Derivatives and Applications

Derivatives and Applications

by Dr. G. Garrido

Applications of the Derivatives

• Case#1 Maximum Air Speed During Cough

• Case #2 Maximizing Profit and Minimizing Cost

Maximum Air Speed During Cough

Consider the Following Function:

 $S\left(r\right)=ar^2\left(r_0-r\right)\ \ Where\ a>0\$  
r=Radius of Trachea(windpipe)
S=Speedy
The function relates Speedy and Radius during a Cough

First Step: Differentiate S(r)

Assuming an interval for the radius :
0< r< r0
S'(r)=-ar2

 $S'\left(r\right)=-ar^2+\left(r_0-r\right)\left(2ar\right)=ar\left[-r+2\left(r_{0_{ }}-r\right)=ar\left(\left(2r_0-3r\right)\right)\right]$  

2. Step: Set factored Derivatives =0

ar((2r0−3r))]=0

  Solution:( r=0 and (2/3)r0)

 $ar\left(2r_0-3r\right)=0$  

3rd. Step

Since 0
Compute S(0) =0
and 

 $S\left(\frac{2}{3}r_0\right)=\frac{4a}{27}\left(r_0\right)^3$  
 $S\left(r_0\right)=0$  

Conclusion

Compare the values and conclude that the speedy of the air is greatestwhen the radius of the trachea is (2/3)r₀