microbook_3e-page19

microbook_3e-page19 - ' to denote a change and start using...

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columns because we are looking at “large” changes in x as opposed to the infinitesimally small changes described in the notes entitled: “What’s the Difference between Marginal Cost and Average Cost?” Why does it make a difference whether we look at small or large changes? Consider the following derivation of the slope of +, x f : +, +, +, + , +, +, +, +, +, +, +, +, +, x 3 x 6 x f x x 3 x x x 6 x x 3 x x 6 x x 3 x 3 x x 6 x 3 x x 3 x x x 2 x 3 x x 3 x x x x 3 x x 3 x x 3 x x f x x f x x f x f 2 2 2 2 2 2 2 2 2 2 2 ' . c ' ' . ' ' ' ' . ' ' 0 ' . ' . ' 0 ' . ' . ' 0 ' . ¡ ' . ' 0 ' . ' 0 ' . ' ' c If we look at one unit changes in the value of x – i.e. 1 x ' – then the slope of +, x f evaluated at each value of x is equal to x 3 x 6 ' . which equals 3 x 6 . since 1 x ' . If we look at changes in x that are so small that the changes are approximately zero – i.e.: 0 x | ' – then the slope of +, x f evaluated at each value of x is approximately equal to x 6 and gets closer and closer to x 6 as the change in x goes to zero. So if +, 2 x 3 x f , then +, x 6 x f c . Since we’ll be looking at infinitesimally small changes in x , we’ll stop using the symbol
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Unformatted text preview: ' to denote a change and start using the letter d to denote an infinitesimally small change. calculus tricks an easy way to find derivatives For the purposes of this course, there are only three calculus rules youll need to know: x the constant-function rule x the power-function rule and x the sum-difference rule. the constant-function rule If + , 3 x f , then the value of + , x f doesnt change x as changes i.e. + , x f is constant and equal to 3. So whats the slope? Zero. Why? Because a change in the value of x doesnt change the value of + , x f . In other words, the change in the value of + , x f is zero. So if + , 3 x f , then + , + , x f x d x f d c . Page 19...
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