Introduction to Calculus: Differential Calculus

Introduction to Calculus

• Calculus is a branch of mathematics focused on continuous change.
• It deals with concepts of limits, derivatives, and integrals.
• Two main branches: differential calculus and integral calculus.

Differential Calculus

• Focuses on rates of change.
• Key concept: derivative.
• The derivative of a function at a point represents the instantaneous rate of change of the function at that point.
• Geometrically, the derivative is the slope of the tangent line to the function at a given point.
• Fundamental Theorem of Calculus: Links differential and integral calculus.
• Techniques for calculating derivatives:
  • Power rule: d(xⁿ)/dx = nx^n-1
  • Product rule: d(uv)/dx = u dv/dx + v du/dx
  • Quotient rule: d(u/v)/dx = (v du/dx - u dv/dx)/v²
  • Chain rule: d(f(g(x)))/dx = f'(g(x)) * g'(x)
• Applications of Derivatives:
  • Finding maximum and minimum values (critical points).
  • Determining the concavity and points of inflection of a function.
  • Solving optimization problems.
  • Analyzing motion (velocity and acceleration).

Integral Calculus

• Focuses on accumulation of quantities.
• Key concept: integral.
• The integral of a function represents the area under the curve of the function over a given interval.
• Indefinite Integral: Represents the family of all antiderivatives of a function.
• Definite Integral: Gives the numerical value of the area under the curve between two limits.
• Fundamental Theorem of Calculus (Part 2): Integrals and derivatives are inverse operations.
• Techniques for evaluating integrals:
  • Substitution method.
  • Integration by parts.
  • Partial fraction decomposition.
• Applications of Integrals:
  • Calculating areas and volumes.
  • Finding displacement and distance traveled.
  • Solving problems related to rates of growth.
  • Determining the average value of a function.

Limits

• Concept of Limit: The limit of a function as x approaches a certain value represents the value that the function approaches as x gets closer and closer to that value.
• Formal Definition: For a function f(x) to have a limit L as x approaches a, for any ε > 0, there exists a δ > 0 such that if 0 < |x-a| < δ, then |f(x) - L| < ε.
• One-sided Limits: Limits as x approaches a from the left or right.
• Rules of Limits: Algebraic operations with limits.

Functions

• Defining a Function: A relationship between a set of inputs (domain) and a set of outputs (range) where each input has exactly one output.
• Types of Functions: Polynomial, rational, exponential, logarithmic, trigonometric, etc.
• Important Properties of Functions: Continuous, differentiable, increasing, decreasing, etc.
• Graphing Functions: Understanding the shape of a function's graph provides insight into its behavior.

Notation

• Common Notation: Understanding the different symbols used for derivatives and integrals.
  • f'(x) = derivative of f(x)
  • f''(x) = second derivative of f(x)
  • ∫f(x)dx = integral of f(x)
  • ∫_a^bf(x)dx = definite integral of f(x) from a to b

This quiz covers the fundamentals of differential calculus, focusing on the concepts of limits, derivatives, and their applications. You will learn about the key techniques for calculating derivatives, such as the power rule and the fundamental theorem of calculus. Test your understanding of rates of change and critical points.