Limits and Continuity

Introduction to Limits

Limits are foundational to calculus, describing the behavior of functions as inputs approach specific values. Understanding limits is essential for defining derivatives and integrals.

• Definition: The limit of a function f(x) as x approaches a value a is the value that f(x) gets closer to as x gets closer to a.

• Notation:

• Indeterminate Forms: Expressions like or require algebraic manipulation to resolve.

• Key Properties:

  • (if denominator ≠ 0)

• Special Limits:

Example: can be simplified by factoring numerator and canceling common terms, yielding a limit of 14.

Differentiation

Differentiation from First Principles

Differentiation measures how a function changes as its input changes. The derivative at a point is the slope of the tangent to the curve at that point.

• Definition: The derivative of f(x) at x = a is

• Example: For , applying first principles yields .

Rules of Differentiation

• Sum Rule:

• Product Rule:

• Quotient Rule:

• Chain Rule:

Example: If , then by the chain rule.

Differentiating Trigonometric, Exponential, and Logarithmic Functions

• Trigonometric:

• Exponential:

• Logarithmic:

Example:

Applications of Differentiation

Slope of a Tangent to a Curve

The derivative at a point gives the slope of the tangent to the curve at that point.

• Formula: If , the slope at is .

• Example: For , at , the slope is .

Finding Maxima, Minima, and Points of Inflection

Critical points occur where the derivative is zero or undefined. The second derivative helps classify these points.

• First Derivative Test: Set to find critical points.

• Second Derivative Test:

  • If , the point is a local minimum.

  • If , the point is a local maximum.

  • If , the point may be a point of inflection.

Example: For , , set to find critical points.

Rates of Change

The derivative represents the instantaneous rate of change of a function with respect to its variable.

• Physical Interpretation: In physics, if is position, then is velocity and is acceleration.

• Example: If , then gives the rate of change of y with respect to t.

Implicit Differentiation

Introduction to Implicit Differentiation

When a function is not given explicitly as y = f(x), but rather as a relationship between x and y, implicit differentiation is used.

• Method: Differentiate both sides of the equation with respect to x, treating y as a function of x (i.e., apply the chain rule to terms involving y).

• Example: For , differentiating both sides gives so .

Summary Table: Key Differentiation Rules

Additional info:

• These notes are based on the Leaving Certificate Higher Level Calculus syllabus, but the content is fully relevant to college-level introductory calculus.

• Topics such as curve sketching, displacement, velocity, and acceleration, as well as further applications of differentiation, are also included in the full notes.