As a review of several important concepts and as an introduction to piecewise functions we are going to study in this page a very simple case: linear functions whose domain is restricted to an open interval. The graph of these functions is a segment.
This is the formula we usually use:
In this formula, m is the slope and c is called the y-intercept.
Using the function notation:
The important concept of derivative function is, in this case, very very simple because the derivative of a linear function is a constant, the slope.
We use these notations:
When the domain of a linear function is restricted to an open interval, the resulting derivative is a step function (with only one piece, a constant horizontal segment):
We can consider vertical translations of a linear function f
The slope of these functions is the same as the slope of f. Then the derivative is the same
In the next applet you can translate up and down (dragging the blue big dot) the function but the derivative (the slope) is the same. Very simple idea.
Now we can think in a similar question from a different point of view.
We start with a constant function (the graph is an horizontal segment)
and we look for functions F such as their derivative are the original constant function
In this simple case the answer is simple
You can play with the following applet to see this idea:
There are a family of functions. Each of these functions is a vertical translation of a linear function. This translation depends on the constant C.
We call these functions antiderivatives or integrals of the function f and we can use this notation
In this context, C is called the constant of integration.
MORE ABOUT ANTIDERIVATIVES
Why we use the word 'integral' if we are talking about derivatives?
'Integral' is related with 'summation' (in general, infinite summation), for example, with the idea of area.
We can see the integral of a constant function as the area of a rectangle (positive or negative).
This is a number. But if a if fixed and b is variable (then, instead of b we can use x as we usually do when talking about variables) we get a function
that is a linear function. The same result that we got when we were looking for an antiderivative of a constant function.
You can play with this basic idea of area with the next applet.
If we move (right and left) the value of a we get the same result as before: a vertical translation.
What happens if we combine these two operations: integration and derivative (in this very simple case)? We get the Fundamental Theorem of Calculus (the simplest case).
If we start with a linear function F(x)
and we take the derivative and then the integral we recover the initial function (up to a constant C)
You can play with the next applet to see this idea:
The next step is to consider a linear function (in general, not a constant function).
The integral function (notice that we need two variables, x and t, for example)
If we remember the formula for the area of a trapezium (US, trapezoid) it is not difficult to calculate (for positive values):
This is a parabola (a quadratic function).
We need some idea of limit to calculate the derivative of a quadratic function. But you can play with the next applet to see that the Fundamental Theorem of Calculus apply:
Tom M. Apostol, Calculus, Second Edition, John Willey and Sons, Inc.
Gilbert Strang, Sums and Differences vs. Integrals and Derivatives, The College Mathematics Journal, January 1990. JSTOR.
Anthony J. Macula, The Point-Slope Formula Leads to the Fundamental Theorem of Calculus, The College Mathematics Journal, Mathematical Association of America, 1995.
Michael Spivak, Calculus, Third Edition, Publish-or-Perish, Inc.
Michael W. Botsko, A Fundamental Theorem of Calculus that Applies to All Riemann Integrable Functions, Mathematics Magazine, 1991.