Differentials and Error Approximation



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Differentials and Error Approximation

  • Approximate using differentials.

  • Compare

  • Estimate propagated error using differentials.

  • Find differentials of a function.

Definition of Differentials

Let represent a differentiable function on an open interval containing x. The differential of x (denoted dx) is any nonzero real number. The differential of y (denoted dy) is


In many types of science and engineering applications that we will study dy is used to approximate When we use this approximation, with physical measuring devices a certain amount of error (called ‘propagated error’) will occur.

Measurement Error



Propagated error/ Exact Value Measured Value

Error in measurement


Tangent Line Approximation:

Example 1: Approximate










Example 2: Approximate







Approximation Using Differentials:

Example 1: Approximate







Example 2: Approximate












Now let’s look at a graph to get an understanding of the relationship between

In the grid provided, carefully graph Next, carefully graph the tangent line to the curve through

Our focus will be from

Label the following on your graph.




Classwork Examples: Page 236, 8, 10, 14, 20, 22, 28, 30, 36

Evaluate and compare




8.

10.




Find the differential dy of the function.




14.

20.





22. Use differentials and the graph of f to approximate (a) and (b) .

(a)
(b)


28. The measurement of the circumference of a circle is found to be 64 centimeters with a possible error of 0.9 centimeters.

(a) Approximate the percent error in computing the area of the circle.

(b) Estimate the maximum allowable percent error in measuring the circumference if the error in computing the area cannot exceed 3%.





30. The radius of a spherical balloon is measured as 8 inches, with a possible error of 0.02 inch.

(a) Use differentials to approximate the possible propagated error in computing the volume of the sphere.


(b) Use differentials to approximate the possible propagated error in computing the surface area of the sphere.



36. A surveyor standing 50 feet from the base of a large tree measures the angle of elevation to the top of the tree as How accurately must the angle be measured if the percent error in estimating the height of the tree is to be less than 6%?



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