derivitaive of sine functions

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:. 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. The derivative, in respect to its calculus definition is defined as a limit and arises when finding slopes of tangents and rates of change on a given graph. Therefore the derivative of the function, , is the limit of the slope of the graph at any given points.

The portfolio will begin by firstly providing a proof and deriving the function to create the derivative. Then the portfolio will continue on by investigating the graph of the function  by analyzing the graph and comparing it to its derivative and also by looking at the behaviour of the gradient of the function as its approaches. The derivative of a function is determined by first determining the equation of the tangent line that passes by the function. The limit of slope of the equation for the tangent line is the function for the derivative of the original graph. Therefore to find the derivative, tangent lines will be drawn and the equations will be determined by using a computer generated graphing program called “Advanced Grapher “. A table of values will then be created to show that the value of the slope of the tangent lines are almost identical to the value of the derivative, as  approaches a value within in a given domain. A conjecture based on the observations will be made for the function and its derivative.  Other various forms of the function  will be also investigated.

The function,, when graphed on the TI-83+ Calculator with the following correct [WINDOWS] settings will produce a graph with a domain of a range of

To access the [WINDOWS], simply press [WINDOW] underneath the screen.

After doing so, change the [WINDOW] to match the above image by doing the following:

Enter:

-2for Xmin=

2 for Xmax=

/2 for Xscl=

-2 for Ymin=

2 for Ymax=

1 for Yscl=

After doing the above tasks the correct graph will be produced by pressing [GRAPH].

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