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NATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA 1505 Mathematics I Mid-Term (Review) 1F o r m u l a This is the formula list that will appear in the mid-term test, so besides these formulas, you need to remember much more. MA1505 Mid-Term Test Formulae List 1. The Taylor series of f at a is ± k =0 f ( k ) ( a ) k ! ( x a ) k = f ( a )+ f ± ( a )( x a ··· + f ( n ) ( a ) n ! ( x a ) n + 2. e x = ± n =0 x n n ! 3. sin x = ± n =0 ( 1) n x 2 n +1 (2 n +1)! 4. cos x = ± n =0 ( 1) n x 2 n (2 n )! 5. ln(1 + x )= ± n =1 ( 1) n 1 x n n 6. tan 1 x = ± n =0 ( 1) n x 2 n +1 2 n +1 54
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2 Questions to Guide Your Review Here are some questions from Thomas’ Calculus, they are not tested in the Mid-term and Final exam, but good for you to examine whether you understand the de±nition correctly. If you are not sure about the answer, check your text-book. All the answers can be found in Thomas’ Calculus. 2.1 Limits and Continuity Remarks: The text-book contains much more than your lecture notes. Based on your lecture notes and syllabus, try to answer the following questions: Qn1-Qn8; Qn15-Qn28. Chapter 2 Questions to Guide Your Review 1. What is the average rate of change of the function g ( t ) over the in- terval from to How is it related to a secant line? 2. What limit must be calculated to find the rate of change of a func- tion g ( t ) at 3. What is an informal or intuitive definition of the limit Why is the definition “informal”? Give examples. lim x : x 0 ƒ s x d = L ? t = t 0 ? t = b ? t = a 4. Does the existence and value of the limit of a function ƒ( x ) as x approaches ever depend on what happens at Explain and give examples. 5. What function behaviors might occur for which the limit may fail to exist? Give examples. 6. What theorems are available for calculating limits? Give exam- ples of how the theorems are used. x = x 0 ? x 0 7. How are one-sided limits related to limits? How can this relation- ship sometimes be used to calculate a limit or prove it does not exist? Give examples. 8. What is the value of Does it matter whether is measured in degrees or radians? Explain. 9. What exactly does mean? Give an example in which you find a for a given and in the pre- cise definition of limit. 10. Give precise definitions of the following statements. a. b. c. d. 11. What exactly do and mean? Give examples. 12. What are ( k a constant) and How do you extend these results to other functions? Give examples. 13. How do you find the limit of a rational function as Give examples. 14. What are horizontal, vertical, and oblique asymptotes? Give ex- amples. 15. What conditions must be satisfied by a function if it is to be con- tinuous at an interior point of its domain? At an endpoint? 16. How can looking at the graph of a function help you tell where the function is continuous? 17. What does it mean for a function to be right-continuous at a point? Left-continuous? How are continuity and one-sided conti- nuity related?
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