Arguably, the three central concepts in the theory of differential calculus are that of a function, that of a tangent and that of a limit. Here we introduce functions and tangents.

## Functions

When looking at differential calculus, two good ways to think about functions are via algebraic geometry and interdependent variables. Neither give the proper, abstract, definition of a function, but both give a nice way of thinking about them.

### Algebraic Geometry Approach

Let us set up the plane, $\Pi$. We choose a distinguished point called the origin and a distinguished direction which we call ‘positive $x$‘. Draw a line through the origin in the direction of positive $x$. This is the $x$-axis. Choose a unit distance for the $x$-direction.

Now, perpendicular to the $x$-axis, draw a line through the origin. This is the $y$-axis. By convention positive $y$ is anti-clockwise of positive $x$. Choose a unit distance for the $y$-direction.

This is the plane, $\Pi$:

Now points on the plane can be associated with a pair of numbers $(a,b)$. For example, the point a distance one along the positive $x$ and five along the negative $y$ can be denoted by the coordinates (1,-5):

Similarly, I can take a pair of numbers, say (-1,3), and this corresponds to a point on the plane.

This gives a duality:

points on the plane $\Leftrightarrow$ pairs of numbers

Now consider the completely algebraic objects

$x=3\,,\,\,y=2\,,\,\,y=x\,,\,\,y=x^2\,,\,\,y=x+2\,,\,\,x^2+y^2=1$.

These objects have as solutions pairs of numbers:

$y=2$ : $\{(3,2),(1,2),(2,2),(0.49,2),(\sqrt{2},2),\dots\}$

$y=x$: $\{(0,0),(1,1),(2,2),(1.2,1.2),(\sqrt{5},\sqrt{5}),\dots\}$

$y=x^2$ : $\{(0,0),(-1,1),(2,4),(10,100),(\sqrt{7},7),\dots\}$

$y=x+2$: $\{(0,2),(-1,1),(2,4),(10,12),(11.3,13.3),\dots\}$

$x=2$ : $\{(2,0),(2,1),(2,2),(2,-1.97),(2,\pi+e),\dots\}$

$x^2+y^2=1$ : $\{(1,0),(0,1),(-1,0),(0,-1),(1/\sqrt{2},1/\sqrt{2}),\dots\}$

Now if we plot all the solutions of an equations we get a curve. When I say curve I just mean a set of points on the page.

Here we see the first four curves above plotted. The horizontal line is $y=2$, the line going through the origin is $y=x$, the line that is two units higher is $y=x+2$ while the parabola is $y=x^2$.

The solutions of the equation $x^2+y^1=1$ form a circle when plotted while the equation $x=2$ gives a vertical line.

To play around with this go to Wolfram Alpha and input “plot some equation”. Some nice curves to plot may be found here.

This gives us another duality:

Points on a Curve (Geometry) $\Leftrightarrow$ Solutions of an Equation (Algebra)

This means we can answer geometric questions using algebra and answer algebraic questions using geometry.

### Algebraic Question to be Answered using Geometry

Let $h_i,\,k_i,\,r_i$ be real numbers. Consider the simultaneous equations

$(x-h_1)^2+(y-k_1)^2=r_1^2$

$(y-h_2)^2+(y-k_2)^2=r_2^2.$

This set of simultaneous equations has __, __, __ or ___________ solutions.

Now there is a difference between the first four curves and the second two curves (above). For the first four curves, for each $x$-coordinate, there is only a single $y$-coordinate. This allows the equations of these curves, in principle, to be written in the form:

$y=$ stuff with $x$s.

Usually mathematicians are lazy so rather than writing stuff… we write

$y=f(x)$.

Such curves are the graphs of functions.

Therefore the circle and the vertical line $x=2$ are not the graphs of functions because, in the case of the circle, associated to $x=0$, are two $y$-coordinates: +1 and -1. With the vertical line $x=2$, associated to $x=2$ is every single $y$-coordinate.

The circle is, however, the graphs of two functions glued together:

$y=+\sqrt{1-x^2}$ and $y=-\sqrt{1-x^2}$.

### Interdependent Variables Approach

Another way to think about functions is as a pair of variables $(x,y)$ such that $y$ depends on $x$ and we say:

$y$ depends on $x$, or

$y$ is a function of $x$, or

$y=f(x)$.

To say that $y$ depends on $x$ is to say that the value of $x$ determines the value of $y$. Equivalently, for each value of $x$, there is a single corresponding value of $y$. $x$ is known as the independent variable and $y$ is known as the dependent variable.

These functions can be graphed by graphing all the points of the form:

$(x_0,y_0)$,

where $y_0$ is the value of $y$ corresponding to $x=x_0$. Therefore we plot the independent variable on the $x$/horizontal axis and the dependent variable on the $y$/vertical axis.

#### Example

The area, $A$, of a square depends on its side-length and we write

$A=A(s)=s^2$.

To plot this function we plot all pairs $(s,s^2)$ as $s$ runs from $s=0$ and up. Note that the $y=x^2$

## Tangents

All of the functions above have one thing in common… when we zoom in on them they look like lines.

For example, consider the function $y= f(t)=\sin t$ plotted over its period $2\pi$:

Take the point $(\pi/6,1/2)\approx (0.5236,0.5)$… let us zoom in on this point:

At the point $(\pi/6,1/2)$, the curve looks just like a straight line. The line that the curve looks like is called the tangent to the curve at $x=\pi/6$ and is the best line approximation to the curve at that point. Here we see it when we zoom out again:

The aim of calculus is to find the slope of these tangents. If we can find the slope of the tangent then we can write down the equation of the tangent.