Detailed Notes on Limits and Derivatives

12.1 Introduction

Calculus is a branch of mathematics that focuses on studying change. This chapter provides an overview of two fundamental concepts in calculus: limits and derivatives. Understanding these concepts is essential for analyzing various real-world problems and mathematical functions.

12.2 Intuitive Idea of Derivatives

Concept of Derivative

• The derivative of a function represents the instantaneous rate of change of a function at a given point. It can be thought of as the slope of the tangent line to the curve at that point.
• Example: If a body is dropped from a height, the distance travelled can be represented by the function (s = 4.9t^2). By finding the average velocity over smaller and smaller time intervals, we can approximate the instantaneous velocity.
• Average velocity between two points (t_1) and (t_2) is given by: [ v = \frac{s(t_2) - s(t_1)}{t_2 - t_1} ]
• As the time interval approaches zero, the limit gives us the instantaneous velocity.

Visualization

• A graphical representation shows that as we calculate the average velocities for points approaching a given time, we find that they converge to the derivative value.

12.3 Limits

Definition of Limit

• A limit describes the behavior of a function as the input approaches a certain value. Denoted by: [ \lim_{x \to a} f(x) = l ]
• This means that as (x) gets closer to (a), the function (f(x)) approaches (l).

Left-Hand and Right-Hand Limits

• Left-Hand Limit is the value of (f(x)) as (x) approaches (a) from the left (denoted (\lim_{x \to a^-} f(x))).
• Right-Hand Limit is the value of (f(x)) as (x) approaches from the right (denoted (\lim_{x \to a^+} f(x))).
• If both limits exist and are equal, then the limit exists at (a): [ \lim_{x \to a} f(x) = l ]

Examples of Limits

1. Polynomials: For a polynomial function like (f(x) = x^2), as (x) approaches 0, (f(0)) is also 0.
2. Rational Functions: For the function (h(x) = \frac{x^2 - 4}{x - 2}), limits need careful handling as (x) approaches 2 (resulting in an indeterminate form).
3. Trigonometric Limits: Examples involving trigonometric functions will illustrate how limits behave near specific angles.

Algebra of Limits

1. (\lim[f(x) \pm g(x)] = \lim f(x) \pm \lim g(x))
2. (\lim[f(x) \times g(x)] = \lim f(x) \times \lim g(x))
3. Quotients and constants follow similar rules, provided the limits exist.

12.4 Derivatives

Definition of the Derivative

• The derivative of a function at a point gives the slope of the tangent line at that point. It is mathematically defined as: [ f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} ]
• This definition captures how a function changes at any point.

Examples

1. For (f(x) = 3x), the derivative is constant: (f'(x) = 3).
2. For (f(x) = x^2), using the limit definition leads to (f'(x) = 2x).
3. Special trigonometric functions like (\sin(x)) and (\cos(x)) also have defined derivatives based on their behavior as defined in calculus rules.

12.5.1 Rules for Finding Derivatives

Algebra of Derivatives

1. The derivative of a sum: ( (u + v)' = u' + v' )
2. The derivative of a product: ( (uv)' = u'v + uv' )
3. The derivative of a quotient: [ \left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2},\text{ where }v eq0 ]

Standard Derivatives

• Basic functions with known derivatives are helpful in calculations:

1. (\frac{d}{dx}(x^n) = nx^{n-1})
2. (\frac{d}{dx}(sinx) = cosx)
3. (\frac{d}{dx}(cosx) = -sinx)

Historical Context

• The development of calculus is attributed to Isaac Newton and G.W. Leibnitz in the 17th century. Their independent work laid the groundwork that allows calculus to be applied in various fields today, such as physics and economics.

• Calculus studies the rate of change and motion.
• A derivative measures the instantaneous rate of change of a function.
• Understanding limits is essential to define derivatives rigorously.
• The limit of a function describes its behavior as the input approaches a specific value.
• Left-hand and right-hand limits help determine a function's limit at a point.
• Derivatives have standard formulas for polynomials and trigonometric functions.
• Theorems like the sum, product, and quotient rules simplify finding derivatives.
• Historical figures in calculus include Newton and Leibnitz.