calc - Outline Calculus Review Oct 7, 2010 Calculus Review...

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Outline Calculus Review Oct 7, 2010 Calculus Review
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Outline Outline 1 Limits 2 Differentiation 3 Integration Calculus Review
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Limits Differentiation Integration Outline 1 Limits 2 Differentiation 3 Integration Calculus Review
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Limits Differentiation Integration Calculus begins with the notion of a limit We care about what happens to a function as we approach some point Sometimes this is really difficult Sometimes it depends on which direction you approach the limit from In this class, we’ll deal with relatively simple cases We’ll typically figure out the limit by plugging the number into the function lim x 2 1 / x Sometimes we can’t plug the number in ( ) but we can look at what happens as x gets bigger and bigger and notice a trend lim x →∞ 1 / x Calculus Review
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Limits Differentiation Integration 1 lim x 0 1 / x 2 lim x →- 1 2 x 2 + 3 3 lim x 1 2 x 2 + 3 4 lim x →∞ 2 x 2 + 3 5 lim x →-∞ 2 x 2 + 3 Calculus Review
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Limits Differentiation Integration Other times, it won’t be clear how to calculate the limit of a function lim x →∞ x 2 + x x One trick is to rewrite the function so you can compute the limit lim x →∞ x + 1 Other times, this trick won’t work either. We’ll introduce other tricks as we need them. Calculus Review
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Limits Differentiation Integration Using Derivatives Outline 1 Limits 2 Differentiation 3 Integration Calculus Review
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Limits Differentiation Integration Using Derivatives Differentiation is all about getting the slope of a line Remember the standard equation for a line: y = mx + b m = rise run = y ( x 1 ) - y ( x 2 ) x 1 - x 2 is the slope Great, the derivative of a straight line is m Notice the derivative (slope) is constant. It doesn’t matter what two points we look at. Calculus Review
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Limits Differentiation Integration Using Derivatives Most functions don’t have constant slope (say, f ( x ) = 3 x 2 + 2) How do we talk about the slope of this line? We have to decide at what point we want to measure the slope Average slope between x and x + h : f ( x + h ) - f ( x ) h derivative (slope) at point x : lim h 0 f ( x + h ) - f ( x ) h Instantaneous slope (derivative) at some point x (more correctly, the slope of the tangent line at point x) We’ll use the notation
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calc - Outline Calculus Review Oct 7, 2010 Calculus Review...

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