Tangent line and slope function

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.

GeoGebra

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.

GeoGebra

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.

GeoGebra