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Solution to homework problems on differentials
- Use differentials to estimate
25 .1 and ln(1.03).
The linear approximation to y at a is
y ∼ y|a + dy = y|a + dy dx
|adx.
In the first case use y =
x, a = 25, and dx = 0.1. Then
y| 25 = 5 dy dx
x−^1 /^2 dy dx |a =
and y ∼ 5 +
In the second case use y = ln(x), a = 1, and dx = 0.03. Then
y| 1 = 0 dy dx
x dy dx
and y ∼ 0 + 1(0.03) = 0. 03.
- The radius of a sphere was measired as 100 cm with possible error of ±.5 cm.
a) Use differentials to estimate the possible error in the calculated volume and the cal- culated surface area.
We have V =
πr^3
so dV dr = 4πr^2.
The estimated possible area is dV = 4πr^2 dr. In this case dr = ± 0. 5 cm so
dV = ± 4 π(100)^2 (0.5)
cubic centimeters.
For the area, A = 4πr^2
so dA dr
= 8πr.
The estimated area is dA = 8πrdr
or dA = ± 8 π(100)(0.5)
square centimeters.
b) For each, estimate the possible percentage error. Using logarithmic differentiation we have dV V
dr r
and dA A
dr r
Since the possible error in r is ± 0 .5%, the estimated percentage error in V is ± 1 .5% and the estimated percentage error in A is ±1%.
In the second part, we’ve used the idea that df /f = 0.02 corresonds to an estimated change in f of 2 percent. More precisely, it corresponds to a 2 percent change in the linear approximation to f. Let me remind you why that is. The linear approximation is
T 1 = f (a) + df = f (a) + f ′(a)dx.
If df /f (a) = 0. 02 , then df = 0. 02 f (a), and so
T 1 = f (a) + 0. 02 f (a),
which is to say T 1 = 1. 02 f (a),
as promised.
- Poiseuille’s law states that flux F (i.e. the volume of blood which flows past a point per unit time) is proportional to the fourth power of the radius R of the blood vessel (i.e. F = kR^4 ). If there is a 5% decrease in the radius of the blood vessel, use differentials to estimate the percentage decrease in the flux.
We have dF F
dR R so a 5% decrease in R leads to an estimated 20% decrease in F.
