Introduction to Calculus Course from Coursera

Calculus is a branch of mathematics that studies continuous change. It is divided into two main parts: differential calculus and integral calculus. Differential calculus focuses on rates of change and slopes of curves, while integral calculus deals with the accumulation of quantities and the areas under and between curves.

Calculus was independently developed by Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century. Newton used calculus to formulate his laws of motion and gravitation, while Leibniz developed much of the notation used today.

Skills you will gain

Skills You’ll Gain from Learning Calculus
Learning calculus equips you with a variety of skills that are valuable in numerous fields. Here’s a breakdown of some key skills you’ll develop:
1. Logical Thinking

  • Critical Analysis: Develop the ability to analyze problems logically and systematically.
  • Problem Solving: Enhance your ability to approach complex problems, break them down into simpler parts, and devise strategies to solve them.
  • Proof and Reasoning: Gain proficiency in constructing mathematical proofs and understanding logical arguments.

2. Mathematics Proficiency

  • Mathematical Concepts: Gain a deep understanding of fundamental mathematical concepts such as limits, derivatives, integrals, and functions.
  • Algebraic Manipulation: Improve skills in manipulating algebraic expressions, which is essential for solving calculus problems.
  • Graphical Interpretation: Learn to interpret and draw graphs of functions, which is crucial for understanding the behavior of functions.

3. Calculus-Specific Skills

  • Differentiation: Learn how to find the derivative of a function, which helps in understanding rates of change and slopes of curves.
    • Applications: Apply differentiation to real-world problems, such as calculating velocity and acceleration in physics.
  • Integration: Master techniques for finding the integral of a function, which is useful for calculating areas under curves and accumulated quantities.
    • Applications: Use integration in various fields, such as finding the total distance traveled given a velocity function or determining the area between curves.
  • Limits and Continuity: Understand the concept of limits and how they are used to define continuity, derivatives, and integrals.
    • Applications: Use limits to solve problems involving instantaneous rates of change and to understand the behavior of functions near specific points.

Practical Applications and Fields of Use

  • Physics: Use calculus to describe motion, forces, and energy.
  • Engineering: Apply calculus in the design and analysis of systems and structures, including electrical circuits, fluid dynamics, and mechanical systems.
  • Economics: Model and predict economic behavior, optimize functions related to cost, profit, and production.
  • Biology and Medicine: Model population dynamics, the spread of diseases, and biological processes.
  • Statistics and Data Science: Use calculus in probability theory, statistical inference, and machine learning algorithms.

What you will learn

Course Outline and Learning Objectives
Week 1-2: Precalculus Foundations

  • Key Ideas: Familiarize with fundamental concepts necessary for calculus, such as manipulating equations and understanding elementary functions.
  • Topics Covered:
    • Algebraic manipulation
    • Functions and their properties
    • Graphs of basic functions (linear, quadratic, polynomial, exponential, logarithmic)
  • Objective: Build a strong base in precalculus to ensure readiness for more advanced calculus topics.

Week 3: Introduction to Tangents, Limits, and Derivatives

  • Key Ideas: Develop an understanding of the preliminary methods in calculus, focusing on the concept of tangents, the computation of limits, and the definition of derivatives.
  • Topics Covered:
    • Concept of a tangent line to a curve
    • Definition and computation of limits
    • Formal definition of the derivative
    • Basic differentiation rules
  • Objective: Gain fluency in the foundational methods that lead to the definition and understanding of derivatives.

Week 4: Differential Calculus

  • Key Ideas: Practice and apply methods of differential calculus to solve problems involving rates of change and optimization.
  • Topics Covered:
    • Techniques of differentiation (product rule, quotient rule, chain rule)
    • Applications of derivatives (velocity, acceleration, optimization problems)
    • Higher-order derivatives
  • Objective: Develop proficiency in differential calculus and its applications to real-world problems.

Week 5: Integral Calculus

  • Key Ideas: Learn and practice the methods of integral calculus, focusing on the computation and application of integrals.
  • Topics Covered:
    • Definition of the definite and indefinite integral
    • Fundamental Theorem of Calculus
    • Techniques of integration (substitution, integration by parts)
    • Applications of integrals (area under a curve, volume of a solid of revolution)
  • Objective: Gain the ability to compute integrals and apply integral calculus to various practical scenarios.

Course Goals
By the end of the course, students will:

  • Master Key Threshold Concepts: Develop a deep understanding of the fundamental concepts in calculus, such as limits, derivatives, and integrals.
  • Balance Theory and Application: Achieve a balance between understanding theoretical principles and applying them to solve practical problems.
  • Appreciate Historical Context: Gain insight into the historical development and motivation behind calculus, enhancing their appreciation of its significance.
  • Apply Mathematics in Various Fields: Be prepared to use calculus in science, engineering, and commerce, leveraging their mathematical skills to tackle real-world challenges.