Differentiation, derivatives and Taylor approximations: Tangent line
Difference quotient and tangent line
In applications we are often interested in the way how quantities change. Speed is a common concept and we make a distinction between average speed and instantaneous speed. The average speed of a moving object over a certain period of time is the speed of a uniform motion, so that the displacement is the same over the period of time. This average speed may differ from the instantaneous speed, i.e., the speed at a certain point in time.
First two illustrative examples.
In the interactive diagram on the right-hand side is shown the graph of the function \(f(t)=2t-1\) with two points on the graph. The bottom point \(A=\bigl(t_A,f(t_A)\bigr)\) on the line can be moved freely and the top point \(B=\bigl(t_B,f(t_B)\bigr)\) on the line is obtained by choosing \(t_B=t_A+{\vartriangle}t\) for a horizontal change \({\vartriangle}t\). Thus, the horizontal change \({\vartriangle}t\) is: \[{\vartriangle}t =t_B-t_A\] 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 \([t_A,t_B]\) is in this example \[\frac{{\vartriangle}f}{{\vartriangle}t}= 2\]
and does not depend on the choice of the point \(A\) and the interval (or alternatively on the horizontal change \({\vartriangle}t\)); Move the slider or the point \(A\) to observe this.
In the interactive diagram on the right-hand side is shown the graph of the function \(f(t)=f(t)=\begin{cases} t^2 & \text{if }t\ge 0\\ 0 & \text{if }t< 0\end{cases}\) with two points on the graph. The bottom point \(A=\bigl(t_A,f(t_A)\bigr)\) on the graph can be moved freely and the upper point \(B=\bigl(t_B,f(t_B)\bigr)\) on the graph is obtained by choosing \(t_B=t_A+{\vartriangle}t\) for a horizontal change \({\vartriangle}t\). Thus, the horizontal change \({\vartriangle}t\) is: \[{\vartriangle}t =t_B-t_A\] 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 alternatively on the horizontal change \({\vartriangle}t\)); Move the slider or the point \(A\) to observe this.
Let's generalise the concept of average change on an interval use the language of mathematical functions.
We have a function \(f(t)\) over the interval \(a\leq t\leq b\). The average rate of change is defined as the difference quotient \[\frac{{\vartriangle}f}{{\vartriangle}t}=\frac{f(b)-f(a)}{b-a}\] where the quantities \({\vartriangle}f\) and \({\vartriangle}t\) are called differences. Thus the difference quotient depends on the values of the function \(f\) at the end points of the interval \([a,b]\) , the length of the interval and, for example, the starting point \(t=a\) of the interval.
The average rate of change of the function \(f(t)\) over the interval \(a\leq t\leq b\) is the slope of the line through the points \(\bigl(a, f(a)\bigr)\) and \(\bigl(b, f(b)\bigr)\).
Mathcentre video
The Gradient of a Straight-Line Segment (20:46)
