Indeterminate Form - Zero Over Zero, 3

Category: Differential Calculus, Algebra

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Evaluate

Solution:

Consider the given equation above

Substitute the value of x which is 0 to the above equation, we have

Since the answer is ⁰/₀, then it is considered as Indeterminate Form which is not accepted as a final answer in Mathematics. Remember this, any number, except zero divided by zero is always equal to infinity. If the Indeterminate form is either ⁰/₀ or ^∞/_∞, then we can use the L'Hopital's Rule in order to solve for these type of Indeterminate Forms. In this case, we can apply the L'Hopital's Rule for the above equation as follows

 

Substitute the value of x which is 0 to the above equation, we have

Since the answer is again ⁰/₀, then we have to apply again the L'Hopital's Rule, as follows

Substitute the value of x which is 0 to the above equation, we have

Since the answer is again ⁰/₀, then we have to apply again the L'Hopital's Rule, as follows

Substitute the value of x which is 0 to the above equation, we have

Therefore,

Derivative - Hyperbolic Functions

Category: Differential Calculus, Algebra

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Find the derivative for

Solution:

Consider the given equation above

Take the derivative of the above equation with respect to x, we have

We can accept the above equation as a final answer but if you wish to substitute the value of hyperbolic cosine, you can also do that one. We know that 

If 

then

becomes

The other way of getting the derivative of the given equation is to eliminate the hyperbolic functions by substituting their equivalent value or identity. We know that 

If

then substitute the value of hyperbolic sine of the above equation as follows

Take the derivative of the above equation with respect to x, we have

Triple Integration

Category: Integral Calculus, Trigonometry

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Evaluate

Solution:

Consider the given equation above

Integrate first the given function with respect to dz, as follows

Integrate the above equation with respect to dr, as follows

Integrate the above equation with respect to dθ, as follows

Substitute the limits and the final answer is