The piecewise function is differentiable on (0, 1) and satisfies f(0) = f(1). Ho

Question

The piecewise function is differentiable on (0, 1) and satisfies f(0) = f(1). However, its derivative is never zero on (0, 1). Does this contradict Rolle's Theorem? Explain. Coughing forces the trachea (windpipe) to contract, which affects the velocity v(r) of air passing through the trachea. The velocity of the air during coughing is v(r) = c(R - r)r^2, 0 r R where c is a constant, R is the normal radius of the trachea, and r is the radius during coughing. Find the radius that maximises the velocity of air through the windpipe. Does this mean that the windpipe expands or contracts when coughing?

Explanation / Answer

a) function is not continous at x=0

limx->0+f(x)=limx->0+1-x=1-0=1

f(0)=0.

rolls theorem cannot be applied as f not continous on[0,1]

so no contradiction

b)v=c(R-r)r2

v=c(Rr2-r3)

velocity is maximum when dv/dr=0

dv/dr=c(2Rr-3r2)=0

==>cr(2R-3r)=0

r cannot be zero

(2R-3r)=0

r=2R/3

radius r=2R/3 maximises velocity of air

windpipe contracts by 1/3 of its radius

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