This preview shows page 1. Sign up to view the full content.
Unformatted text preview: en there exists a sequence of polynomials that converges
uniformly to g on the interval 1, 1 .
This part follows at once from part b, in the light of the fact that if g is the function defined in
part b then for each x
1, 1 we have 377 gx f1
d. x2 1 1 x2 x . Use Scientific Notebook to calculate some nth Maclaurin polynomials of the function f defined in
part a. For each chosen value of n, if f n is the nth Maclaurin polynomial, and h n x f n 1 x 2 for each
x, ask Scientific Notebook to sketch the graph of the function h together with the graph of the absolute
value function and observe graphically that the sequence h n is converging uniformly to the absolute
value function on the interval 1, 1 . The case n 35 is illustrated in the following figure.
1
0.8
0.6
0.4
0.2 1 0.5 0 0.5 1 This exercise is of considerable importance because it may be used as the starting point for a major
theorem known as the StoneWeierstrass . You can find an elementary presentation of the
StoneWeierstrass theorem in Rudin reference starting with Corollary 7.27. Exercises on the Trigonometric Functions
1. Given any real number x, prove that sin x 0 if and only if x is an integer multiple of . Prove that cos x 0
if and only if x is an odd multiple of /2. Prove that if n is any integer then cos n 1 n . Solution: We know that sin sin 0 0 and that sin x 0 whenever 0 x . We also know that whenever x and y are numbers and sin x sin y 0 we have
sin x y sin x cos y sin y cos x 0
and from this fact we see that sin n 0 for every integer n. Now suppose that x is any number for which
the equation sin x 0 holds. If we define n to the the greatest integer that does not exceed the number x/
then we have
0 x n
and since sin x n 0 we conclude that x n. A similar argument may be used to show that
n
n is an integer .
2
Finally, if G is the set of integers n for which the equation cos n 1 n is true then we know that 1
and that the sum and difference of any two members of G must belong to G. Therefore the equation
cos n 1 n holds for every integer n.
x R cos x 0 2. Prove that if is any real number then the equation
sin x sin x
holds for every real number x if and only if is an even multiple of . Solution: We already know that the equation
sin x 2n sin x
holds for every number x and every integer n. Now suppose that is a number for which the equation
sin x sin x 378 G holds for every real number x. Since
sin sin 0 sin 0 0
we know that is an integer multiple of . We can therefore express as n for some integer n. Since
1 sin sin n cos n 1 n
2
2
we know that n is even.
3. Prove that the restriction of the function sin to the interval /2, /2 is a strictly increasing function from
/2, /2 onto the interval 1, 1 . Prove that if the function arcsin is now defined to be the inverse function
of this restriction of sin then for every number u
1, 1 we have
u
1
arcsin u Þ
dt.
0
1 t...
View
Full
Document
 Fall '08
 STAFF
 Math, Calculus

Click to edit the document details