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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:. 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.

Middle

Once again since therefore  and is able to be reduced to by substituting 0 in for h, which will create the following limits formula:

When simplified will become:

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 function is.

To prove that the above conjecture, that the derivative of the function is, we will also solve for the derivative graphically as well. The derivative is defined as the slope at the point, and therefore where the derivative is based solely on the slope or the gradient of the original function of. The slope for the function of  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 is at 0 when the function reaches its maximum and minimum amplitude which means that the derivative of is equal to 0 at these points which are. Using this information, we can plot the points in which x-intercepts of the derivative of is.

Conclusion

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

-0.654

-0.990

-0.990

-0.654

(2π, 0)

1

(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 , then it will equal at 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.

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

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