4 8 differentials
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4.8 - Differentials. Linear (or Tangent Line) Approximations. For values close to a , . Linear Approximation – Examples. Determine the linearization (another name for linear approximation) of f ( x ) = ln x at a = 1. Find the linear approximation of the function .

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4.8 - Differentials

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4 8 differentials

4.8 - Differentials


4 8 differentials

Linear (or Tangent Line) Approximations

For values close to a,


4 8 differentials

Linear Approximation – Examples

Determine the linearization (another name for linear approximation) of f (x) = ln x at a = 1.

Find the linear approximation of the function

And use it to approximate the numbers

and


4 8 differentials

Differentials

Up to now, we’ve thought of dy/dx as notation for a derivative. We can think of dx and dy as separate quantities called differentials.


4 8 differentials

Differentials

We can now think of dy/ dx as a ratio of two quantities (the differential of y and the differential of x). So for a given change in x (dx) we can calculate a change in y (dy).


4 8 differentials

Differentials – Example 1

Find the differential dy and evaluate dy for the given values of x and dx.


4 8 differentials

Differentials – Example 2

You may have used this concept when calculating errors in measurements or calculations.

The radius of a circular disk is given as 24 cm with a maximum error in measurement of 0.2 cm. (a) Use differentials to estimate the maximum error in the calculated area of the disk. (b) What is the relative errors (dA / A)


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