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Unformatted text preview: 40 then 8x 3 6x 1 0. We need to show that if x cos 80 then 8x 3 6x 1 0. To prove the first of these assertions, we assume that x 10 cos 40 . In order to show that 8x 3 6x 10 we shall use the trigonometric identity
cos 3 4 cos 3 3 cos
that holds for every number . We now observe that
8x 3 6x 1 8 cos 40 3 6 cos 40
2 4 cos 3 40 2 cos 3 40 3 cos 40 1
1 1 2 cos 120 1
2 1
10
2
4. A theorem in elementary calculus, known as Fermat’s theorem, says that if a function f defined on an interval
has either a maximum or a minimum value at a number c inside that interval then either f c 0 or f c
does not exist. Give a brief outline of a strategy for approaching the proof of this theorem. Solution:
Case 1: Suppose that f is a function defined on an interval and that has a maximum value a number c
inside that interval, and suppose that f c exists. We need to show that f c 0.
Case 2: Suppose that f is a function defined on an interval and that has a minimum value a number c
inside that interval, and suppose that f c exists. We need to show that f c 0. 5. A well known theorem on differential calculus that is known as L’Hôpital’s rule may be stated as follows:
Suppose that f and g are given functions, that c is a given number, that
fx
lim
L,
xcg x
and that one or other of the following two conditions holds
a. Both f x and g x approach 0 as x
b. g x Ý as x c. c. Then
fx
L.
gx
Describe how the proof of L’Hôpital’s rule can be broken down into two parts. For each part of the proof, say
what is being assumed and what is being proved.
lim
xc The purpose of this exercise is not to dig into the underlying ideas of L’Hôpital’s rule. Nor do we
need to concern ourselves with the question as to whether it is actually necessary to break the
proof into two parts. Instead, we assume that the two parts are needed. The student should be
able to see that, approached this way, the theorem requires two separate proofs. In one proof we
assume that both f x and g x approach 0 as x c and in the other proof we assume that g x
Ý
as x c. Exercises on the Symbol
1. a. Outline a strategy for proving an assertion that has the form P Q R. Solution: Assume that P is true. Then write a proof that Q is true. Then write a proof that R is
true.
b. Write down the assertion P Q
this form. R in its contrapositive form and outline a strategy for proving it in 11 Solution: The contrapositive form says that Q
R P. Once again we have two jobs
to do. We assume that the statement Q is false and prove that P is false. Then we assume that the
statement R is false and again prove that P is false. 2. a. Outline a strategy for proving an assertion that has the form P Q R. We need to show that if P is true than at least one of the assertions Q and R is true. We could
begin by writing: “Suppose that P is true." Then we could try to prove that at least one of the
assertions Q and R is true. One way to do this is to assume that Q is false and then sho...
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 Fall '08
 STAFF
 Math, Calculus

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