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Unformatted text preview: MATH 137 Assignment 4 Due: 11 am, Friday, October 14 Your assignment consists of two parts. Part 1 is available online at http://mapleta.uwaterloo.ca and is due on—line by 4 pm on Thursday, October 13.
Part 2 consists of the problems below. Place your solutions to the problems in part 2 into the correct slot in the drop boxes
outside MC 4066, corresponding to the class section in which you are registered. If your assignment is put into the wrong slot, it may be lost for a long time and
may never get marked. You may copy and use the assignment templates that are available on the D2L
website for MATH 137. Make sure your name and your UW user ID are clearly
written at the top of the ﬁrst page, and underline your last name. While it is okay to get help if you are stuck, you are required to include, at the
top of your assignment, the names of students who helped you or with whom
you collaborated. This acknowledgement will no effect on your assignment mark.
Any outright copying of assignments, in whole or in part, is an act of cheating,
and it will be reported to the associate dean. Hand in your solutions to the following 5 problems. Your solutions must have
legible handwriting, and must be presented in clear, concise and logical steps that
fully reveal what you are doing. 1. Let f be a function such that sin(7r:z:) 3 ﬂat) < 1 _ for all CE in the interval (0, 1) Find 1113132 f (:11), and explain your reasoning.
02—) Decide if f continuous at a 2 1/2, and explain your reasoning. 2. Leta be any real number and let f($)= 2 1—zvzwhen$<a
ac wheana V Find the values of a that make f a continuous function on all of R. Then,
for each suitable value of a, sketch the resulting continuous function. 3. Evaluate the following limits if they exist. (a) lirn arctan (cc —— C(74)
x—aoo . , 92:5 — :1:
(b) .2le gas—+1 (c) lim az—>+oo 4. The function f 2: 1 2 has the :13—axis as a horizontal asymptote. In
:5
1
deed, f(x) = Um + x —> 0 as :1: —> 00, by inspection. So, we expect that for every 6 > 0 there is a large number K such that
I f < 6 when x > K. For 6 > 0, ﬁnd a suitable such K, and show that it
is suitable. 5. Let f 9:3 + bx? + cm + d be a general cubic polynomial with the
coefﬁcient in front of 51:3 adjusted to be a 1. (a) Explain why f > 0 when a: > 0 and very large, and why f < 0
when :1: < 0 and very large. Hint. Rewrite f as ﬁx) 2 5133(1 + b/a: + (2/222 + d/x3) and then in—
spect the sign of the part in brackets when a: is very large. (b) Use .the above information to show that every cubic must cut the x—axis
in at least one place, i.e. prove every cubic has a real root. (c) Does every degree 4 polynomial cut the :IJ—axis? Explain your answer. EXTRAS Once you have seen the concept of derivative, you should work on these extra
problems, but do not hand these in. Problems such as these could be on exams. 1. Suppose f is a function with the property that If g 2:2 for every real
number 3:. (a) Show that f(0) = 0.
(b) Show that f’(0) = O. (C) Let ﬂit) = {
Does f’(0) exist? If so, what is f’(0)? :52 sin 1 when x 74 0
Cl: 0 whenx = O. 2. (a) Sketch the graph of the function f = (b) Show that f is differentiable for all :13. (c) Find a formula for f’ Hint. For a: > 0 and for a: < 0, you can ﬁnd f’ by just looking at
f (:13), and remembering a bit of high school calculus. To get f’ (0), use
the deﬁnition of derivative. 3. (a) If g(;v) 2 1162/3, use the deﬁnition of the derivative to show that g’(0)
does not exist. (b) Is the function 9 continuous at 0? Explain your answer. The following problems from Stewart’s book are strongly recommended for you
to practice on your own. Problems such as these are eligible for the mid—term
exam. Do not hand these in but make sure you know how to solve problems such as these. If you get stuck, solutions can be found in your Student Solutions
Manual. 0 Section 2.5, pp. 127—130 # 41, 45, 49, 51, 53, 55, 67
0 Section 2.6, pp. 140—143 # 19, 25, 27, 35, 37, 53, 57
0 Section 2.7, pp. 150—153 # 19,21, 23, 25, 29, 53 0 Section 2.8, pp. 162—165 # 51, 53, 57
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 Fall '08
 ANDREWCHILDS
 Algebra

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