As we know, the relationship between an independent variable and a dependent variable is denoted by
As we also know, the geometrical interpretation of this relation takes the form of a curve in an plane as illustrated in Figure 4.
We know how to calculate a value of given a value of . We can either do this graphically (which is inaccurate) or else use the function itself. So, at an value of the corresponding value is where
Let us examine the curve in the neighbourhood of the point . There are two important constructions of interest
- the tangent line at
- the normal line at
These are shown in Figure 5.
We note the geometrically obvious fact: the tangent and normal lines at any given point on a curve are perpendicular to each other.
The curve is drawn below. On this graph draw the tangent line and the normal line at the point :
From your graph, estimate the values of and in degrees. (You will need a protractor.)
Returning to the curve we know, from the geometrical interpretation of the derivative that
(the notation means evaluate at the value )
Here is the angle the tangent line to the curve makes with the positive -axis. This is highlighted in Figure 6: