**Mathematical analysis solution:**

homework assignment help is most useful online portal for students providing all type of Online Mathematical Analysis assignment help Services .Mathematical analysis Assignment Help, 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.

**Example 1:**

Evaluate lim_(x->1) x^{3} − 1 / x− 1

**Solution:**

`lim_(x->1)` x^{3} − 1/x− 1

= 3(1)^{3 − 1} = 3(1)^{2} = 3 [ lim_(x->a) x^{n }− a^{n}/ x − a=na^{n − 1 ]}

**Example 2:**

Evaluate`lim_(x->0)` e^{tan x − 1}/ tanx

**Solution:**

Put tanx = y. Then y → 0 as x → 0

Therefore `lim_(x->0)`e^{tan x − 1}/tanx

=`lim_(y->0)`e^{y − 1}/ y = 1

**Example 3: **

Compute the value of Δy and dy if y = f(x) = x^{3} + x^{2} − 2x + 1 where x changes (i) from 2 to 2.05 and (ii) from 2 to 2.01

**Solution:**

(i) We have f(2) = 2^{3} + 2^{2 }− 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 = (3x^{2} + 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:**

=a^{x} / loga +x^{a + 1}/ a + 1 + 10x +cosec 2x / 2 + c

**1.Integrate the following with respect to x**

(1) 5x^{4} + 3(2x + 3)^{4} − 6(4 − 3x)^{5}

(2) p cosec^{2} (px − q) − 6(1 − x)^{4 }+ 4e^{3} − 4x

**Answer:** (1) x^{5} +3/10 (2x + 3)^{5} +1/3 (4 − 3x)^{6}

(2) − cot(px − q) +6/5 (1 − x)^{5} − e^{3} − 4x

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

**Answer:**log3 -1

**Introduction to advanced mathematical analysis:**

In particular, you resolve have considered sequences and functions, limits and continuity. This subject continues learn in advanced mathematical analysis, extending the objects in Abstract Mathematics analysis. The importance is on series, functions and sequences in n-dimensional real space, and we shall as well study the common concept of a metric space.

For example, as we shall observe, compactness is an extremely main idea in optimization. Further generally, a course of this nature, by means of the importance on abstract reasoning and proof, resolve help you to assume in an analytical way, and be there able to prepare mathematical arguments in an exact, logical manner.

Let us see the topic sequence and series in advanced mathematical analysis.

**Definition of sequence:**

A sequence be a function from the set of natural numbers to the set of real numbers.

If the sequence is denoted by the letter ‘`a` ’, then the image of n`in` N under the sequence a is *`a(n)=a_(n).` *

Since the domain for every sequence be the set of natural numbers, the images of 1, 2, 3 …*n…* under the sequence is the set of natural *a* are denoted by * `a_(1), a_(2),a_(3)......a_(n)` ,*…. Respectively.

Here `a_(1),a_(2),a_(3).......a_(n)` from the sequence.

“A sequence is denoted in its range”.

**Terms of a sequence:**

The various numbers occurring in a sequence are known as its terms. We denote the terms of a sequence by a_(1), a_(2), a_(3), ...., a_(n),...., the subscript represent the location of the term. The *n*^{th} term be known as a common term of the sequence.. For example, in the sequence 1, 3, 5, 7, … 2*n *− 1, …

The 1^{st} term is 1, 2^{nd} term is 3, … … and *n*^{th} term is 2*n *− 1.

**For example: **

Find the 8^{th} term of the sequence whose n^{th} term is `(-1)^(n+1)((n+1)/(n))`

**Solution:**

Given an=(-1)^ n+1(n+1 /n)

Substituting n=8 we get

an = (-1)^8(8+1/8) =9/7.

**Definition of series:**

If `a_(1), a_(2), a_(3), ...a_(n) ...` is an infinite sequence then `a_(1) + a_(2) + ...+ a_(n) + ...` is

known as an infinite series. It is also denoted bysum_(k=1)^oo a_(k)

If S_{n} = a_{1} + a_{2} + … + a_{n} then Sn is called the n^{th} partial sum of the series sum_(k=1)^oo a_(k)

This is one of the topic in advanced mathematics analysis.

** Introduction to fundamentals of mathematical analysis:**

Fundamentals of mathematical analysis it is helpful for all higher grade students. Mathematical analysis referred to both differential and integral calculus. To solve calculus problems, we need to remember the formulas. Some important math topics are calculus, differential functions. In this article we shall discuss about fundamentals of mathematical analysis.

**Example problem1:**

To find f’(x) the function of f(x) = x^2+2x+3, when x=2.

**Solution:**

f(x) = x^2+2x+3

f^'(x)=2x+2`

f^'(2) = 2(2)+2

f^'(2) = 4+2

f^' (2) =6

Answer is 6.

**Example problem2:**

To find f^'(x) the function of f(x) = x^2+2x+3 , when x=3.

**Solution:**

f(x) = x^2+2x+3

f^'(x)=2x+2

f^'(2) = 2(3)+2

f^'(2) = 6+2

f^'(2) =8

Answer is 8.

**Example problem 3:**

To find f^'(x) the function of f(x) = x^2+2x+3 , when x=4.

**Solution:**

f(x) = x^2+2x+3`

f^'(x)=2x+2`

f^'(2) = 2(4)+2

f^'(2) = 8+2`

f^'(2) =10

Answer is `10.`

**Problem 1:- Fundamentals of mathematical analysis:**

Integrate the known expression with respect to x: int 12x^4 - 11x^5 dx

**Solution:**

Given int 12x^4 - 11x^5 dx.

**Step 1:-**

int 12x^4- 11x^5 dx = int 12x^4 dx. - int 11x^5 dx.

**Step 2:-**

= int 12x^4dx. - 11 int x^5 dx.

**Step 3:-**

= (12x^5)/ (5) - (11x^6)/ (6) + c.

**Step 4:-**

int 12x^4 - 11x^5 dx = (12x^5)/ (5) - (11x^6)/ (6) + c.

**Answer:**

int 12x^4 - 11x^5 dx = (12x^5)/ (5) - (11x^6)/ (6) + c

**Problem 2:- Fundamentals of mathematical analysis:**

Integrate the known exponential function: int tan x + e^ (2x) dx

**Solution:**

**Step 1:-**

int tan x + e^ (2x) dx = int tan x dx + int (e^(2x)) dx

= int tan x dx + e^ (2x)/ (2)

**Step 2:-**

= - log (cos x) + e^ (2x)/ (2) + c

**Answer:**

- log (cos x) + e^ (2x)/ (2) + c

**Introduction to maximum rate of change:**

Rate of change is a concept that lays the foundation for the branch of calculus. Simply put, rate of change is a measure of how fast a dependent variable changes (with respect to the independent variable).

For any function y = f(x), rate of change can be expressed as: Rate of change = dy/dx = f’(x)

The concept of rate of change has an array of applications in the real world. For example, the concept may be used to measure how fast a body is moving (i.e. velocity), how fast velocity is increasing (i.e. acceleration) etc. In the context of real world applications, it may be interesting to figure out what is the fastest rate at which a variable may change. This can be measured by computing the maximum rate of change for a variable.

Let us assume a function y = f(x). We know that the rate of change at any x for this function is defined as dy/dx. Let the rate of change be expressed by a function g(x) as follows:

g(x) = dy/dx

To measure the maximum rate of change, we essentially need to compute the maxima for g(x).

The condition for maxima of a function g(x) is that first derivative of g(x) (i.e. g’(x)) should be zero and the second derivative of g(x) (i.e. g’’(x)) should be negative.

Therefore, condition for the maximum rate of change for function y = f(x) is that:

g’(x) = d^{2}y/dx^{2} = 0, and

g’’(x) = d^{3}y/dx^{3} should be negative.

Graphically, maximum rate of change signifies that the tangent at the corresponding point is the steepest.

Consider the following example:

y = -4x^{3} + 22x^{2} - 5x + 9

Let us find the value of x at which the maximum rate of change happens and compute the corresponding rate of change.

dy/dx = -12x^{2} + 44x - 5

d^{2}y/dx^{2} = -24x + 44

d^{3}y/dx^{3} = -24

We see that the third derivative is negative, hence equating second derivative to zero would yield the point of maximum rate of change for y.

d^{2}y/dx^{2} = -24x + 44 = 0

Therefore, x = 44/24 = 1.83

Hence, maximum rate of change for y occurs at x = 1.83 and the value of maximum rate of change is dy/dx (at x = 1.83).

Maximum rate of change = -12(1.83)^{2} + 44(1.83) – 5 = 33.3

Graphically, this can be represented as:

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