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derivitaive of sine functions

Extracts from this document...

Introduction

International Baccalaureate

Mathematics Portfolio - Standard Level Type I

Derivatives of Sine Functions

Student Names: Nam Vu Nguyen

Set Date: Monday, January 14, 2008

Due Date: Tuesday, February 05, 2008

School Name: Father Lacombe Senior High School

Teacher: Mrs. Gabel

I CERTIFY THAT THIS PORTFOLIO ASSIGNMENT IS ENTIRELY MY OWN WORK

                                         Nam Vu Nguyen: ___________________________________

International Baccalaureate

Mathematics Portfolio - Standard Level - Type I

Derivatives of Sine Functions

Mathematics the study of the concepts of quantity, structure, space and change, is a type of science that draws conclusions and connections to the world’s analytical problems around us. Mathematicians call the study of mathematics, a science of patterns that is discovered in numbers, space, science, computers, imaginary abstractions, and everything that is contained in the universe itself. Mathematics is also found in numerous natural phenomena that occur around us each and every day. Today math is used to be applied and developed into numerous evolving educational fields, inspiring humans to discover and make use of their mathematical knowledge, which will in turn lead to entirely new discoveries.

The purpose of this mathematical paper is to explore and analyze the derivative of the trigonometric function:image00.png. In this portfolio, the derivative is a function that is used to find the instantaneous rate of change, or gradient, of another function at a given point along the x-axis.

...read more.

Middle

Once again since image72.pngtherefore image73.png and image83.pngis able to be reduced to image84.pngby substituting 0 in for h, which will create the following limits formula:

image85.png

When simplified will become:

image86.png

image87.png

image88.png

Therefore the derivative which is also called “differential quotient”, is a function that is derived from an original function, which can create the instantaneous rate of change at any given point on the original function.

Therefore the derivative of the functionimage00.png isimage54.png.

To prove that the above conjecture, that the derivative of the functionimage00.png isimage54.png, we will also solve for the derivative graphically as well. The derivative is defined as the slope at the pointimage89.png, and therefore where the derivative is based solely on the slope or the gradient of the original function ofimage00.png. The slope for the function of image00.png at 0, or in other words, when the tangent line remains horizontal on the graph; which are the respective maximum and minimum points, the derivative of sine is at 0 for its f (x) value in (x, f (x)).          
        

We can note the slope of image00.pngis at 0 when the function reaches its maximum and minimum amplitude which means that the derivative of image90.pngis equal to 0 at these points which areimage91.png. Using this information, we can plot the points in which x-intercepts of the derivative of image00.pngisimage92.png.

...read more.

Conclusion

2828/html/images/image15.png" style="width:69.33px;height:32px;margin-left:0px;margin-top:0px;" alt="image15.png" />

image16.pngimage17.png

        -0.654

image18.png

image19.png

image20.png

image21.png

image22.png

image16.png-0.990

image23.png

image24.png

image25.png

image26.png

image27.png

image14.png

image28.png

image29.png

image30.png

image31.png

image32.png

image05.png

image33.png

image34.png

image30.png

image35.png

image36.png

image14.png

image38.png

image16.pngimage39.png

image16.pngimage25.png

image40.png

image16.pngimage41.png

image16.png-0.990

image43.png

image16.pngimage44.png

image16.pngimage20.png

image16.pngimage45.png

image16.pngimage46.png

image16.png-0.654

image16.pngimage47.png

image16.pngimage48.png

image16.pngimage14.png

image16.pngimage49.png

image16.pngimage50.png

image16.pngimage11.png

image16.pngimage51.png

image16.pngimage52.png

image16.pngimage08.png

image16.pngimage53.png

image16.png(2π, 0)

image16.png1

image16.png(2π, 1)

         After verifying that the slope of the original function does equal to the y-value of the derivative at a given point (x, f (x)), we can make a second conjecture:

Conjecture:

         “If the slope is a given number of the tangent line for the function image00.png, then it will equal image54.pngat the given point (x, f (x)) as the y-value.”

         We can now confirm that the derivative of sine is cosine, as the properties and given ordered points match.

...read more.

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