Differentiation, derivatives and Taylor approximations: Tangent line
Tangent line and slope function
We have already seen that for a smooth function the difference quotient can be used to approximate the slope of the graph at a fixed point. If we connect the end points and on the interval with a straight line and decrease the length of the interval, we see the connecting line approaching a certain line tangent to the graph of at the point . Play with the interactive diagram in the example below.
The interactive diagram on the right-hand side shows the graph of the function with two points on the graph. The bottom point on the graph can be move freely by dragging the magenta coloured point on the horizontal axis and the top point on the graph is obtained by choosing as horizontal change . The horizontal change is therefore:
This brings us to the concept of a tangent line. This is a straight line through a point on the graph of the function with a slope equal to the slope of the graph of at that point. A tangent line at a point no longer intersects the graph in the neighbourhood of that point; the tangent line only touches the graph. In the interactive example below you can view the tangent line at any point on the graph of a given function.
Tangent line
A line tangent to the graph of a function at a point is called a tangent line. In the diagram on the right-hand side, the graph of the function is drawn, a freely movable point is placed on the graph of , and the tangent line is drawn at the point . The slope of the line is always equal to the slope of the graph of at the point .
The point can be moved freely in this example and there is always a tangent line at this point. This is what is meant by a smooth function. In mathematical language, the function is then called differentiable. This means that the function is differentiable at any point.
Slope function and slop graph With a given smooth function you can determine the slope of the graph at any point in the domain, i.e., the slope of the tangent line at any point. In this way a new function is created: the slope function. The graph of the slope function is called the slope graph. It is easy to read the behaviour of the function from the slope graph:
- A descending part of the graph of a function has negative slopes, so the slope graph is there below the horizontal axis
- An increasing part of the graph of a function has positive slopes, so the slope graph is there above the horizontal axis
- In a local maximum and minimum of a function, the slope is zero and so the slope graph intersects at such point the horizontal axis.
Below on the right-hand side is shown the slope graph of the function from the previous example: the point on the slope graph corresponds to the tangent line at the point on the graph at the left-hand side. When you move , the tangent line to the graph changes and takes on a different slope. This slope is the vertical coordinate of the point and in this way the slope graph on the right-hand side of the interactive diagram is created.