Introduction to Calculus: Differential Calculus

Introduction to Calculus

• Calculus is a branch of mathematics that deals with continuous change.
• It encompasses two main branches: differential calculus and integral calculus.
• Differential calculus focuses on rates of change, slopes of curves, and tangents.
• Integral calculus focuses on areas under curves, volumes of solids, and accumulation of quantities.

Differential Calculus

• Derivatives: The derivative of a function at a point represents the instantaneous rate of change of the function at that point.
• Concepts:
  • Limit: A fundamental concept in calculus, describing the behavior of a function as its input approaches a certain value.
  • Derivatives of basic functions: The calculus provides formulas for finding the derivatives of common functions, such as polynomials, trigonometric functions, exponential functions, and logarithmic functions.
• Applications:
  • Finding slopes of tangents to curves.
  • Determining maximum and minimum values of functions.
  • Analyzing the rate of change of quantities in various fields (e.g., physics, economics, engineering).
• Techniques:
  • Power rule, product rule, quotient rule, chain rule, and implicit differentiation are common approaches to finding derivatives of various functions.
• Rates of Change: Derivatives allow the calculation of the instantaneous rate of change, a crucial concept for understanding how quantities change over time.
  • Examples: Velocity (derivative of displacement), Acceleration (derivative of velocity), slope (derivative of a function), marginal cost (derivative of total cost function).

Integral Calculus

• Integrals: Integrals are used to find the area under a curve or to accumulate quantities over a range.
• Concepts:
  • Antiderivatives: Finding the function whose derivative is the given function. Used to find indefinite integrals.
  • Definite integrals: Finding the exact area under a curve between two specific points.
• Applications:
  • Calculating areas and volumes of shapes.
  • Determining displacement from velocity, work done by a force.
  • Finding the accumulated effect of a rate (e.g., total cost from marginal cost).

Fundamental Theorem of Calculus

• The Fundamental Theorem of Calculus connects differentiation and integration.
• It states a critical relationship between derivatives and integrals.
• This theorem establishes the relationship between the two core concepts of calculus.
  • Part 1: Describes how to find the definite integral using antiderivatives.
  • Part 2: Connects the derivative of an integral to the original function.

Applications of Calculus

• Physics: Analysing motion, forces, and energy.
• Economics: Calculating marginal cost, marginal revenue, and optimal production levels.
• Engineering: Optimizing designs, finding areas for construction.
• Computer Science: Curve fitting, image processing.
• Other Fields: Calculus is applicable to various other fields, highlighting its broad applicability. Examples include biology (population growth models), finance (modeling financial instruments), and many more science and technology applications.

Limits and Continuity

• Limits: The limit of a function describes its behavior as the input approaches a certain value. Understanding limits is essential for the development of derivative concepts and in calculating areas via integration.
• Continuity: A function is continuous when there are no breaks or jumps in its graph, so the graph can be drawn without lifting the pen. Continuity is often a requirement for applying calculus concepts.

This quiz covers the fundamentals of differential calculus, including key concepts such as derivatives, limits, and their applications. Understand how to find slopes of curves and determine maximum and minimum values using calculus principles.