Mathematical Analysis Solution

Introduction :

Mathematical analysis, which the mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of calculus. It is the branch of the pure mathematics most explicitly concerned with the notion of a limit, whether the limit of a sequence or the limit of a function.                                                                                                                                                                                                                

Mathematical analysis degree of such closeness cannot be described in terms of basic algebraic operations of addition and multiplication and their inverse operations subtraction and division respectively.

                                                                                                                                                                                                                       

 

Examples for mathematical analysis solution:

 

Example 1:

 Evaluate `lim_(x->1)` x3 − 1 / x− 1

Solution:

`lim_(x->1)` x3 − 1/x− 1

= 3(1)3 − 1 = 3(1)2 = 3           [      `lim_(x->a)` xn − an/ x − a=nan − 1  ]

Example 2:

 Evaluate`lim_(x->0)` etan x − 1/ tanx

Solution:

 Put tanx = y. Then y → 0 as x → 0

Therefore `lim_(x->0)`etan x − 1/tanx

=`lim_(y->0)`ey − 1/ y = 1

Example 3:

Compute the value of Δy and dy if y = f(x) = x3 + x2 − 2x + 1 where x changes (i) from 2 to 2.05 and (ii) from 2 to 2.01

Solution:

(i) We have f(2) = 23 + 22 − 2(2) + 1 = 9

f(2.05) = (2.05)3 + (2.05)2 − 2(2.05) + 1 = 9.717625.

and Δy = f(2.05) − f(2) = 0.717625.

In general dy = f ′(x) dx = (3x2 + 2x − 2)dx

When x = 2, dx = Δx = 0.05 and dy = [(3(2)2+2(2)−2] 0.05 = 0.7

(ii) f(2.01) = (2.01)3 − (2.01)2 − 2(2.01) + 1 = 9.140701

∴ Δy = f(2.01) − f(2) = 0.140701

When dx = Δx = 0.01, dy = [3(2)2 + 2(2) − 2]0.01 = 0.14

Example 4:

Integrate the following with respect to x.  ax + xa + 10 − cosec 2x cot2x

Solution:

`int` (ax + xa + 10 − cosec 2x cot2x)dx

=`int` axdx + `int` xadx + 10 `int` dx −`int` cosec 2x cot 2x dx

=ax / loga +xa + 1/ a + 1 + 10x +cosec 2x / 2 + c

 

Practice problem for mathematical analysis solution:

 

1.Integrate the following with respect to x

(1)    5x4 + 3(2x + 3)4 − 6(4 − 3x)5

(2)    p cosec2 (px − q) − 6(1 − x)4 + 4e3 − 4x

Answer: (1) x5 +3/10 (2x + 3)5 +1/3 (4 − 3x)6

                     (2)  − cot(px − q) +6/5 (1 − x)5 − e3 − 4x

2.Evaluate`lim_(x->0)`[ 3x + 1 − cos x − ex]  /  x .

Answer:log3 -1