Derivative in Real Life Context

Note: This article is the second part of the derivative concept series. The first part is  The Algebraic and Geometric Meaning of Derivative and the third part is Derivative and the Maximum Area Problem.

If we change the labels of the Figure 2 in the The Algebraic and Geometric Meaning of Derivative article – its x-axis to time and the y-axis to distance – the graph of the secant line is the difference in distance y_2 - y_1 =  100 km – 50 km over the difference  in timex_2 - x_1 = 10:00 – 8:00.  This is shown in the figure below.  This is equivalent to 50 km/2 hrs or 25 km/hr. From the computation above, it is clear that the interpretation of the slope of the secant line is the total distance over the total time or the average speed.

On the other hand, the slope of the tangent line is the speed of the bicycle at exactly 4 o’clock. At exactly 4:00 o’clock the bicycle was traveling 50 km per hour, a lot faster than its average speed. Now, this is reasonable because in real life, the speed of travel is not always constant.  The slope of the tangent line or the speed of travel at a particular point is called the instantaneous speed.

We have also observed from the previous article that as Q approaches P, the secant line’s approximation of the tangent line becomes better and better. This means that as Q approaches P, the average speed becomes closer and closer to the instantaneous speed at P.

The Algebraic and Geometric Meaning of Derivative

Note: This is the first part of the Derivative Concept Series. The second part is Derivative in Real Life Context and the third part is Derivative and the Maximum Area Problem.

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If we want to get the slope of a line, we need two points. Suppose the points have coordinates (x_1,y_1) and (x_2,y_2), we have learned that the slope is described by the formula \displaystyle\frac{y_2-y_1}{x_2-x_1}.

In Figure 1, we have line l tangent to the function f at point P where the coordinates of P are (x,f(x)).  The problem that gave birth to calculus is getting the slope of this tangent line. There is, however, a problem.  We need two points to compute for the slope but we have only one point.

Note that the word tangent in this problem is different from the definition of tangent on a circle because it is clear that line l will intersect the graph in more than one point.

Figure 1 - Line l tangent to the function f at point P.

Using the concept of limits we can remedy this problem. First, we create point Q with x coordinate h units to the right of the x-coordinate of P. We then draw line PQ, a secant line to the function f.

Figure 2 - A secant line is drawn through P.

In effect, the coordinates of Q would be (x+h, f(x+h)) and it is clear that the slope of the secant line PQ is described by the formula

\displaystyle\frac{f(x+h)-f(x)}{x+h - x} = \frac{f(x+h)-f(x)}{h}

If we want to approximate the slope of the tangent line, it is reasonable that we move Q towards P with P fixed. Click here to explore the diagram above using GeoGebra.

From the GeoGebra exploration above, if we move Q towards P, we observe the following:

1.)    The value of h approaches 0.

2.)    The inclination of the secant line approaches the inclination of the tangent line.

3.)    The slope of the secant line approaches the slope of the tangent line.

4.)    If point Q coincides with point P, then the slope of the secant line and is equal to the slope of the tangent line.

If we let m be the slope of the secant line andf'(x) be the slope of the tangent line, focusing on observations 1 and 4, we can say the following equivalent statements:

  • The limit of the slope of the secant line m as Q approaches P is equal to f'(x).
  • The limit of the slope of the secant line m as h approaches 0 is equal to f'(x).
  • The limit of \displaystyle \frac{f(x+h)-f(x)}{h} as h approaches 0 is equal to f'(x).

Using the limit notation, we can say that

f'(x) = \lim_{h \to 0} \displaystyle\frac{f(x+h)-f(x)}{h}

From the above discussion, we can see that the derivative of a function at  a particular point is the slope of the line tangent to that function at that particular point.

In the next post, we will discuss the meaning of derivative in real life situations.

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