Find the linearization of at x = 1.
Use a linear approximation to estimate e­0.015.
Last time we looked at how to approximate function values using the idea of a linear approximation.
Today we will focus on the derivative and small changes.
We have a function, f(x ).
We can predict the ACTUAL change in Δy
Δx
We may not always have the original function, but we can measure rates of change.
dy
dx
y.
In Section 3.7 we discussed the Marginal Cost:
3000
C(101) C(100)
0 120
C'(100)
Fudge balls are coated with chocolate 2mm thick.
If the fudge balls are 3cm in diameter, then use the differential to estimate the amount of chocolate needed to coat one fudge ball.
From an algebra viewpoint, finding the differential is easy:
Error Estimates
Every measurement has error.
If we use a measurement in a calculation of volume or area, then the error is transmitted to the next calculation.
Propagation of Error
dx = dy = Ball bearings must be 0.7 cm in radius. They must be manufactured to within 0.01 cm.
How will volume be affected if there is error in the radius?
Relative error in the radius:
Use the differential to estimate the value of Relative error in volume:
1
4.002