Section 4.12

Differentials

How does a tiny error in measurement explode into a larger error in calculation? Differentials allow us to propagate errors and predict changes without re-calculating everything.

1

The Notation of Change

Definitions

  • dx
    The independent variable change (same as ).
  • Δy
    The Actual Change in the function.
  • dy
    The Differential (Linear Change).

Calculus treats fractions like as separable parts!

2

Worked Example: Area Error

Circle Measurement

The radius of a circle is measured as with a possible error of . Estimate the max error in the Area.

Differential Scanner

dy (Tangent): 1.000Δy (Actual): 1.125
dy is the change along the line. Δy is the change along the curve. The difference is the error (Red).

For small dx, dy is a great approximation of Δy.

1. Formula
2. Differentiate
3. Plug in Values

Error .

3

Level Up Examples

Example A: Cube Root

Use differentials to estimate the change in when changes from 8 to 7.9.

1. Identify Parts:
. .
2. Differential dy:
.
.
3. Result:

The value decreases by roughly 0.0083.

Example B: Relative Error

If radius error is 2%, what is Area error?

1. Setup:
.
2. Form Ratio dA/A:
.
3. Conclusion:

(2%).
(4%).

Area error is double the radius error.

5

Practice Quiz

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