geno3141

To find the derivative of  u = sin(2x),  we need to use the chain rule:  u' = cos(2x)·2  --->  u' =  2cos(2x)

Similarly,  v = cos(2x)  --->  v' = -sin(2x)·2  --->  v' = -2sin(2x)

To find the derivative of  y = sin(2x)·cos(2x), we need to use the product rule:  y = u·v  --->  y' = u·v' + v·u'

y' = sin(2x)·-2sin(2x) + cos(2x)·2cos(2x)     --->  y' = -2sin²(2x) +2cos²(2x)

|y'| = |-2sin²(2x) +2cos²(2x)|  =  2|cos²(2x) - sin²(2x)|

2/3 of 7/10                               

=  2/3 x 7/10                        "of" means "times"

=  14/30                               multiplying numerator times numerator; denominator times denominator

=  7/15                                 reducing by dividing both the numerator and denominator by 2

The part 100+200 equals 300 but it does not equal 300 + 300.

If you look at the far left side:  100+200,  it does not equal the far right side  600.

Using 27.7 quarts per day for 6 days means that 27.7 x 6 = 166.2 quarts.

180 - 166.2 = 13.8 quarts remaining. (About enough for another half-day.)

This can be done by first calculating the whole "slice" of the circle and then subtracting the area of the triangle.

Since there are 360° in a triangle and the central angle is 30°, the area of the slice will be 30/360 or 1/12 of the circle.

The formula for the area of a circle = πr².  

Since the diameter is 6, the radius will be 3 and the area of the circle = π·3², which is 9π.) = The area of the slice will be (1/12) of 9π which is (3/4)π .

The area of the triangle can be found various ways. One way is to use the formula:  A = .5·a·b·sinC.

a = 3     b = 3    sin(30°) = 0.5                    A = .5·3·3·.5  =  2.25

Area of the shaded area  =  (3/4)π - 2.25

It appears that you went from Philadelphia to New York City by using Amtrak's Acela while I went by way of Hong Kong -- could you use some frequent flyer miles?

If your variable is an exponents, use logarithms (either common or natural).

7^x  =  5                    Take the log (or ln) of both sides:

Using common logs:

log(7^x)  =  log(5)       Now, since you have a log of an exponent, the exponent comes out as a multiplier:

x·log(7)  =  log(5)         Now, divide both sides by log(7):

x  =  log(5) / log(7)       Calculator time:

x  ≈  0.827

[If you used ln instead of log, your answer would be the same.]

If you want to simplify the power of a quotient, take the numerator to the exponent and take the denominator to the exponent, also:  for example:  (2/3)^4  becomes  2^4 / 3^4  which becomes 16/81.

The 'E12' refers to scientific notation; thus, 8.712121212E12 means 8.712121212 x 10^12, or

8 712 121 212 000

It takes approximately 28.3495 grams to make one ounce.

Dividing  100  by  28.3495, the answer is (approximately) 3.5 ounces.