Rolle's Theorem

Rolle's Theorem:

Let f be a function that satisfies the following three conditions:

1. f is continuous on the closed interval [a, b].

2. f is differentiable on the open interval (a, b).

3. f(a) = f(b).

Then there exists at least one c in the open interval (a, b) such that $f'(c) = 0$.

Lagrange's Mean Value Theorem (Without Proof)

Lagrange's Mean Value Theorem:

Let f be a real-valued function that satisfies the following conditions:

1. f is continuous on the closed interval [a, b].

2. f is differentiable on the open interval (a, b).

Then there exists at least one c in the open interval (a, b) such that

$$f'(c) = \frac{f(b) - f(a)}{b - a}$$

Improper Integrals

Improper Integrals:

Improper integrals are integrals where the integrand or the limits of integration are infinite.

1. Improper Integrals with Infinite Limits:

   Consider the integral $\int_{a}^{\infty} f(x) \, dx$. This is defined as

   $$\int_{a}^{\infty} f(x) \, dx = \lim_{b \to \infty} \int_{a}^{b} f(x) \, dx,$$

   Similarly, for $\int_{-\infty}^{b} f(x) \, dx$ , we have

   $$\int_{-\infty}^{b} f(x) \, dx = \lim_{a \to -\infty} \int_{a}^{b} f(x) \, dx,$$

   For integrals with both limits infinite, we define

   $\int_{-\infty}^{\infty} f(x) \, dx = \lim_{a \to -\infty} \lim_{b \to \infty} \int_{a}^{b} f(x) \, dx,$

2. Improper Integrals with Discontinuous Integrands:

   Consider the integral $\int_{a}^{b} f(x) \, dx$ where f(x) is discontinuous at a point c in (a, b) . We define this as

   $\int_{a}^{b} f(x) \, dx = \int_{a}^{c} f(x) \, dx + \int_{c}^{b} f(x) \, dx,$

   provided both integrals on the right exist as improper integrals.

Beta and Gamma Functions

Beta Function

The Beta function B(x, y) is defined for x > 0 and y > 0 as

$$B(x, y) = \int_{0}^{1} t^{x-1} (1-t)^{y-1} \, dt$$

It can also be expressed in terms of Gamma functions as

$$B(x, y) = \frac{\Gamma(x) \Gamma(y)}{\Gamma(x + y)}$$

Gamma Function

The Gamma function $\Gamma(z)$ is defined for z > 0 as

$$\Gamma(z) = \int_{0}^{\infty} t^{z-1} e^{-t} \, dt$$

The Gamma function satisfies the recurrence relation

$$\Gamma(z+1) = z \Gamma(z)$$

Function of Several Variables

A function of several variables is a rule that assigns a unique value to a set of input variables. For example, a function f of n variables can be written as $f(x_1, x_2, \ldots, x_n)$.

Examples:

• Two variables: $f(x, y) = x^2 + y^2$
• Three variables: $g(x, y, z) = x + y - z$
• General form: $h(x_1, x_2, \ldots, x_n)$

Limits and Continuity (𝛿 - 𝜖 approach)

Limits and continuity are fundamental concepts in calculus and mathematical analysis.

Limits

A limit describes the value that a function (or sequence) approaches as the input (or index) approaches some value. Limits are essential for defining derivatives, integrals, and continuity.

The limit of a function $f(x)$ as x approaches a is L, denoted as:

$$\lim_{x \to a} f(x) = L$$

if for every $\epsilon > 0$ , there exists a $\delta > 0$ such that whenever $0 < |x - a| < \delta$, it follows that $|f(x) - L| < \epsilon$

Continuity

A function is continuous if it does not have any abrupt changes in value, meaning the function’s graph can be drawn without lifting the pen from the paper.

A function $f(x)$ is continuous at a point a if:

1. $\lim_{x \to a} f(x)$ exists.

2. $f(a)$ is defined.

3. $\lim_{x \to a} f(x) = f(a)$

In other words, $f(x)$ is continuous at a if the limit of $f(x)$ as x approaches a equals the value of $f(a)$.

Partial Derivatives

Given a function $f(x, y)$ of two variables, the partial derivative of f with respect to x is denoted by $\frac{\partial f}{\partial x}$ and is defined as:

$$\frac{\partial f}{\partial x} = \lim_{\Delta x \to 0} \frac{f(x + \Delta x, y) - f(x, y)}{\Delta x}$$

Similarly, the partial derivative of f with respect to y is denoted by $\frac{\partial f}{\partial y}$ and is defined as:

$$\frac{\partial f}{\partial y} = \lim_{\Delta y \to 0} \frac{f(x, y + \Delta y) - f(x, y)}{\Delta y}$$

Partial derivatives can be represented in various notations:

• $\frac{\partial f}{\partial x}$ or $f_x$ for the partial derivative with respect to x
• $\frac{\partial f}{\partial y}$ or $f_y$ for the partial derivative with respect to y

Total Derivatives

Euler's theorem (Homogeneous Functions)

Euler's theorem in multivariable calculus relates to homogeneous functions and their partial derivatives. A function $f(x, y)$ is said to be homogeneous of degree n if for all $\lambda \in \mathbb{R}$,

$$f(\lambda x, \lambda y) = \lambda^n f(x, y)$$

Euler's Theorem on Homogeneous Functions

For a homogeneous function f(x, y) of degree n, Euler's theorem states:

$$x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} = n f(x, y)$$

Jacobian

Maxima and Minima

Lagrange's Method of Multipliers

Taylor & Maclaurin's Theorem & Series

Taylor's Theorem and Taylor Series

Taylor's Theorem:

Let f(x) be a function that is (n+1)-times differentiable on an interval containing $x_0$. Then, for each x in that interval, there exists c between $x_0$ and x such that

$f(x) = f(x_0) + f'(x_0)(x - x_0) + \frac{f''(x_0)}{2!}(x - x_0)^2 + \cdots + \frac{f^{(n)}(x_0)}{n!}(x - x_0)^n + R_n(x),$

where the remainder term $R_n(x)$ is given by

$$R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!}(x - x_0)^{n+1}$$

Taylor Series:

The Taylor series of f(x) centered at $x_0$ is the power series

$$\sum_{n=0}^{\infty} \frac{f^{(n)}(x_0)}{n!}(x - x_0)^n$$

The Taylor series converges to f(x) in some interval around $x_0$ if f(x) is analytic (i.e., equal to its Taylor series in that interval).

Examples of Taylor Series:

1. Exponential Function: $$e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} \quad \text{for all x},$$
2. Sine and Cosine Functions: $$\sin(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!} \quad \text{for all x},$$

Maclaurin's Theorem & Series

Maclaurin's Theorem:

Let f(x) be a function that is (n+1)-times differentiable at x = 0. Then, for each x near 0,there exists c between 0 and x such that

$f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \cdots + \frac{f^{(n)}(0)}{n!}x^n + R_n(x),$

where the remainder term $R_n(x)$ is given by

$$R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!}x^{n+1}$$

Maclaurin Series:

The Maclaurin series of f(x) is the Taylor series centered at x = 0, given by

$$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n.$$

The Maclaurin series converges to f(x) in some interval around 0 if f(x) is analytic (i.e., equal to its Maclaurin series in that interval).

Examples of Maclaurin Series:

1. Exponential Function: $$e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} \quad \text{for all } x$$
2. Sine and Cosine Functions: $$\sin(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!} \quad \text{for all x}$$ $$\cos(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} \quad \text{for all x}$$