Indeterminate Form - Zero Over Zero, 8

Category: Differential Calculus, Trigonometry

"Published in Vacaville, California, USA"

Evaluate

Solution:

To get the value of a given function, let's substitute the value of x to the above equation, we have 

   
Since the answer is 0/0, then it is an Indeterminate Form which is not accepted as a final answer in Mathematics. We have to do something first in the given equation so that the final answer will be a real number, rational, or irrational number. 

Method 1:

Since the answer is Indeterminate Form, then we have to apply the double angle formula at the numerator and then simplify as follows

Substitute the value of x to the above equation, we have 

Therefore, 

Method 2:

Another method of solving Indeterminate Form is by using L'Hopital's Rule. This is the better method especially if the rational functions cannot be factored. L'Hopitals Rule is applicable if the Indeterminate Form is either 0/0 or ∞/∞. Let's apply the L'Hopital's Rule to the given function by taking the derivative of numerator and denominator with respect to x as follows 

Substitute the value of x to the above equation, we have 

  

Therefore,

 

Indeterminate Form - Zero Times Infinity, 2

Category: Differential Calculus, Trigonometry

"Published in Newark, California, USA"

Evaluate:

Solution:

To get the value of a given function, let's substitute the value of x to the above equation, we have

Since the answer is 0•∞, then it is an Indeterminate Form which is not accepted as a final answer in Mathematics. We cannot use the L'Hopital's Rule because the Indeterminate form is 0•∞. L'Hopital's Rule is applicable if the Indeterminate Form is either 0/0 or ∞/∞. We have to do something first in the given equation so that the Indeterminate Form becomes 0/0 or ∞/∞.

Let's rewrite the above equation as follows

Substitute the value of x to the above equation, we have

Since the Indeterminate Form is  ∞/∞, then we can use the L'Hopital's Rule to the above equation as follows

 
Substitute the value of x to the above equation, we have

  
Therefore,

In this case, the above equation has no limit. 

Indeterminate Form - Zero Over Zero, 6

Category: Differential Calculus, Trigonometry

"Published in Newark, California, USA"

Evaluate

Solution:

To get the value of a given function, let's substitute the value of x to the above equation, we have  

Since the answer is 0/0, then it is an Indeterminate Form which is not accepted as a final answer in Mathematics. We have to do something first in the given equation so that the final answer will be a real number, rational, or irrational number.

Method 1:

Since the answer is Indeterminate Form, then we have to use the principles of trigonometric identities. Let's convert the double angle function into single angle function and then simplify as follows

Substitute the value of x to the above equation, we have 

Therefore,

Method 2:


Another method of solving Indeterminate Form is by using L'Hopital's Rule. This is the better method especially if the rational functions cannot be factored. L'Hopitals Rule is applicable if the Indeterminate Form is either 0/0 or ∞/∞. Let's apply the L'Hopital's Rule to the given function by taking the derivative of numerator and denominator with respect to x as follows

Substitute the value of x to the above equation, we have

Therefore,

 

Rectangular Parallelepiped Problem, 14

Category: Solid Geometry, Plane Geometry, Trigonometry

"Published in Newark, California, USA"

Find the angles that the diagonal of a rectangular parallelepiped 2 in. by 3 in. by 4 in. makes with the faces.

Solution:

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

Photo by Math Principles in Everyday Life

Next, we need to draw a diagonal line at the top and bottom of the rectangular parallelepiped as well as its diagonal and angles as follows

Photo by Math Principles in Everyday Life

By using Pythagorean Theorem, the length of x is

By using Pythagorean Theorem, the length of d is 

Therefore, the value of an angle is

or

Next, we need to draw a diagonal line at the left and right of the rectangular parallelepiped as well as its diagonal and angles as follows

Photo by Math Principles in Everyday Life

By using Pythagorean Theorem, the length of y is 

Therefore, the value of an angle is

or

Next, we need to draw a diagonal line at the front and back of the rectangular parallelepiped as well as its diagonal and angles as follows

Photo by Math Principles in Everyday Life

By using Pythagorean Theorem, the length of z is 

 

Therefore, the value of an angle is 

or