Maximum Minimum Problem, 3

Category: Differential Calculus, Trigonometry, Algebra

"Published in Suisun City, California, USA"

Find the maximum and minimum values of the function

Solution:

Consider the above equation

Take the derivative of the above equation with respect to x as follows

Since we want to solve for the maximum and minimum values, equate the above equation to zero as follows

To solve for the value of Sin x, use the Quadratic Formula as follows

If you choose the positive sign

Therefore,

If you choose the negative sign

Therefore,

More Integration Procedures, 3

Category: Integral Calculus, Algebra

"Published in Suisun City, California, USA"

Evaluate the integral for

Solution:

The above equation has two radicals with different indexes. We need to substitute each radicals with another variable with their common index as follows

Let

Substitute the above values to the given equation, we have

Since the exponent at the numerator is greater than the exponent at the denominator, then we need to do the division of polynomial as follows

The above equation becomes

but

Therefore, the final answer is

Graphical Sketch - Three Leaf Rose

Category: Analytic Geometry, Trigonometry, Algebra

"Published in Newark, California, USA"

Sketch the graph for a Three Leaf Rose

Solution:

The given equation above is also another type of Polar Equation where the parameters are r and θ. The purpose of converting the equations of Rectangular Coordinate to Polar Coordinate is to simplify the equation. Most of the equations of Polar Equations are complicated if they are written in Rectangular Coordinate. The graphs of Polar Equations are mostly leaves, flowers, and spiral shapes. Let's consider the above equation again

We need to assign the values of θ from 30° to 360° in order to get the values of r as well. All angles must be expressed in radians because radians are unitless values. You can use calculator in order to get the values of r as follows

Photo by Math Principles in Everyday Life

Next, you need a polar paper in order to plot the graph. If you don't have a polar paper, you can use an ordinary graphing paper but you need a compass, protractor, and ruler. As you noticed at the above table that some values of r are negative. We will not use the parameters that have negative values of r. If you plot the values of r and θ and then connect all the points, the graph of the given equation will be like this

Photo by Math Principles in Everyday Life