• January 31st, 2016

MAT 137Y: Calculus! 5 questions

Paper, Order, or Assignment Requirements

MAT 137Y: Calculus!

Problem Set 6.

Need to be done within 30 hours

Need at least 80%

Note: Question 0 is a warm-up question to make sure you understand the definition of lower/upper integral and the definition of supremum and infimum. Do not submit it. Only submit the other five questions. 0. Let f be a bounded function on the real interval [a, b]. (a) Prove that I b a (f) satisfies the following two properties: i. Lf (P) ≤ I b a (f) for every partition P of [a, b]. ii. For every ε > 0, there exists a partition P of [a, b] such that I b a (f) − ε < Lf (P) ≤ I b a (f). (b) Let J be a real number. Assume J satisfies the following two properties: i. Lf (P) ≤ J for every partition P of [a, b]. ii. For every ε > 0, there exists a partition P of [a, b] such that J − ε < Lf (P) ≤ J. Prove that J = I b a (f) Note: This problem is very short once you are very comfortable with the definition of supremum and of lower integral. 1. Write the statement (without proof) of the results equivalent to Question 0 for I b a (f) instead of I b a (f). 2. Let f be a bounded function on the interval [a, b]. (a) Assume that f satisfies the following property: ∀ε > 0, ∃ partition P of [a, b], such that Uf (P) − Lf (P) < ε. Prove that f is integrable on [a, b]. (b) Assume that f is integrable on [a, b]. Prove that f satisfies the following property: ∀ε > 0, ∃ partition P of [a, b], such that Uf (P) − Lf (P) < ε. Hint: Use the definition of integrability: f is integrable on [a, b] if and only if I b a (f) − I b a (f) = 0. Also use the definitions of I b a (f) and I b a (f). Finally, remember that we always know that I b a (f) − I b a (f) ≥ 0, whether f is integrable or not. 3. Let f and g be two bounded functions on the interval [a, b]. (a) Let P be a partition of [a, b]. Only one of the following two inequalities is always true: Lf+g(P) ≤ Lf (P) + Lg(P), Lf+g(P) ≥ Lf (P) + Lg(P) Determine which one is always true, prove it, and then show the other one is not always true with an example. (b) Repeat Question 3a with upper sums instead of lower sums. (c) Assume that f and g are integrable on [a, b]. Prove that f +g is also integrable on [a, b]. Hint: Use Problem 2 repeatedly. 4. Give an example of two bounded functions f and g on an interval [a, b] such that I b a (f + g) 6= I b a (f) + I b a (g). 5. In this question, you are going to compute the exact value of Z 5 2 (5x − x 2 ) dx using Riemann sums. Let us call f(x) = 5x − x 2 . Since f is continuous on [1, 3], we know it is integrable. Hence, its value can be computed using any Riemann sums via equation (5.2.7) in the book. For every natural number n, let us call Pn the partition that splits [2, 5] into n equal sub-intervals. Notice that limn→∞ ||Pn|| = 0. Hence, we can write Z 5 2 5x − x 2  dx = limn→∞ S(Pn), where S(Pn) is any Riemann sum for f and Pn. In particular, to make things simpler, we are going to choose the Riemann sum S(Pn) where at every subinterval we use the righ-endpoint to evaluate f. (a) Let us write Pn = {x0, x1, . . . , xn}. Find a formula for xi in terms of i and n. (b) What is the length of each sub-interval in Pn? (c) Since we are using the right-endpoint, it means we are picking x ? i = xi . Use your above answers to obtain an expression for S(Pn) in the form of a sum with sigma notation. (d) Using the formulas X N i=1 i = N(N + 1) 2 , X N i=1 i 2 = N(N + 1)(2N + 1) 6 , X N i=1 i 3 = N2 (N + 1)2 4 if needed, add up the expression you got to obtain a nice, compact formula for S(Pn) without any sums or sigma symbols. (e) Calculate limn→∞ S(Pn). This number will be the exact value of Z 5 2 (5x − x 2 )dx. Hint: Your final answer should be 27 2

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