Derivative - Trigonometric Functions, 2

Category: Differential Calculus, Algebra, Trigonometry

"Published in Newark, California, USA"

Find the second derivative for

Solution:

Consider the given equation above

Since the denominator of the given equation contains a radical sign, then we have to rationalize the denominator in order to eliminate the radical sign as follows

But

Hence, the above equation becomes

Using the Half Angle Formula, the above equation becomes

Take the derivative on both sides of the equation with respect to x, we have

Take the derivative on both sides of the equation again with respect to x, we have

Therefore, the final answer is

Note: You must memorize or remember the trigonometric formulas and identities as much as you can so that it will be easier for you to take the derivative of trigonometric functions and equations.

Integration - Powers

Category: Integral Calculus, Algebra

"Published in Newark, California, USA"

Evaluate

Solution:

Consider the given equation above

The terms inside the radical sign has a common factor, which is x. Factor the terms inside the radical sign, we have

The other factor is a perfect trinomial square. Rewrite the above equation in terms of a square of a binomial

Take the square root of the above equation

Apply the distributive property of multiplication over addition to the above equation

Integrate the above equation by power with respect to x, we have

Therefore,

where C is the constant of integration.

Centroid - Masses, Weights

Category: Algebra, Physics, Mechanics

"Published in Newark, California, USA"

Find the centroid of the following weights: 2 lbs at (3, 0, 1), 5 lbs at (-2, 3, 2), and 9 lbs at (4, 1, 4).

Solution: 

To illustrate the problem, it is better to draw the figure as follows

Photo by Math Principles in Everyday Life

The total weight of the three weights is 

The x value of the centroid is calculated as follows

The y value of the centroid is calculated as follows

The z value of the centroid is calculated as follows

Therefore, the centroid of the three weights is