Differentiation, derivatives and Taylor approximations: Tangent line
Tangent line and slope function
We have already seen that for a smooth function \(f\) the difference quotient can be used to approximate the slope of the graph at a fixed point. If we connect the end points \(\bigl(a,f(a)\bigr)\) and \(\bigl(a+{\vartriangle}t,f(a+{\vartriangle}t)\bigr)\) on the interval \([a, a+{\vartriangle}t]\) with a straight line and decrease the length \({\vartriangle}t\) of the interval, we see the connecting line approaching a certain line tangent to the graph of \(f\) at the point \(\bigl(a,f(a)\bigr)\). Play with the interactive diagram in the example below.
The interactive diagram on the right-hand side shows the graph of the function \(f(t)= \tfrac{1}{3}e^t\) with two points on the graph. The bottom point \(A=\bigl(t_A,f(t_A)\bigr)\) on the graph can be move freely by dragging the magenta coloured point on the horizontal axis and the top point \(B=\bigl(t_B,f(t_B)\bigr)\) on the graph is obtained by choosing \(t_B=t_A+{\vartriangle}t\) as horizontal change \({\vartriangle}t\). The horizontal change \({\vartriangle}t\) is therefore: \[{\vartriangle}t =t_B-t_A\] The corresponding vertical change is: \[\begin{aligned} {\vartriangle}f&=f(t_B) -f(t_A)\\[0.2cm] &= f(t_A+{\vartriangle}t)-f(t_A)\end{aligned}\] The average rate of change \(\frac{{\vartriangle}f}{{\vartriangle}t}\) over the interval \([t_A,t_B]\) depends in this example on the choice of the point \(A\) and the interval (or if you prefer the horizontal change \({\vartriangle}t\)); just move the slider or the point \(A\) to observe this. As you go towards a smaller horizontal change \({\vartriangle}t\) with a fixed choice of the point \(A\) the average rate of change \(\frac{{\vartriangle}f}{{\vartriangle}t}\) goes towards a constant. The average rate of change over the interval \([t_A,t_B]\) is equal to the slope of the line connecting the two points \(A\) and \(B\). When you bring the points closer together by decreasing \({\vartriangle}t\), you see this connecting line approaching a fixed line with a certain slope.
This brings us to the concept of a tangent line. This is a straight line through a point on the graph of the function \(f\) with a slope equal to the slope of the graph of \(f\) 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 \(\ell\) tangent to the graph of a function \(f\) at a point \(P\) is called a tangent line. In the diagram on the right-hand side, the graph of the function \(f(t)=t^5-5t^3-t^2+4t+2\) is drawn, a freely movable point \(P\) is placed on the graph of \(f\), and the tangent line \(\ell\) is drawn at the point \(P\). The slope of the line \(\ell\) is always equal to the slope of the graph of \(f\) at the point \(P\).
The point \(P\) 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 \(f\) 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 \(P'\) on the slope graph corresponds to the tangent line at the point \(P\) on the graph at the left-hand side. When you move \(P\), the tangent line to the graph changes and takes on a different slope. This slope is the vertical coordinate of the point \(P'\) and in this way the slope graph on the right-hand side of the interactive diagram is created.
