رياضيات

Take-Home Quizzes

1) Show that if t is not equal to x and n is a natural number greater than 1, then

( tn – xn ) / ( t- x )

= tn-1 + tn-2 x +tn-3 x2 +………………………+ xn-1

2) Use part(1) to deduce that the limit of the function

ξ(t) = ( tn – xn ) / ( t- x ) at the point t = x is equal to n xn-1

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Let f be the function defined as follows:

{ 8 /x ; x ≥ 4

{ √ x ; 1 ≤ x < 4

f(x) = { √ (- x) ; – 4 < x < 0

{ 1 / 4×2 ; x ≤ – 4

a) Graph f and determine its domain and range

b) Find the value of f , if exists, at each of the following points:

x = 16 , x = 4 , x = 3 , x = 9/4 , x=3/2 , x = 1, x = 9/8 , x = 4/9 , x = 0 , x = – 4/9 , x = -1 , x = -9/4 , x = -3 ,

x = -4 , x= -5 , x = -6 , x =- 10
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For each of the following cases:

1. Graph the function f and determine its domain and range

2. Find, if exists, each of the following values: f(-5) , f(2) , f(3) and f(5)

Case A

Let

{ x – 2 ; x < 1

f(x) = 3x ; 1 ≤ x < 3

{ 2 ; x >3

Case B

Let

{ x + 2 ; x < 1

f(x) = – x ; 1 < x < 3

{ 1 ; x ≥ 3

Case C

Let

{ 3x ; x ≤ 1

f(x) = – 2 ; 1 < x < 3

{ 2x +4 ; x ≥ 3

1. Graph the function f and determine its domain and range

2. Find, if exists, each of the following values: f(-5) , f(2) , f(3) and f(5)

Case D

Let

{ 2 ; x < -1

f(x) = 2x – 2 ; -1 < x < 1

{ 2x +4 ; x > 1

Case E

Let

{ 2 x ; x ≤ -1

f(x) = 2x – 4 ; -1 < x < 1

{ 4 ; x ≥ 1