Doppler Effect calc. problem: Find rate of change of F with respect to v, help please?

Question

Frequency F of a fire truck siren heard by stationary observer is F = (132, 400)/(331 +or- v) where +or- v is the velocity of the truck. Find the rate of change of F with respect to v when:

when the truck is approaching at teh velocity of 30m/s (use -v)
and when truck is moving away at a velocity of 30m/s (use +v)

Thank you!

Answer 1

Use the quotient rule for differentiation. Taking the case where the firetruck is approaching:

The numerator of the derivative has the denominator of the original function times the derivative of the numerator minus the numerator of the original function times the derivative of the denominator.

This is pretty easy, since the numerator of the original function is a constant. The derivative of a constant is 0. When approaching, you only have to worry about the original numerator times the derivative of (331 - v). The 331 is a constant with a derivative of 0 leaving you the derivative of -v. That's -1.

Therefore, the numerator of your derivative is -132400 * -1, or 132400 when approaching.

The denominator of the derivative is just the square of the original denominator, or (331 - v)^2 when approaching. When approaching, that would give you 301^2 (301 squared), or 90601.

When approaching, the rate of change would be 132,400/(301^2), or about +1.46.

Finding the rate of change when moving away works the same, but with a +v instead of a -v.