GeoGebra Tutorial Video 2 – Constructing a Square

This is the second video tutorial on using GeoGebra. This is the video of the article GeoGebra Tutorial 3 – Constructing a Square. Sorry for some grammar lapses (I am not very good in impromptu), and  I am really having a hard time imitating American accent to make most of the viewers understand this video. Sometimes my tongue gets twisted — so twisted that it automatically adds “s” to singular nouns (chuckles) like squares instead of square.

Anyway, this is a math blog, so I suppose it is appropriate for us to leave the grammar stuff to language teachers, although you are very much welcome to comment on errors, grammar may it be.

My next video is on Graphs and Sliders, so keep posted.

Derivative and the Maximum Area Problem

Note: This is the third part of the Derivative Concept Series. The first part is The Algebraic and Geometric Meaning of Derivative and the second part is Derivative in Real Life Context.

Introduction

The computation of derivative is often seen in maximum and minimum problems.  In this article, we will discuss why do we get the derivative of a function and equate it to 0 when we want to get its maximum or minimum. To give you a concrete example, let us consider the problem below.

Find the maximum area a rectangle with perimeter 10 units.

Without using calculus, we can substitute values for the rectangle’s length, compute for its width and its corresponding area. If we set the interval to 0.5, then we can come up with the table shown in Figure 1.

Figure 1 - Table showing the length, width, and area of a rectangle with perimeter 10.

Looking at the table above, we can observe that a rectangle of length of 2.5, a square, has the maximum area. If we have prior calculus  knowledge, however, we know that whatever the value of our perimeter, a square having the given perimeter will always have the maximum area.

Using elementary algebra, if we let x be the width of our rectangle, it follows that the length is 5-x. Let f(x) be the area of the rectangle. In effect, the area of the rectangle is described by the equation f(x) = 5x - x^2. We want to maximize the area, which implies that we want to find the maximum value of f(x).

Figure 2 – A rectangle with Perimeter 10 and width x units.

In elementary calculus, to compute for the maximum value of f(x), we get its derivative, which is equal to 5 - 2x, which we will denote f'(x). We then equate the f'(x) to 0 resulting to the equation 5-2x=0 \Rightarrow x = 2.5 which is exactly the maximum value in the table above.

Derivative and Equation to 0

In the article the Algebraic and Geometric Meaning of Derivative, we have learned that the derivative of a function is the slope of the line tangent to that function at a particular point. From elementary algebra, we also have learned the properties of slopes. If a line is rising to the right, the slope is greater than 0; if the line is rising to the to the left, then the slope is less than 0. We have also learned that a horizontal line has slope 0 and the vertical line has an undefined slope.

Figure 3 – Properties of slope of a straight line.

In the problem above, we calculated by getting the derivative (the slope of the line tangent to a function at a particular point) and equate it to 0. But a line with slope 0 is a horizontal line. In effect, we are looking for a horizontal tangent of f(x) = 5x-x^2. To give a clearer picture let us look at the graph of f(x) = 5x - x^2.

Figure 4 – Tangent lines of 5x – x2.

From the graph it is clear that the maximum point of the function is where the tangent line (red line) horizontal. In fact, there are only three possible cases that tangent line could be horizontal as shown in Figure 5: first, the minimum of a function (blue graph); second, the inflection point (red graph); and the third is the maximum of the function (green graph).

It should also be noteworthy to say that all the ordered pairs (length, area) or(width, area) in Figure 1 will be on the blue curve in Figure 4.

Figure 5 – Cases of a graph where the tangent is horizontal.

The derivative has many applications and it is seen in many topics in calculus.  In the next Derivative Tutorial, we are going to discuss how the derivative is used in other context.

Summary

  • The derivative of a function is the slope of the line tangent to a function at a particular point.
  • The horizontal line has slope zero.
  • In solving maxima and minima problems, we get the derivative of a function and equate to zero to get the minimum or maximum. We do this because geometrically, we want to get the line tangent to a function at a particular point that is horizontal.

CaR Tutorial 1- Constructing an Isosceles Triangle

CaR or Compass and Ruler is a free dynamic geometry software written in Java by Rene Grothman. The CaR window is shown below.

Figure 1 - The Compass and Ruler window.

The upper part of the window contains the menu bar and toolbar. The toolbar contain tools in constructing and editing mathematical objects. The left window below the toolbar is the Objects window and the right pane is the Drawing pad where we construct drawings.

Tutorial 1 – Constructing an Isosceles Triangle

In the first tutorial, we are going to create an isosceles triangle by using the center of the circle and two points on its circumference.

1.) We will not need the Coordinate axes so click the Show grid icon until the Show the Grid icon until the grid or axes is not shown.
2.)Click the Circle tool, then click the drawing pad to determine the center of the circle, and click another location to determine its radius.
3.) Click the Point tool and click another location on the circumference circle. After step 3, your drawing should look like the figure below.

Figure 2 - Circle with 2 points on its circumference.

4.) Click the Segment tool and two points to construct a side of the triangle. Continue until the triangle is formed.
5.) Click the Move button to drag the points and observe what happens to the triangle. Explain why the triangle is always isosceles.
6.) Click the Hide object button and click the circle to hide it.
7.) Next, we will change the name of the points and display their names. To do this, right click a point to display the Edit Point dialog box. In the Name text box, type A, then click the Show Object Names icon.

Figure 1 - The Edit Point Dialog box.

Use the same process to change the name of the other points.

Congratulations, you have finished the first Compass and Ruler tutorial.

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