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MA1012_HW18_201012

# MA1012_HW18_201012 - Mathematics I Homework 18...

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Mathematics I Homework 18: Differentials [email protected] Linear approximation formula: f ( x 0 + Δ x ) = f ( x ) = f ( x 0 ) + ʹஒ f ( x 0 )( x x 0 ) Solved example: Find the approximate value of the square root of 99: 1. Identify the function: f ( x ) = x 2. Identify the value that, when substituted in the function, is easy to calculate. x 0 =100 (because the square root of 100 is 10) 3. The value you are asked to calculate is x , so x = 99 3. Calculate ʹஒ f ( x ) = 1 2 x 1 2 = 1 2 x 4. Substitute back into the linear approximation formula: f (99) = f (100) + ʹஒ f (100)(99 100) , therefore, 99 = 100 + 1 2 100 ( 1) that is, 99 = 10 1 20 99 = 199 20 If you compare this result (approximately 9.95) with the number given by a calculator, you will notice that both are almost the same. Now, try to solve the following exercises. Solutions appear in at the end of this document. 5. Employ differentials to find an approximated value for the volume of a thin spherical shell (like an egg) of external diameter of 200 mm and 1 mm thickness.

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MA1012_HW18_201012 - Mathematics I Homework 18...

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