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Sunday, March 31, 2013

Derivative - Exponential Functions

Category: Differential Calculus, Algebra

"Published in Suisun City, California, USA"

Find the derivative for 



Solution:

Consider the given equation above



Apply the derivative of a rational function, we have
















Saturday, March 30, 2013

Derivative - Algebraic Functions, Radicals, 2

Category: Differential Calculus, Algebra

"Published in Suisun City, California, USA"

Find the derivative for



Solution:

Consider the given equation above



Apply the derivative by power, we have



























Friday, March 29, 2013

Proving Trigonometric Identities, 3

Category: Trigonometry, Algebra

"Published in Suisun City, California, USA"

The lower right-hand corner of a long piece of paper 6 in. wide is folded over to the left-hand edge as shown. The length L of the fold depends on the angle θ. Show that 




Photo by Math Principles in Everyday Life

Solution:

Consider the given figure above and label further using the principles of Plane Geometry, we have


Photo by Math Principles in Everyday Life

By Pythagorean Theorem, 




We need to use another equations in order to eliminate x and y. By using trigonometric functions of right triangles, we have

Using a small right triangle













Using a large right triangle





Consider again the equation,



Substitute the values of x and y, we have

















Take the square root on both sides of the equation, we have




Multiply both the numerator and the denominator by 1 - sin2 θ, we have





Therefore,





Thursday, March 28, 2013

Logarithmic Expression

Category: Algebra

"Published in Suisun City, California, USA"

Rewrite the expression for



Solution:

Consider the given equation above



Expand the logarithmic expression for the main fraction, as follows



Expand the logarithmic expression for the second term, as follows





Rewrite the logarithmic expression for the square root and cube root functions and therefore,