How do you make Calculus easy and enjoyable for kids who are just starting to learn it? Well, I am pondering over this as a niece of mine has just started taking her first calculus lessons. So I am writing this article, especially for her. Calculus is an ubiquitous branch of mathematics having applications in many fields including physics, electrical engineering, mechanical engineering, civil engineering, chemical engineering, biology, economics etc. In fact all the technological advancements that we enjoy today would not have been possible without this important branch of mathematics!
Historically the credit for inventing calculus goes to Sir Isaac Newton and Gottfried Wilhelm Leibniz. There was some work done in this field by earlier people including, the Indian mathematician, Bhāskara of 12th century.
Calculus can be simply defined as study of change. What change? Any change for example, movement is a change. Movement is change of the position (and shape) of an object with time.
Limit: The following example from Wikipedia nicely illustrates Tangent line as a limit of secant lines. The derivative f′(x) of a curve at a point is the slope of the line tangent to that curve at that point. This slope is determined by considering the limiting value of the slopes of secant lines.
Historically the credit for inventing calculus goes to Sir Isaac Newton and Gottfried Wilhelm Leibniz. There was some work done in this field by earlier people including, the Indian mathematician, Bhāskara of 12th century.
Calculus can be simply defined as study of change. What change? Any change for example, movement is a change. Movement is change of the position (and shape) of an object with time.
Limit: The following example from Wikipedia nicely illustrates Tangent line as a limit of secant lines. The derivative f′(x) of a curve at a point is the slope of the line tangent to that curve at that point. This slope is determined by considering the limiting value of the slopes of secant lines.
Here the function involved (drawn in red) is f(x) = x3 − x. The tangent line (in green) which passes through the point (−3/2, −15/8) has a slope of 23/4. Note that the vertical and horizontal scales in this image are different.
Differentiation: When two (or more) related variables change, how does a change in one variable affects the other variable? What is the change rate at any given point? This change rate is defined as a derivative. And the process of obtaining derivative of a function f(x) with respect x is called differentiation.
Integration: Suppose we have an object that moves at a speed (rate of change of distance) that itself changes with time. What will be the cumulative distance traveled by the object over a period of time? Then we have integration that can answer this question.
Here are some of the useful sites to understand the concepts of calculus.
- General introduction to calculus (Wikipedia)
- Definition of the Derivative includes explanation, with illustrations, of the physical and geometrical concepts of the derivative.
- Calculus on Wikibooks has simple and good online tutorials on calculus.
- For advanced calculus students, MIT open course ware has many calculus courses part of its mathematics courses. All these lessons have online and downloadable versions including videos lessons. These courses cover a spectrum of subjects ranging from single variable calculus to multi-variable partial differential equations to mathematical analysis. and its applications
