Advance Engineering Maths: Function of real Variables

Advance Engineering Maths: Function of real Variables

Functions of Real Variables – Advanced Engineering Mathematics

In Advanced Engineering Mathematics, the study of functions of real variables is fundamental. It deals with real-valued functions, their properties, limits, continuity, and differentiability.

1. Definition of a Function of a Real Variable

A real function is a rule that assigns a unique real number $f(x)$ to each real number $x$ in its domain.

Mathematical Notation:

$$f:D\to \mathbb{R},x\mapsto f(x)$$

where $D$ is the domain of $f(x)$, and $f(x)$ is the range.

Example:
If f(x)=x2f(x) = x^2f(x)=x2, then for $x=2$, we get $f(2)=4$.

2. Types of Functions

Functions of a real variable can be classified into:

Algebraic Functions – Polynomial, rational, root functions
Trigonometric Functions – $\sin x,\cos x,\tan x$, etc.
Exponential and Logarithmic Functions – ex,ln⁡xe^x, \ln xex,lnx
Piecewise Functions – Defined in different intervals
Even & Odd Functions –

• Even: $f(-x)=f(x)$ (e.g., x2x^2x2)
• Odd: $f(-x)=-f(x)$ (e.g., x3x^3x3)

3. Important Properties of Functions

(i) Domain and Range

• Domain: Set of all values for which $f(x)$ is defined.
• Range: Set of all possible values of $f(x)$.

Example:
For f(x)=xf(x) = \sqrt{x}f(x)=x​,

• Domain: $x\ge 0$ (since square root of a negative number is not real).
• Range: $y\ge 0$.

(ii) Limit of a Function

The limit of $f(x)$ as $x\to a$ is written as:

lim⁡x→af(x)=L\lim_{x \to a} f(x) = Lx→alim​f(x)=L

Example:

lim⁡x→2(3x+5)=3(2)+5=11\lim_{x \to 2} (3x + 5) = 3(2) + 5 = 11x→2lim​(3x+5)=3(2)+5=11

(iii) Continuity

A function is continuous at $x=a$ if:
lim⁡x→a−f(x)=lim⁡x→a+f(x)\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x)limx→a−​f(x)=limx→a+​f(x)
$f(a)$ is defined
lim⁡x→af(x)=f(a)\lim_{x \to a} f(x) = f(a)limx→a​f(x)=f(a)

Example:
f(x)=x2f(x) = x^2f(x)=x2 is continuous everywhere, but f(x)=1xf(x) = \frac{1}{x}f(x)=x1​ is not continuous at $x=0$ (because it is undefined).

(iv) Differentiability

A function $f(x)$ is differentiable at $x=a$ if the derivative exists:

f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x) = \lim_{h \to 0} \frac{f(x+h) – f(x)}{h}f′(x)=h→0lim​hf(x+h)−f(x)​

Example:
For f(x)=x2f(x) = x^2f(x)=x2,

f′(x)=2xf'(x) = 2xf′(x)=2x

$f(x)$ is differentiable everywhere.

(v) Mean Value Theorem (MVT)

If $f(x)$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then there exists a point $c$ such that:

f′(c)=f(b)−f(a)b−af'(c) = \frac{f(b) – f(a)}{b – a}f′(c)=b−af(b)−f(a)​

Example:
For f(x)=x2f(x) = x^2f(x)=x2 on $[1,3]$,

f′(c)=9−13−1=4⇒2c=4⇒c=2f'(c) = \frac{9 – 1}{3 – 1} = 4 \Rightarrow 2c = 4 \Rightarrow c = 2f′(c)=3−19−1​=4⇒2c=4⇒c=2

4. Applications in Engineering Mathematics

Signal Processing – Continuous functions in Fourier Analysis
Control Systems – Stability of real functions in system response
Thermodynamics – Behavior of state functions like entropy
Fluid Mechanics – Velocity profile functions

5. Summary Table of Key Formulas

Conclusion

Functions of real variables form the foundation of Advanced Engineering Mathematics, and their concepts are widely applied in engineering fields. Understanding their properties helps in solving real-world problems.

Would you like practice problems or detailed proofs for any theorem?

Advance Engineering Maths: Function of real Variables

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Advanced Engineering Mathematics