Calculus

The word Calculus originated from Latin, meaning "Little Pebble". In the ancient times, pebbles were used to work out sums, hence the adoption of the word.

Who invented Calculus?
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The share of the credits goes to Sir Issac Newton (British) newton.jpeg, Gottfried Wilhelm Leibniz (German)leibniz.jpeg, and Joseph Louis Lagrange (Italian)lagrange.jpeg.

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Gradient

As learnt in Coordinate Geometry, gradient of a straight line is defined as rise/run.
line_graph.jpggrad.jpg, where m reps the gradient.
What about the gradient of a curve?
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In a curve, the gradient is always changing. It can go from positive, to negative and even become zero at some point. The point at which m = 0 is called the stationary point.

To find the gradient of a particular point on a curve, we could draw a tangent at that point and calculate its gradient (which could lead to an inaccurate answer), or we could use the gradient function.


To find the gradient at P, we start by identifying a point Q (close to P) and finding the gradient of the line PQ. Next, we move to point R (closer to P) and find the gradient of line PR. Finally, we move to a point S (even closer to P) and find the gradient of PS. Now, by examining the various lines and gradients, we see that as Q approaches P, the lines approach the tangent at P, which is the gradient we want AND the values of the gradients approach 4!

This method of deriving the gradient is called "1st Principles".


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Here is an example of how to obtain the gradient of a curve using 1st Principles:




How do we get the general formula to obtain the gradient of any function?

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From the pattern, we can deduce the general formula:

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So, basically, when we differentiate a function, we are finding it's gradient. According to the power rule above, when we differentiate a function x^n, we "bring down" the power as the coefficient of the gradient and reduce it's resulting power by 1.

Graphically, when we differentiate a parabola, we obtain a straight line. The following will explain this in greater detail:
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Try these Geogebra applet to trace the gradient of a quadratic & cubic graphs:

Drag Point A to trace the gradient function.

This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com
This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com


You can download the above applets here:



More on Notations:

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Before moving on, it will be useful to go through lessons 1 - 4 of Chapter 2: Finding Derivatives at
http://www.calculus-help.com/tutorials/

The following summarises the concepts from Lessons 1 - 4:

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OR, we can simply expand the product before differentiating:
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