notes_logs_and_implicit_diff

notes_logs_and_implicit_diff - OPMT 5701 Lecture Notes 1...

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1 OPMT 5701 Lecture Notes 1.1 Natural Logarithm and the Exponential e 1. The Number e if y = e x then dy dx = e x if y = e f ( x ) then dy dx = e f ( x ) · f 0 ( x ) 2. Examples 1. (a) y = e 3 x dy dx = e 3 x (3) (b) y = e 7 x 3 dy dx = e 7 x 3 (21 x 2 ) (c) y = e rt dy dt = re rt 2. Logarithm (Natural log) ln x (a) Rules of natural log If Then y = AB ln y =ln( AB )=ln A +ln B y = A/B ln y =ln A ln B y = A b ln y A b )= b ln A NOTE: ln( A + B ) 6 A B EVER!!! (b) derivatives IF THEN y x dy dx = 1 x y f ( x )) dy dx = 1 f ( x ) · f 0 ( x ) (c) Examples i. y =l n ( x 2 2 x ) dy/dx = 1 ( x 2 2 x ) (2 x 2) ii. y n ( x 1 / 2 1 2 ln x dy/dx = μ 1 2 ¶μ 1 x = 1 2 x 1
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1.2 Di f erentials Given the function y = f ( x ) the derivative is dy dx = f 0 ( x ) However, we can treat dy/dx as a fraction and factor out the dx dy = f 0 ( x ) dx where dy and dx are called di f erentials .I f dy/dx can be interpreted as ”the slope of a function”, then dy is the ”rise” and dx is the ”run”. Another way of looking at it is as follows: dy = the change in y dx = the change in x f 0 ( x ) = how the change in x causes a change in y Example 1 if y = x 2 then dy =2 xdx Lets suppose x and dx =0 . 01 . What is the change in y ( dy ) ?
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notes_logs_and_implicit_diff - OPMT 5701 Lecture Notes 1...

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