 - 32x^7] / (1 + 0.5x^4)^2
= [-6x^3] / (1 + 0.5x^4)^2
When x = 2, we have:
sensitivity
= R'(2)
= [-6(2)^3] / (1 + 0.5(2^4))^2
= -48 / 9^2
= -48 / 81
= -16/27
