Derivatives and Applications Lesson

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Derivatives and Applications

by Dr. G. Garrido

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Applications of the Derivatives

Case#1 Maximum Air Speed During Cough
Case #2 Maximizing Profit and Minimizing Cost

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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

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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]

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2. Step: Set factored Derivatives =0

ar((2r0−3r))]=0

Solution:( r=0 and (2/3)r0)

ar\left(2r_0-3r\right)=0

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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

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Conclusion

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

Derivatives and Applications

by Dr. G. Garrido

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