The 0.0001 Time Rate Approach, an Extended Application

In the previous post, I introduced to you the 0.0001 Time Rate approach This post is an extended application of the said technique on curvatures and radius of curvatures.

Curvature is defined (by The Facts on File Dictionary of Mathematics) as the rate of change of the slope of the tangent to a curve. For each point on a smooth curve there is a circle that has the same tangent and the same curvature at that point. The radius of this circle called the “radius of curvature”, is the reciprocal of the curvature, and its center is known as the center of curvature. If the graph of a function y=f(x) is a continuous curve, the slope of the tangent at any point is given by the derivative dy/dx and the curvature is given by

C = \displaystyle\frac{\displaystyle\frac{d^2y}{dx^2}}{(1+(\displaystyle\frac{dy}{dx})^2)^{3/2}}

and the radius of curvature is given by

RC = \frac{1}{C} = \displaystyle \frac{(1+(\displaystyle \frac{dy}{dx})^2)^{3/2}}{\displaystyle \frac{d^2y}{dx^2}}

where: \frac{dy}{dx} is the derivative of y with respect to x and \frac{d^2x}{dx^2}  is the second derivative of y with respect to x or in simpler terms, it is the derivative of \frac{dy}{dx}. » Read more

GeoGebra Tutorial 17 – Functions, Tangent Lines and Derivatives

This is the 17th tutorial of the GeoGebra Intermediate Tutorial Series. If this is your first time to use GeoGebra, I strongly suggest that you read the GeoGebra Essentials Series.    In this tutorial, we are going to use slider control a, b, c, d and e and graph the function f(x) = ax4 + bx3 + cx2 + dx + e.

Figure 1

We then  construct a line tangent to the function and passing through  point A and trace the graph of the point whose x-coordinate is the x-coordinate of A, and whose y-coordinate as the  slope of the tangent line. We compute for the derivative of f(x), and see if  there is a relationship between the trace and the derivative. If you want to follow the this tutorial step by step, you can open the GeoGebra window. Before following the tutorial, you may want to see the final output.

1. Open GeoGebra. We will need the Algebra view and the Axes so be sure that they are displayed. If not, use the View menu from the menu bar to show them.
2. To label points only, click the Options menu, click Labeling, and then click New Points Only.
3. To create slider a, type a = 1 in the input box and press the ENTER key. Right click the equation a = 1 in the algebra window (leftmost window pane) and click Show object from the context menu. Slider a should appear on your drawing pad.
4. Using step 3, create 4 more sliders namely b, c, d and e.
5. To graph the function f(x) = ax4 + bx3 + cx2 + dx + e, type f(x) = a*x^4 + b*x^3 + c*x^2 + d*x + e in the input box, then press the ENTER key.
6. Move the sliders and observe what happens.

Figure 2

7. To construct point A on function f, select the New Point tool and click graph of the function.
8. To construct a line tangent to f and passing through A, select the Tangents tool, click point A and click the graph of f. A tangent line should appear passing through point A.
9. Move point A on the function, and move the sliders. What do you observe?
10. To get the slope of the tangent line, select the Slope tool and then click the tangent line. This will produce m (see Algebra window). Given similar values of the numerical coefficients and the right place for point A, your graph should look like Figure 3.

Figure 3

11. We now create point D, which will trace the ordered pair (x(A),m) where m is the slope of the tangent line. Note that x(A) means the x-coordinate of A and m was automatically assigned by GeoGebra to the value of the slope.  To create the point type D = (x(A), m).
12. We now change the color of point D. To do this, right click point D, and click Object Properties from the context menu. In the dialog box, select the Color tab and select a color you want from the color palette. Next, select the Basic tab, be sure to check the Show trace check box, then click the Close button.

Figure 3

13. Move point A along the function. What do you observe about the traces of point D?
14. To graph the derivative of f(x), type f’(x) = derivative[f], then press the ENTER key. What do you observe about the derivative of the function f?
15. Right click the derivative function and click Properties. In the dialog box, go to the Color tab and select the color you want, preferably the same color as point D. Drag point A. After this step, your drawing should look like the one shown in Figure 1.
16. What can you say about the relationship of the derivative and the path traced by point D and the derivative function?
17. How can you relate the tangent function, its slope, the derivative function and the line traced by point D?
18. Based on the activity above, how will you describe the derivative of a function at a particular point and derivative in general?

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.
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