Posted: November 14th, 2015

# Calculus

Calculus

MATH HOMEWORK ASSIGNMENT

Problem 14. If x0 and x1 are positive numbers and
sn =
1
2
(xn + xn−1),
prove that the sequence converges. (Hint : use the ”Nested closed intervals

Theorem”)
Problem 15. Prove the identity (which appears in the Weierstrass Approximation
Theorem): x is a real variable and n is a positive integer,
Xn
k=0

n
k

x
k
(1 − x)
n−k

x −
k
n
2
=
x(1 − x)
n
.
Problem 16. (TextBook : p.92 No.7) Let P(x) = x
n + a1x
n−1 + · · · + an where n is an
even positive integer, the A’s are real , and an < 0. Show that the equation P(x) =

0 has
at least two real roots. What more can you say about them?
Problem 17. (Textbook, p. 92 No. 9.) Let
f(x) = A
a
2 + x
+
B
b
2 + x
+
C
c
2 + x
− 1,
where A, B, and C are positive and a > b > c > 0. Discuss the nature of the graph
of y = f(x) and explain why the equation f(x) = 0 has exactly three roots x1, x2,

x3 ,
satisfying the inequalities −a
2 < x1 < −b
2 < x2 < −c
2 < x3.
1
2 PROFESSOR CARLOS J MORENO
Problem 18. (TextBook p. 104, No 6.) Show that
e
x = 1 + x +
x
2
2! + · · · +
x
n
n!
+ Rn+1
with
0 < Rn+1 < ex x
n+1
(n + 1)!, if 0 < x
and
|Rn+1| <
|x|
n+1
(n + 1)!, if x < 0.
Problem 19 .Suppose N > 0, x0 > 0, and
xn+1 =
1
2
 N
xn
+ xn

, n ≥ 0.
Prove: For n ≥ 1, xn > xn+1 >

N and
xn −

N ≤
1
2
n
·
(x0 −

N)
2
x0
.
Note: This is Newton’s algorithm to approximate the square root of N starting with

the
approximation x0 .

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