Calculus-Introductory(lecture notes, assginmet, exam review etc)

Calculus-Introductory(lecture notes, assginmet, exam review etc)

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MATH1010: Chapter 2 cont… 1 LIMITS AND DERIVATIVES cont… Calculating Limits Using the Limit Laws (2.3, pg. 104) Recall: Last day, we talked about how to evaluate limits. Example: x x x 5 cos lim 2 0 The Squeeze Theorem: If ) ( ) ( ) ( x h x g x f when x is near a (except possibly at a ) and L x h x f a x a x = = ) ( lim ) ( lim then L x g a x = ) ( lim Returning to our Example : x x x 5 cos lim 2 0
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MATH1010: Chapter 2 cont… 2 Example: + 2 4 0 1 sin 6 lim x x x Example: If 3 ) 2 ( ) ( 3 2 + x x f for all x , find ) ( lim 2 x f x . Example: 3 1 cos lim 3 2 0 x x x
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MATH1010: Chapter 2 cont… 3 Precise Definition of a Limit (Section 2.4 of Stewart, pg. 114) Recall: Last day, we had the following definition for the limit of a function: Definition: We write L x f a x = ) ( lim and say that “the limit of ) ( x f , as x approaches a , equals L if we can make the values of ) ( x f arbitrarily close to L by taking x to be sufficiently close to a (on either side of a ), but not equal to a . Application:
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Unformatted text preview: Suppose that we need to manufacture a metal sphere with volume 700cm 3 . If the maximum allowable tolerance in the volume is 3 cm 3 , then what is the maximum error that we can make in the radius of the sphere? Example: = + = 1 1 , 5 , 7 2 ) ( x x x x f Suppose we want to be within 0.1 away from the value of the limithow close should we get to 1 = x ? Formally, we want to find a small number, which well call , so that < < 1 x guarantees that 1 . 9 ) ( < x f . MATH1010: Chapter 2 cont 4 So, what is the value of that we should use? Now suppose we want to be within 0.01 of the limithow close must we be to 1 = x ? Now, we could keep playing this game, requiring ) ( x f to be even closer to the limit, but the point is for us to be able to do it for any small distance away, say ....
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Calculus-Introductory(lecture notes, assginmet, exam review etc)

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