Calculus What is it about?

Calculus It derived from two simple geometric problems: The problem of tangents – find the slope of a tangent to a curve at point P The problem of areas – calculate the area under a graph between lines a and b.

Why? Why are these two questions important? They have far-reaching applications in Science, Engineering and Technology. History starts the beginnings of calculus with Archimedes, later by Fermat and Barrow (who was Newton’s professor at Cambridge!). The Fundamental Theorem of Calculus was put together, independently, by Newton and Leibniz (using Descartes ideas). Mathematicians from various countries, over years of work, created a problem-solving tool of immense power and versatility. Limits of functions weren’t in the start of Calculus, but have been recognized as being highly important in its study. (Added by Cauchy, Weierstrass and others).

Secants to tangents A line on a curve connecting two points is called a secant. If the secant points are brought closer together until they coincide at the same point, then the line becomes a tangent.

Areas under a Curve Trying to find an area under a straight line is easy, using rectangles, triangles and trapezoids. Curves are a different matter, and approximation methods can work well in earlier grades. To find the area “S” in the above diagram, we can make estimations.

Areas under a Curve Drawing two lines, we can determine the areas of 2 rectangles and 2 triangles (roughly) if we had values on the axes. A better method is to draw a series of vertical lines, evenly spaced and find those areas.

Areas under a Curve infinite A better method is to draw a series of vertical boxes, evenly spaced and find those areas. As we make those rectangles smaller and smaller, (the limit that the width approaches zero), the more accurate the area under the curve. The number of rectangles with zero width is ________________. infinite