ABE425 Engineering Measurement Systems ABE425 Engineering Measurement Systems Math Refresher Dr. Tony E. Grift Dept. of Agricultural & Biological Engineering University of Illinois

Differentiation Example:

Integration Dummy variable integrates out (you put in the limits). Lower limit (write x=..) Upper limit (write x=..)

Product rule. Examples: Prime notation:

Quotient rule. Examples: Prime notation:

If a function is a function of another function you have to use the chain rule. Examples: Prime notation:

When you need to integrate the product of two functions, you need to integrate by parts. Product rule Example: Yellow frame: choice or definition

8 Complex numbers in “standard” notation.

9 To add or subtract complex numbers use the “standard” notation. Addition Subtraction

10 Multiplication To multiply or divide complex numbers do NOT use standard notation. Division

11 To multiply or divide complex numbers use Euler’s notation instead. Multiplication Division

12 Multiplying a complex number by its complex conjugate gives a real number. Complex conjugate Standard notation Euler’s notation Standard notation Euler’s notation

13 Taylor and MacLaurin series expansions are used extensively in math proofs. Taylor series Special case: MacLaurin series

14 Example of MacLaurin expansion (most important one in fact):

15 Express cos x in a MacLaurin series

16 Express sin x in a MacLaurin series

17 Add them up while putting the complex operator before the sine

18 Answers Euler’s formula

19 Sine and cosine in complex exponential (Euler’s) notation Adding and subtracting give:

The complex form of sine and cosine can make calculations a lot easier.

Differentiation and integration: Remember these! Prime notation

22 Euler’s formula Relationship sin/cos with complex numbers Complex conjugate Taylor series MacLaurin series

ABE425 Engineering Measurement Systems ABE425 Engineering Measurement Systems Math Refresher Dr. Tony E. Grift The End Dept. of Agricultural & Biological Engineering University of Illinois