Math 242 Tutorial Problems
Tutorial #1 (Jan 13)
Chapter 1: 11a, 27, 28
Tutorial #2 (Jan 20)
Show that the sup and inf (least upper bound and greatest lower bound) of a set, if it exists, are unique.
Prove that if a_n=(sqrt(n^2 + 1)/n, then a_n --> 1 as n --> infinity.
Tutorial #4 (Feb 3)
Chapter 4: 1a, 1b, 3
Tutorial (Feb 24)
(1) Prove that limit x --> 2, f(x) = (x^2+1)/2x is 5/4
Chapter 6: 9, 11
Tutorial (March 2)
Chapter 6: 9, 11
Chapter 7: 23
(A): If | f ' (x)| < 1, then f(x) is uniformly continuous.
Tutorial (March 10)
Chapter 8: 4, 19
(A): Show that L(f,P) <= U(f,Q) for any bounded function f on [a,b] and for any partitions P and Q of [a,b]. Here, L and U are the lower and upper Riemann sums.
(B): Prove the Fundamental Theorem of Calculus for a continuous function f on [a,b] where int(f) is the Riemann integral. (See Theorem 8.6.)
Tutorial (March 24)
Chapter 9: 6 (go back to Ch 3), 10
Let fn(x) = 1/(1+(nx)^2) and gn(x)=nx(1-x)^n on [0,1]. Prove that {fn} and {gn} converge pointwise (to what?) but not uniformly.
Last updated: March 22