Introduction to calculus

Calculus (from Latin calculus, literally "small pebble used for counting on an abacus")is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations. It has two major branches, differential calculus(concerning rates of change and slopes of curves),and integral calculus (concerning accumulation of quantities and the areas under and between curves); these two branches are related to each other by thefundamental theorem of calculus. Both branches make use of the fundamental notions of convergence of infinite sequencesand infinite series to a well-defined limit. Generally, modern calculus is considered to have been developed in the 17th century byIsaac Newton and Gottfried Leibniz. Today, calculus has widespread uses in science,engineering and economics.

Calculus is a part of modern mathematics education. A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study offunctions and limits, broadly calledmathematical analysis. Calculus has historically been called "the calculus ofinfinitesimals", or "infinitesimal calculus".Calculus (plural calculi) is also used for naming some methods of calculation or theories of computation, such aspropositional calculus, calculus of variations,lambda calculus, and process calculus.

Plane analytical geometry

Plane Analytical Geometry

By M Bourne

The curves that we learn about in this chapter are called conic sections. They arise naturally in many situations and are the result of slicing a cone at various angles.

Depending on where we slice our cone, and at what angle, we will either have a straight line, a circle, a parabola, an ellipse or a hyperbola. Of course, we could also get a single point, too.

An interesting application from nature:

Why study analytic geometry?

Science and engineering involves the study of quantities that change relative to each other (for example, distance-time, velocity-time, population-time, force-distance, etc).

It is much easier to understand what is going on in these problems if we draw graphs showing the relationship between the quantities involved.

The study of calculus depends heavily on a clear understanding of functions, graphs, slopes of curves and shapes of curves. For example, in the Differentiation chapter we use graphs to demonstrate relationships between varying quantities.

We begin with some basic definitions of slope, parallel lines and perpendicular lines.

cylinder facts

Cylinder Facts

Notice these interesting things:

• It has a flat base and a
  flat top
• The base is the same as the top
• From base to top the shape stays the same
• It has one curved side
• It is not a polyhedron as it has a curved surface

An object shaped like a cylinder is said to be cylindrical.

Surface Area of a Cylinder

Surface Area = 2 × π × r × (r+h)

Which is made up of:

• Surface Area of One End = π × r²
• Surface Area of Side = 2 × π × r × h

Example: h = 7 and r = 2

Surface Area		= 2 × π × r × (r+h)
		= 2 × π × 2 × (2+7)
		= 2 × π × 2 × 9
		= 36 π
		≈ 113.097

Volume of a Cylinder

Volume = π × r² × h

Just multiply the area of the base by the height of the cylinder:

• Area of the base: π × r²
• Height: h
• Volume = Area × Height = π × r² × h

Example: h = 7 and r = 2

Volume		= π × r2 × h
		= π × 22 × 7
		= 28 π
		≈  87.96

How to remember: Volume = pizza

Imagine you just cooked a pizza.
The radius is "z", and the thickness "a" is the same everywhere ... what is the volume?

Volume equals   pi × z × z × a

(we would normally write "pi" as π, and z × z as z², but you get the idea!)

Volume of a Cone vs Cylinder

The volume formulas for cylinders and cones are very similar:

The volume of a cylinder is:	π × r2 × h
The volume of a cone is:	1 3 π × r2 × h

So a cone's volume is exactly one third ( 1 3 ) of a cylinder's volume.
In future, order your ice creams in cylinders, not cones, you get 3 times as much!

It Doesn't Have to Be Circular

Usually when we say Cylinder we mean a Circular Cylinder, but you can also have Elliptical Cylinders, like this one:

And we can have stranger cylinders!
So long as the cross-section is curved and is the same from one end to the other, then it is a cylinder. But the area and volume calculations will be different than for the above.

More Cylinders

facts about torus

Torus Facts

Notice these interesting things:

• It can be made by revolving a
  small circle (radius r) along a line made
  by a bigger circle (radius R).
• It has no edges or vertices
• It is not a polyhedron

Torus in the Sky.
The Torus is such a beautiful solid,
this one would be fun at the beach !

Surface Area

Surface Area = 4 × π² × R × r

Example: r = 3 and R = 7

Surface Area		= 4 × π2 × R × r
		= 4 × π2 × 7 × 3
		= 4 × π2 × 21
		= 84 × π2
		≈ 829

Volume

Volume = 2 × π² × R × r²

Example: r = 3 and R = 7

Volume		= 2 × π2 × R × r2
		= 2 × π2 × 7 × 32
		= 2 × π2 × 7 × 9
		= 126 π2
		≈ 1244

Note: Area and volume formulas only work when the torus has a hole!

And did you know that Torus was the Latin word for a cushion? (This is not a real roman cushion, just an illustration I made)

When we have more than one torus they are called tori
 

As the small radius (r) gets larger and larger, the torus goes from looking like a Tire to a Donut:

shapes

Cube	Cubes have 6 faces, 12 edges and 8 vertices. All sides on a cube are equal length. All faces are square in shape. A cube is a type of cuboid.
Cuboid	Cuboids have 6 faces, 12 edges and 8 vertices. All the faces on a cuboid are rectangular.
Sphere	Spheres have either 0 or 1 faces, 0 edges and 0 vertices.
Cylinder	Cylinders have either 2 or 3 faces, 0 or 2 edges, and 0 vertices.
Cone	Cones have either 1 or 2 faces, 0 or 1 edges, and 1 apex (which is described by some mathematicians as a vertex).
Triangular Prism	Triangular Prisms have 5 faces, 9 edges, and 6 vertices. The two faces at either end are triangles, and the rest of the faces are rectangular.
Hexagonal Prism	Hexagonal Prisms have 8 faces, 18 edges, and 12 vertices. The two faces at either end are hexagons, and the rest of the faces are rectangular.
Triangular-based Pyramid	Triangular-based pyramids have 4 faces, 6 edges and 4 vertices. The base is a triangle. All of the faces are triangular. If the triangular faces making up the prism are all equilateral, then the shape is also called a Tetrahedron.
Square-based Pyramid	Square based pyramids have 5 faces, 8 edges and 5 vertices The base is a square. All the other faces are triangular.
Hexagonal Pyramid	Hexagonal pyramids have 7 faces, 12 edges, and 7 vertices. The base is a hexagon. All of the other faces are triangular.