Posted: November 14th, 2015

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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