M AT H 1 2 0 Differential and Integral Calculus Fa l l 2 0 1 6 - P r o b l e m S e t # 1 1 Due Wednesday, November 30. Write your name and student number very clearly in the upper right hand corner of your front sheet and staple your sheets if necessary. You are encouraged to collaborate with others when working on your homework assignments, but you must write up solutions independently, on your own. Do not copy the work of others. In accordance with academic integrity regulations, you must acknowledge in writing the assistance of any students, professors, books, calculators, or software. Be sure that your final write-up is clean and clear and effectively communicates your reasoning to the grader. 1) Review the -notation and proofs by induction in Appendix E of the textbook. Prove that for every positive integer n the following identities are true n (a) i ( i + 1) = i =1 2n (b) i3 = i=n n(n + 1)(n + 2) . 3 3n2 (n + 1)(5n + 1) . 4 n (c) (2i 1) = n2 i =1 2) Let Fn be the nth Fibonacci numbers. That is, F1 = 1, F2 = 1, F3 = F2 + F1 = 2, F4 = F3 + F2 = 3, F5 = F4 + F3 = 5, . . . . The general formula for n 3 is Fn = Fn1 + Fn2 . Let = 1+2 5 . This number is called the golden ratio, and satisfies the equation 2 = + 1. (a) Prove that for every n 1 we have Fn n2 . [Hint: use a proof by induction. First check that F1 12 and that F2 22 . Then prove that if Fm m2 for every 1 m k, then Fk+1 k+12 .] (b) Prove that the 1000th Fibonacci number F1000 has at least 209 decimal digits. (c) Let N be the number of decimal digits of the ( F1000 )th Fibonacci number FF1000 . Prove that N has at least 208 decimal digits. (For comparison, there are approximately 8 1067 ways of shuffling a deck of 52 cards, and approximately 1080 atoms in the observable universe.) 3) Find the derivative (with respect to x) of the following functions: !4 Z 2 x (a) F ( x ) = (b) G ( x ) = (c) H ( x ) = 0 (t + x2 )dt Z cos( x) ex . ln(w3 )dw. Z sin( x) cos( x ) 1 1 2 d. \f