Understanding Engineering Mathematics

3.3.6 Inequalities

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Find the ranges of values ofxfor which the following are satisfied:

(i) 2x− 3 > 2 (ii)

2 x

x− 3

< 1

(iii) x^2 −x+ 1 ≥3(iv)x^2 + 2 x+ 2 < 5

(v) | 2 x− 1 |≤2(vi)

2

|x− 4 |

< 4

(vii)

1

x− 1

<− 3

3.3.7 Inverse of a function

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Find the inverse functions for each of the following functions, specifying the values for

which they exist.

(i) 2x+ 1 (ii)

x− 1

x+ 2

x=− 2

(iii) x^2 + 1 ,x≥ 0

3.3.8 Series and sigma notation

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A.Write the following in sigma notation:

(i) 1+ 8 + 27 + 64 + 125 (ii) 3+ 6 + 9 + 12 + 15 +···+ 99

(iii)

1

2

+

1

3

+

1

4

+

1

5

+···+

1

50

(iv) 1−

1

3

+

1

9

−

1

27

+···

B.Write down the first four terms in the series:

(i)

∑∞

r= 1

1

r

(ii)

∑^20

r= 0

r!

(iii)

∑∞

r= 1

1

r(r+ 1 )

3.3.9 Finite series

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Sum the geometric series:

(i) 1+

1

2

+

1

4

+

1

8

+···+

1

32

(ii)

∑^6

n= 1

( 0. 1 )n

(iii)

∑^8

n= 1

2 n (iv)

∑^4

n= 1

(

1

3

)n

(v) 1+ 0. 1 + 0. 01 + 0. 001 + 0. 0001