If r denotes the reaction of the body to some stimulus of...

Name: 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 44408
Uploaded: 2022-04-29T13:03:33-08:00
Duration: 4 min 37 s
Channel: Kumareshwaran Rathinasabapathy
Description: 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 44408

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=(39+18x^5)/(1+0.5x^5) 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 =

Transcript

00:01 We now determine the sensitivity which is defined to be the rate of change of reaction r with respect to x.

00:08 That is s equals dr by dx.

00:14 So we differentiate this r with respect to x.

00:18 D differentiating r with respect to x, we get dr over dx.

00:23 And this equals on this right side we have this quotient.

00:27 We have to use the quotient rule of derivatives to find.

00:31 Derivative.

00:32 So first we put the denominator expression that is 1 plus 0 .3 x power 6 and this has to multiply with the derivative of 43 plus 17 x power 6 which is the derivative of 43 is 0 plus the derivative of 17 x power 6 is 17 times the derivative of x per 6 is 6 x power 5 when we use the power rule of derivatives.

00:58 We then put minus.

01:00 Now we put the numerator expression that is 43 plus 17 x power 6 and this has to be multiplied with the derivative of the denominator expression that is the derivative of 1 is 0 plus the derivative of 0 .3 x .6 is 0 .3 times the derivative of x power 6 which is 6 x .5 using the power rule and this has to be divided by the square of the denomination expression that is 1 plus 0 .3 x power 6 this is quantity square.

01:38 Now we want to find the stimulus sensitivity at x equal to 2.

01:44 So basically we find the dr by d x at x equals 2.

01:52 So let's do that in fact before that we can simplify this before we apply x equal to 2.

01:59 So let's simplify this.

02:01 And so this equals 1 plus 0 .3 x power 6 times this is 17 times of 6, which equals 1002 and x power 5 minus we then have 43 plus 17 x power 6.

02:26 And this is multiplied with 0 .3 times 6 is 1 .8.

02:32 So it is 1 .8 x power 5...