Continuity And Differentiability - An Introduction

Continuity and Differentiability - An introduction

Definition of continuity: A function is said to be continuous at a point if the limit of the function at that point exists
Types of discontinuities: Removable, jump, and infinite discontinuity
Definition of differentiability: A function is said to be differentiable at a point if the derivative of the function exists at that point
Differentiable implies continuous, but continuous does not imply differentiable
Common examples of continuous and differentiable functions: Polynomial functions, exponential functions, and trigonometric functions
Common examples of continuous and differentiable functions: Polynomial functions, exponential functions, and trigonometric functions2
Common examples of continuous and differentiable functions: Polynomial functions, exponential functions, and trigonometric functions3
Common examples of continuous and differentiable functions: Polynomial functions, exponential functions, and trigonometric functions4
Common examples of continuous and differentiable functions: Polynomial functions, exponential functions, and trigonometric functions5
Common examples of continuous and differentiable functions: Polynomial functions, exponential functions, and trigonometric functions6
Common examples of continuous and differentiable functions: Polynomial functions, exponential functions, and trigonometric functions7

The Derivative

The derivative of a function measures how the function changes as its input changes
Definition of the derivative: The derivative of a function f(x) at a point x = a is defined by the limit: $\lim_{h\to 0} \frac{f(a + h) - f(a)}{h}$
Notation for derivative: $f'(x)$ , $f\'(x)$ , or $\frac{df}{dx}$

Differentiation Rules

Constant rule: The derivative of a constant c is 0: $\frac{d}{dx}(c) = 0$
Power rule: If $\frac{d}{dx}(x^n)$ exists, then $\frac{d}{dx}(x^n) = n x^{n-1}$ for all real numbers n
Sum/Difference rule: If $\frac{d}{dx}(f(x) \pm g(x))$ exist, then $\frac{d}{dx}(f(x) \pm g(x)) = \frac{df}{dx}(f(x)) \pm \frac{dg}{dx}(g(x))$

Chain Rule

The chain rule allows us to differentiate composite functions
Statement of the chain rule: If g(x) is differentiable at x=a and f(x) is differentiable at g(a), then the composite function (f ∘ g)(x) is differentiable at x=a and its derivative is given by: $\frac{d}{dx}(f(g(x)))=\frac{df}{dg}(g)\cdot\frac{dg}{dx}(x)$

Examples

Example 1: Differentiate $f(x) = x^2 + 3x + 5$
Example 2: Differentiate $f(x) = \sin(x)$
Example 3: Find the derivative of the function $f(x) = e^{2x}$
Example 4: Use the chain rule to differentiate $g(x) = \sin(x^2)$

Higher Derivatives

The second derivative of a function is the derivative of its first derivative
Notation for second derivative: $f''(x)$ , $f\'\'(x)$ , or $\frac{d^2f}{dx^2}$
Higher derivatives can also be defined by taking successive derivatives

Trigonometric Functions

Definitions of trigonometric functions: sine, cosine, tangent, cosecant, secant, and cotangent
Trig identities: Pythagorean identities, double angle identities, and reciprocal identities
Derivatives of trigonometric functions:
- $\frac{d}{dx}(\sin(x)) = \cos(x)$
- $\frac{d}{dx}(\cos(x)) = -\sin(x)$
- $\frac{d}{dx}(\tan(x)) = \sec^2(x)$
- $\frac{d}{dx}(\csc(x)) = -\csc(x) \cot(x)$
- $\frac{d}{dx}(\sec(x)) = \sec(x) \tan(x)$
- $\frac{d}{dx}(\cot(x)) = -\csc^2(x)$

Limits and Continuity

Definition of a limit: A function f(x) approaches a limit L as x approaches a point c, denoted by $\lim_{x\to c} f(x) = L$
Basic properties of limits: Sum/difference, product, quotient, and power rules
Definition of continuity: A function f(x) is continuous at a point c if:
- $\lim_{x\to c} f(x)$ exists
- $f(c)$ is defined
Types of discontinuities: Removable, jump, and infinite discontinuity

Continuity and Differentiability - An introduction

Definition of continuity: A function is said to be continuous at a point if the limit of the function at that point exists.
Types of discontinuities: Removable, jump, and infinite discontinuity.
Definition of differentiability: A function is said to be differentiable at a point if the derivative of the function exists at that point.
Differentiable implies continuous, but continuous does not imply differentiable.
Common examples of continuous and differentiable functions: Polynomial functions, exponential functions, and trigonometric functions.

The Derivative

The derivative of a function measures how the function changes as its input changes.
Definition of the derivative: The derivative of a function f(x) at a point x = a is defined by the limit: $\lim_{h\to 0} \frac{f(a + h) - f(a)}{h}$ .
Notation for derivative: $f'(x)$ , $f\'(x)$ , or $\frac{df}{dx}$ .

Differentiation Rules

Constant rule: The derivative of a constant c is 0: $\frac{d}{dx}(c) = 0$ .
Power rule: If $\frac{d}{dx}(x^n)$ exists, then $\frac{d}{dx}(x^n) = n x^{n-1}$ for all real numbers n.
Sum/Difference rule: If $\frac{d}{dx}(f(x) \pm g(x))$ exist, then $\frac{d}{dx}(f(x) \pm g(x)) = \frac{df}{dx}(f(x)) \pm \frac{dg}{dx}(g(x))$ .

Chain Rule

The chain rule allows us to differentiate composite functions.
Statement of the chain rule: If g(x) is differentiable at x=a and f(x) is differentiable at g(a), then the composite function (f ∘ g)(x) is differentiable at x=a and its derivative is given by: $\frac{d}{dx}(f(g(x)))=\frac{df}{dg}(g)\cdot\frac{dg}{dx}(x)$ .

Examples

Example 1: Differentiate $f(x) = x^2 + 3x + 5$ .
Example 2: Differentiate $f(x) = \sin(x)$ .
Example 3: Find the derivative of the function $f(x) = e^{2x}$ .
Example 4: Use the chain rule to differentiate $g(x) = \sin(x^2)$ .

Higher Derivatives

The second derivative of a function is the derivative of its first derivative.
Notation for second derivative: $f''(x)$ , $f\'\'(x)$ , or $\frac{d^2f}{dx^2}$ .
Higher derivatives can also be defined by taking successive derivatives.

Trigonometric Functions

Definitions of trigonometric functions: sine, cosine, tangent, cosecant, secant, and cotangent.
Trig identities: Pythagorean identities, double angle identities, and reciprocal identities.
Derivatives of trigonometric functions:
- $\frac{d}{dx}(\sin(x)) = \cos(x)$
- $\frac{d}{dx}(\cos(x)) = -\sin(x)$
- $\frac{d}{dx}(\tan(x)) = \sec^2(x)$
- $\frac{d}{dx}(\csc(x)) = -\csc(x) \cot(x)$
- $\frac{d}{dx}(\sec(x)) = \sec(x) \tan(x)$
- $\frac{d}{dx}(\cot(x)) = -\csc^2(x)$

Limits and Continuity

Definition of a limit: A function f(x) approaches a limit L as x approaches a point c, denoted by $\lim_{x\to c} f(x) = L$ .
Basic properties of limits: Sum/difference, product, quotient, and power rules.
Definition of continuity: A function f(x) is continuous at a point c if:
- $\lim_{x\to c} f(x)$ exists
- $f(c)$ is defined.

Types of discontinuities

Removable discontinuity: A point at which a function is undefined, but the limit as x approaches that point exists and can be made equal to the value obtained by defining the function at that point.
Jump discontinuity: A point at which the function has a finite jump in value.
Infinite discontinuity: A point at which the limit of the function as x approaches the point is either infinite or does not exist.

Example: Removable Discontinuity

The function $\frac{x^2 - 4}{x - 2}$ has a removable discontinuity at x = 2.
By simplifying the expression, we can rewrite the function as f(x) = x + 2.
The limit of f(x) as x approaches 2 is equal to the value of the function f at x = 2, which