Maths for Engineers and Scientists 2

Maths for Engineers and Scientists 2

Linear approximation, small variations and errors

• Maclaurin series
• Taylor series
• +
• Example 1. Find the Maclaurin series for up to and including the terms.

•  

Using the Taylor series only up to the term involving is called “linear approximation”. It gives a quick (but sometimes quite rough) way to approximate functions when is near the point .

• Using the Taylor series only up to the term involving is called “linear approximation”. It gives a quick (but sometimes quite rough) way to approximate functions when is near the point .
• Linear approximation when ( is approximately or close to the value )
• .
• Example 2. Find the linear approximation of near .
• Example 3.Find the linear approximation of near to .

•  

Example 4. The Lorentz factor , where m/s is the speed of light and is the speed of the object under consideration, is used in special relativity to calculate time dilation. Evaluate and and simplify the results. Use the Taylor expansion to work out the linear approximation of valid for speeds about .

• Example 4. The Lorentz factor , where m/s is the speed of light and is the speed of the object under consideration, is used in special relativity to calculate time dilation. Evaluate and and simplify the results. Use the Taylor expansion to work out the linear approximation of valid for speeds about .

•  

Small variations and errors

• We can use the linear approximation above in another way. Keeping the value close to , we write the change or error in and the change or error in as

•  

Example 5. The radius of a (spherical) star was observed to increase by from metres. Use linear approximation to estimate the percentage its volume increases by.

• Example 5. The radius of a (spherical) star was observed to increase by from metres. Use linear approximation to estimate the percentage its volume increases by.
• Example 6. We want to drill a circular hole in a sensitive piece of equipment. The cross
• sectional area of the hole has to be accurate to better than . Estimate the accuracy we require in the radius of the drill bit.

•