Approximation - Error Problem, 2

Category: Differential Calculus, Arithmetic

"Published in Newark, California, USA"

Evaluate using the Approximation and Error Method for

without using a calculator.

Solution:

Consider the given number

Let

If        x = 1
then dx = 0.001

Substitute the values of x and dx to the above equation to solve for dy as follows

Therefore,

Check: 

If you will use a scientific calculator,

which is almost the same as the final answer.

Binomial Theorem

Category: Algebra, Arithmetic

"Published in Newark, California, USA"

Evaluate 

without using a calculator.

Solution:

Consider the given number

Rewrite and expand the given number as follows

It is better to convert the decimal number into its equivalent fraction for the given number 

Next, use the Binomial Theorem to expand the given number as follows

Get the Least Common Denominator (LCD) for all the fractions and simplify as follows

Convert the fraction into its equivalent decimal number as follows

Therefore,

Check:

If you will use a scientific calculator,

which is almost the same as the final answer.

More Integration Procedures, 6

Category: Integral Calculus, Trigonometry

"Published in Suisun City, California, USA"

Evaluate

Solution:

Consider the given above equation

Convert the double angle function into a single angle function as follows

Since the denominator contains trigonometric functions, then we have to use the Miscellaneous Substitution method. On December 17, 2012, we discussed and derived the equations for Miscellaneous Substitution as follows

Let 

Substitute the above values to the given equation above, we have

but

Therefore,