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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:

Linear functions:  | matematicasVisuales

In this formula, m is the slope and c is called the y-intercept.

Using the function notation:

Linear functions:  | matematicasVisuales

REVIEW

Polynomial Functions (1): Linear functions
Two points determine a stright line. As a function we call it a linear function. We can see the slope of a line and how we can get the equation of a line through two points. We study also the x-intercept and the y-intercept of a linear equation.

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:

Linear functions: formula | matematicasVisuales

REVIEW

Polynomial functions and derivative (1): Linear functions
The derivative of a lineal function is a constant function.

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):

Linear functions: step function with only one step, a constant horizontal segment | matematicasVisuales

We can consider vertical translations of a linear function f

Linear functions: vertical translations | matematicasVisuales

The slope of these functions is the same as the slope of f. Then the derivative is the same

Linear functions: slopes | matematicasVisuales

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.

Linear functions: translations up and down, the derivative is the same | matematicasVisuales

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)

Linear functions: formula, constant function | matematicasVisuales

and we look for functions F such as their derivative are the original constant function

Linear functions: antiderivative of a constant function | matematicasVisuales

In this simple case the answer is simple

Linear functions: antiderivative | matematicasVisuales

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

Linear functions: antiderivatives or integrals | matematicasVisuales

In this context, C is called the constant of integration.

MORE ABOUT ANTIDERIVATIVES

Polynomial functions and derivative (5): Antidifferentiation
If the derivative of F(x) is f(x), then we say that an indefinite integral of f(x) with respect to x is F(x). We also say that F is an antiderivative or a primitive function of f.

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).

Step functions: The definite integral of a constant function is the area of a rectangle (positive or negative) | matematicasVisuales
Linear functions:  | matematicasVisuales

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

Linear functions: integral function | matematicasVisuales

Linear functions: integral function | matematicasVisuales

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

Linear functions: Fundamental Theorem of Calculus | matematicasVisuales

Then

Linear functions: Fundamental Theorem of Calculus | matematicasVisuales

In general

Linear functions: Fundamental Theorem of Calculus | matematicasVisuales
Linear functions: Fundamental Theorem of Calculus | matematicasVisuales

Linear functions: Fundamental Theorem of Calculus | matematicasVisuales

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)

Linear functions: Fundamental Theorem of Calculus | matematicasVisuales
Linear functions: Fundamental Theorem of Calculus | matematicasVisuales

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)

Linear functions: integral function | matematicasVisuales

If we remember the formula for the area of a trapezium (US, trapezoid) it is not difficult to calculate (for positive values):

Linear functions: integral, area of a trapezium or trapezoid  | matematicasVisuales

This is a parabola (a quadratic function).

Linear functions: integral function, parabola | matematicasVisuales

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:

REFERENCES

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.

MORE LINKS

Continuous Piecewise Linear Functions
A continuous piecewise linear function is defined by several segments or rays connected, without jumps between them.
Non continuous Piecewise Linear Functions
Graphs of these functions are made of disconnected line segments. There are points where a small change in x produces a sudden jump in the value of the function.
Powers with natural exponents (and positive rational exponents)
Power with natural exponents are simple and important functions. Their inverse functions are power with rational exponents (a radical or a nth root)
Polynomial Functions (2): Quadratic functions
Polynomials of degree 2 are quadratic functions. Their graphs are parabolas. To find the x-intercepts we have to solve a quadratic equation. The vertex of a parabola is a maximum of minimum of the function.
Polynomial Functions (3): Cubic functions
Polynomials of degree 3 are cubic functions. A real cubic function always crosses the x-axis at least once.
Polynomial Functions (4): Lagrange interpolating polynomial
We can consider the polynomial function that passes through a series of points of the plane. This is an interpolation problem that is solved here using the Lagrange interpolating polynomial.
Polynomial functions and derivative (1): Linear functions
The derivative of a lineal function is a constant function.
Polynomial functions and derivative (2): Quadratic functions
The derivative of a quadratic function is a linear function, it is to say, a straight line.
Polynomial functions and derivative (3): Cubic functions
The derivative of a cubic function is a quadratic function, a parabola.
Polynomial functions and derivative (4): Lagrange polynomials (General polynomial functions)
Lagrange polynomials are polynomials that pases through n given points. We use Lagrange polynomials to explore a general polynomial function and its derivative.
Polynomial functions and derivative (5): Antidifferentiation
If the derivative of F(x) is f(x), then we say that an indefinite integral of f(x) with respect to x is F(x). We also say that F is an antiderivative or a primitive function of f.
Polynomial functions and integral (1): Linear functions
It is easy to calculate the area under a straight line. This is the first example of integration that allows us to understand the idea and to introduce several basic concepts: integral as area, limits of integration, positive and negative areas.
Polynomial functions and integral (2): Quadratic functions
To calculate the area under a parabola is more difficult than to calculate the area under a linear function. We show how to approximate this area using rectangles and that the integral function of a polynomial of degree 2 is a polynomial of degree 3.
Integral of powers with natural exponent
The integral of power functions was know by Cavalieri from n=1 to n=9. Fermat was able to solve this problem using geometric progressions.
Polynomial functions and integral (3): Lagrange polynomials (General polynomial functions)
We can see some basic concepts about integration applied to a general polynomial function. Integral functions of polynomial functions are polynomial functions with one degree more than the original function.
Definite integral
The integral concept is associate to the concept of area. We began considering the area limited by the graph of a function and the x-axis between two vertical lines.
Monotonic functions are integrable
Monotonic functions in a closed interval are integrable. In these cases we can bound the error we make when approximating the integral using rectangles.
Indefinite integral
If we consider the lower limit of integration a as fixed and if we can calculate the integral for different values of the upper limit of integration b then we can define a new function: an indefinite integral of f.
The Fundamental Theorem of Calculus (1)
The Fundamental Theorem of Calculus tell us that every continuous function has an antiderivative and shows how to construct one using the integral.
The Fundamental Theorem of Calculus (2)
The Second Fundamental Theorem of Calculus is a powerful tool for evaluating definite integral (if we know an antiderivative of the function).
Archimedes' Method to calculate the area of a parabolic segment
Archimedes show us in 'The Method' how to use the lever law to discover the area of a parabolic segment.