**Unformatted text preview: **DUE: WEEK OF FEBRUARY 24TH, START OF GROUP LAB THE COMPARISON TEST FOR IMPROPER INTEGRALS 1. Announcements • Midterm 2: Friday 27-March-2020, 6pm-8pm,
• Last Day to Drop: Friday 3-April-2020, 4pm.
• Final Exam: Saturday 9-May-2020, 12:30pm-3pm.
PRELAB: In the lab, we will learn the following theorem.
f and g be continuous, and suppose that 0 ≤ g (x) ≤ f (x)
R∞
R∞
Ra∞ f (x) dx is convergent, then so isR ∞a g (x) dx.
g (x) dx is divergent, then so is a f (x) dx.
a Theorem. Let (1)
(2) If
If for all a ≤ x. Then To work with this theorem, we need to practice working with inequalities. In particular, we
would like to develop a technique for contructing useful inequalities.
Consider the following:
Exercise 1. For 2 ≤ x, it is true that
1
1
≤
x
x (1.1) r
1+ 1
.
x4 Let's work through the proof to see why (1.1) is true. We will start with the assumption that
2 ≤ x. We then use a more simple, true inequality, along with a step-by-step process, to build up
to (1.1).
(1) To start, we know that when 2 ≤ x we must have 0 <
(2) adding 1 to both sides of 0 < 1
x4 1
x4 . (Why? Convince yourself!) we obtain (3) after taking the square root of both sides of your answer in (b) (since they're positive
(why?)), we have
(4) Now multiply both sides of your answer in (c) by x1 , which is positive (why?) to obtain the
desired result
We have successfully shown that (1.1) is true using familiar mathematical operations. TURN PAGE OVER
1 DUE: WEEK OF FEBRUARY 24TH, START OF GROUP LAB Exercise 2. 2 [On the back of this page] Using a similar process, show that for 2 ≤ x
r
1+ √
4
≤
5.
x6 That
with the inequality 2 ≤ x and through a series of manipulations, conclude that
q is, START
√
4
1 + x6 ≤ 5. It may be useful to use simple inequalities that we know are true, such as 12 < 1. It
will also be benecial to recall the transitive property of inequalities. (If a<b and b<c then
a<c.) Note that this time you need an UPPER bound, not a LOWER bound. (Hint: Start with
2 ≤ x and divide both sides by 2x). Exercise 3. [On the back of this page] Determine if 1
1+x is larger or smaller than 1
x + sin2 x for x ≥ 1. Justify your conclusion with the methods illustrated in Exercise 1. ...

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