Section 13.5
Differentials
Estimating change using partial derivatives.
1
Introduction
Recall that approximates the change in .
In 3D, the Total Differential approximates the change in based on changes in both and .
2
The Total Differential
Formula
Where and are small changes in and .
3
Geometric Interpretation
Interactive: Tangent Plane Approximation
represents the change in height on the Tangent Plane (Yellow), while is the actual change on the Surface (Blue).
4
Worked Examples
Example 1: Calculating dz
Find the total differential for .
1. Find .
2. Find .
3. Combine: .
Example 2: Estimating Change
Estimate the change in as changes from to .
Actual change: .
Estimate is very close!
Example 3: Error Propagation
A cone's radius is measured as 10cm and height as 20cm, with max error 0.1cm each. Estimate max error in volume.
. Max error uses differentials with .
Substitute :
Max volume error is .
5
Practice Quiz
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