Midterm-Exam Aid

# Midterm-Exam Aid - MATH138 Midterm Exam-AID SOS Prepared by...

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MATH138: Midterm Exam-AID SOS Prepared by Vincent Chan [email protected] February 21, 2010 I include numbers after each section heading that correspond to the textbook section headings for your convenience. Proofs of theorems are included in your textbook, and will be omitted. Finally, I strongly recommend going through the examples and exercises in your textbook, as they are very helpful in prepar- ing for your exams and understanding the material. The material that follows will cover the most dif±cult material that is presented in Math 138, while assuming knowledge from Math 137 (such as basic integration and areas). 1 Review of Integration (5.1 - 5.5) Recall: if a function ° is de±ned on an interval ° ±²³ ± , we divide the interval into ´ subintervals of equal length ² µ ³´ ³ ° ± µ ¶´ . Letting ± ³ µ ° ²µ ± ²···²µ ° ³ ³ be the endpoints of these intervals and µ ° ± ± ° µ ± ± ± ± ± , we de±ne the defnite integral oF ° From ± to ³ as ° ² ³ ° ´ µ µ ¸µ ³¶ ·¸ ° ²³ ° ± ± ²± ° ´ µ ° ± µ² µ provided this limit exists. If so, we say ° is integrable on ° ± . The integral can be thought of as the area under the graph ° , if ° is non-negative. Some properties of integration: (i) Linearity: ° ² ³ ¹° ´ µ µ¹ º ´ µ µ ¸µ ³ ¹ ° ² ³ ° ´ µ µ ¸µ ¹ ° ² ³ º ´ µ µ ¸µ² for any constant ¹ and integrable functions °²º . (ii) Decomposition: ° ² ³ ° ´ µ µ ¸µ ³ ° ´ ³ ° ´ µ µ ¸µ ¹ ° ² ´ ° ´ µ µ ¸µ² for any » ± ° ± and integrable function ° . (iii) Monotonicity: ° ² ³ ° ´ µ µ ¸µ ² ° ² ³ º ´ µ µ ¸µ² if ° ´ µ µ ² º ´ µ µ on µ ± ° ± , for any integrable functions . (iv) ° ² ³ ° ´ µ µ ¸µ ³ ° ° ³ ² ° ´ µ µ ¸µ² for any integrable function ° . 1

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WATERLOO SOS EXAM-AID: MATH138 MIDTERM Of great importance is the Fundamental Theorem of Calculus, which shows us that integration is a sort of anti-differentiation, to some extent: THEOREM 1.1 [FUNDAMENTAL THEOREM OF CALCULUS] . Suppose ° is continuous on ° ±²³ ± . (i) Then function ´ deFned by ´ ² µ ³´ ° ° ± ° ² ³ ·¶ for µ ° ° ± is continuous on ° ± , differentiable on ² ³ , and ´ ° ² µ ° ² µ ³ . (ii) Then ° ² ± ° ² µ ³ ·µ ´ ¸ ² ³ ³ ± ¸ ² ± ³ where ¸ is any antiderivative of ° , i.e. ¸ ° ´ ° . Recall: since the Fundamental Theorem gives a connection between integrals and antiderivatives, we use the notation ± ° ² µ ³ ·µ to denote the antiderivative of ° , and call it an indeFnite integral . Recall the most general antiderivative of a given integral is obtained by adding a constant to a particular antiderivative , giving rise to a term of the form “ µ ¹ ” when evaluating inde±nite integrals. Furthermore, we will adopt the convention that a formula for a general inde±nite integral is only valid on some interval , not necessarily the whole real line. Based on the Fundamental Theorem and our knowledge of differentiation, we can produce
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