**The New Calculus Course.**

**Lesson 1: All straight lines have slope.**

**The only geometric object that has slope as an attribute, is the straight line. In Ancient Greece, slope was measured in terms of right (90 degree) angles. After Newton and Leibniz, slope was measured in terms of a trigonometric ratio - the tangent (tan) ratio. The following applet explains the differences:**

**All straight lines have slope.**

**Lesson 2: There is no change in x or y.**

**The rise and run of any straight line are finite differences, not changes in x and y. Neither the inclination angle nor the trigonometric tangent ratio ever change for a given straight line.**

**The coordinates of any given point never change - no such thing as 'instantaneous rate'.**

**Lesson 3: What is a tangent line?**

**A tangent line is a finite line segment that intersects another curved line in one point only, and crosses it nowhere. The slope of this tangent line, expressed in terms of variables, is called the general derivative, and we will discuss it in subsequent lessons. The derivative is a special kind of slope because it pertains to a tangent line, rather than just any straight line which has a slope.**

**Webster dictionary has this definition:**

**meeting a curve or surface in a single point if a sufficiently small interval is considered.****First known use: 1582**

**This was the understanding that both Newton and Leibniz had. Given these facts, it's not possible for any straight line to be tangent to any other straight line. If Newton's understanding were incorrect, even his famous root approximation method would fail to work. The following applet demonstrates tangent lines:**

**Tangent lines never cross a curve.**

**Lesson 4: What is a secant line and how do we find its slope?**

**A secant line is a finite line segment whose endpoints lie on a given curve. It is possible for a secant line to intersect the curve in more points, besides the end two points of the secant line. The slope of a secant line is found from its endpoints.**

**Secant lines and their slopes.**

**Lesson 5: Secant and tangent lines.**

**If secant lines are parallel to each other over a given interval, then there is a unique tangent line that is parallel to all these secant lines in the same interval. This will be proved in a later lesson. For a demonstration, run the following applet.**

**Secant and tangent lines explained.**

**Lesson 6: The derivative: finding the slope of a tangent line.**

**It is easy to find the slope of any tangent line given that we know a parallel secant line. The following applet explains the concept. Pay careful attention to the notes.**

**Slope of secant lines parallel to a given tangent.**

**Lesson 7: What is a natural average?**

A natural average (arithmetic mean) is a special kind of average that applies to continuous and smooth curves. In the applet that follows, this concept is clearly explained. Be sure to click on the check-box called "Show quadrature in yellow shading". The yellow shading is the area given as a rectangle. Move the green points and observe what happens to the blue line lengths.

A natural average (arithmetic mean) is a special kind of average that applies to continuous and smooth curves. In the applet that follows, this concept is clearly explained. Be sure to click on the check-box called "Show quadrature in yellow shading". The yellow shading is the area given as a rectangle. Move the green points and observe what happens to the blue line lengths.

**Mastery of this concept is essential in understanding the mean value theorem which will be proved in the final lesson.**

**Natural averages explained.**

**Lesson 8: Area as a product of natural averages.**

**The Ancient Greeks defined areas in terms of plane numbers (Book VII, Def. 16). The sides are natural averages that correspond respectively to the average length (arithmetic mean) of all the vertical lines and the average length of all the horizontal lines in a given rectangle. The following applet explains:**

**Area is the product of two averages.**

**Lesson 9: The mean value theorem.**

**The mean value theorem (mvt) is one of the least well understood theorems in calculus. Until the New Calculus was discovered, there was no easy way to prove it. The Encyclopeadia Britannica has this to say:**

*Although the mean-value theorem seems obvious geometrically, proving the result without appeal to diagrams involves a deep examination of the properties of real numbers and continuous functions. (*

**Article on mean value theorem**)**After the arrival of the New Calculus, the Britannica entry is no longer correct. In fact, it was never correct, because real numbers do not exist. The mvt works because of the property of continuous and smooth functions, that have the attribute of natural average (mean). Through the mvt, we also realise a connection between the derivative and the integral.**

**The following applet explains the meaning of the mean value theorem and demonstrates its proof:**

**Proof of the mean value theorem.**

**Conclusion:**

**To learn much more and acquire a deeper understanding, you should study the full lessons (pdf files) at the main New Calculus Course Site.**

**There are also many other interesting and useful applets available for download here.**

**The New Calculus is also available in German here: Die Neue Analysis**

**And also in Chinese here:**

**新微积分**

**Please report any errors and/or typos you may notice in the applets. Use Contact Tab.**

(C) John Gabriel, 2014

(C) John Gabriel, 2014

**All rights reserved.**