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Unformatted text preview: assume that the function f is increasing and we want to prove that f is a convex
function. Suppose that a and x and b lie in the interval S and that a x b. Using the mean value
theorem we choose a number c between a and x and a number d between x and b such that
fx fa
fb fx
and
f c
f d.
xa
bx
Since f is increasing we have
fx fa
fb fx
fd
f c
.
xa
bx
17. Prove that if f is a convex function on an open interval S then f must be continuous on S.
a. Solution: Suppose that x S. We shall show that f must be continuous at x. Choose numbers a,
b, c and d in S such that a b x c d. a b x c d c d Now given any number t between x and c a b x t we have
fx
x
from which we deduce that fb
b ft
t fx
x 245 fc
c ft
t fd
d fc
c fx
x fb
b t x ft lim f t fd
d fx
fx fc
c t x and the fact that
t 0 0 follows at once from the sandwich theorem. We can show similarly that
lim f t f x 0
t0 and we therefore know that f is continuous at the number x.
18. By clicking on the icon you can reach some exercises that introduce Newton’s method for approximating roots of an equation. Some Exercises on Taylor Polynomials
1. Suppose that f is a polynomial whose degree does not exceed a given positive integer k and that f
for every j 0, 1, 2, , k. Prove that f is the constant function zero.
The fact that f is the constant function 0 follows at once from the equation
n fx f j 0
j! j0 j 0 0 xj that holds for every real number x.
2. Suppose that f and g are two polynomials whose degrees do not exceed a given positive integer k and that
f j 0 g j 0 for every j 0, 1, 2, , k. Prove that f x g x for every number x. Hint: Apply the preceding exercise to the polynomial f g. 3. Prove that if f is a polynomial whose degree does not exceed a given positive integer k and n is an integer
satisfying n k then the nth Taylor polynomial of f is f itself.
Since k n, we can find numbers a 0 , a 1 , , a n such that the equation
n fx ajxj
j0 holds for every number x.
4. Given a nonnegative integer n and that
f x 1x n
for every number x, work out the nth Taylor polynomial of f and obtain a simple proof of the binomial
theorem
n 1x n
j0 n
j As you may know, the expression
equation
n
j n xj
j , which is called the n, j binomial coefficient, is defined by the
nn 1n 2n
j! j1 n n!
j !j! whenever n and j are integers and 0 j n. We shall define binomial coefficients more generally later.
All we have to observe is that if x is any number and j is a nonnegative integer then
f j x n n 1 n j1 1x n j
and that, consequently, 246 j 0
n! f nn 1n
n! j1 Some Exercises on Indeterminate Forms
1. Evaluate each of the following limits. In each case, use Scientific Notebook to verify that your limit
value is correct.
lim x
x0 sin x
x3 lim tan x
x0 lim tan x x
sin x
x0 x sin x lim x3 lim 1 x x0 1/x lim x0 log 1 x
x
e x0 1x
x 1/x To find the limit
lim x
x0 sin x
x3 we observe first that
lim x
x0 and so L’Hôpital’s rule guarantees that
lim x
x0 sin x lim x 3 0...
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 Fall '08
 STAFF
 Math, Calculus

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