# a6 - b if x ≥ 2 Find the values of a and b so that f has...

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University of Toronto at Scarborough Department of Computer and Mathematical Sciences MATA37 Calculus II for Mathematical Sciences Summer 2007 Assignment #6 Work on the text material and the problems below. You will have a quiz on the material in Assignment 5 or 6 (at your TA’s discretion) in your tutorial on Tuesday, July 10. STUDY: Chapters 9-11 for this assignment. Chapters 13, 14 and 19 for future assignments and lectures (Omit Chapters 12, 13-15, and 18). PROBLEMS: 1. Page 163-164 # 16, 22, 28(a). 2. Page 179-185 # 1, 2, 11, 16(a, b), 26, 27. 3. This problem is about ﬁnding a formula for higher-order derivatives. Find a formula for the nth-derivative, f ( n ) ( x ), for these functions. As extra practice, verify a couple of your ﬁndings using the PMI. (a) f ( x ) = 1 /x (b) f ( x ) = e bx where b is a non-zero constant (c) f ( x ) = xe x (d) f ( x ) = sin ( x ) 4. Deﬁne a function f on all of R by f ( x ) = x 3 if x < 2 and f ( x ) = ax
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Unformatted text preview: + b if x ≥ 2. Find the values of a and b so that f has a continuous derivative on all of R . 5. Verify that ( tan-1 ( x )) = 1 / (1 + x 2 ). 6. Page 202 - 208 # 9, 14, 36, 36, 40. 7. Let f ( x ) = sin ( x )-2 x + 1 . Find the exact number of roots of f and fully justify your ﬁndings. 8. Let g ( x ) = xe 2 x . Find a formula for g ( n ) ( x ) for a natural number n and then verify your claim with a good proof using the PMI. 9. Let a,b > 0 be constants and let h ( x ) = x a (1-x ) b where x ∈ [0 , 1]. Use the EVT to justify that h has an absolute max on [0 , 1] and then ﬁnd this value and where it occurs. 10. Prove that tan ( x ) > x for all x ∈ (0 , π 2 ). NOTE: Reading Week is Monday, July 2 to Friday, July 6, so there are no classes that week. MATA37 will resume on Tuesday, July 10....
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