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William (Student) December 5 2016
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There are two different types of wave: longitudinal and transverse. When you hear the word "wave" you are probably thinking of a transverse wave.
Imagine you are floating in the sea. The waves of water are coming towards you and you bob up and down over the waves as they pass you. This type of wave is a transverse wave, the movement of any point in the wave (you in this case) is at right angles to the direction the wave is moving (towards the beach). When you are at the top of the wave you are at a peak and when you are at the bottom of the wave you are at a trough. Another example of this type of wave is a light wave (although the waves are a lot smaller).
A longitudinal wave is a one where the movement of any point in the wave is in the same direction the wave is moving. These are harder to visualise. If you push and pull a slinky in a straight line you will see areas where the slinky is more bunched up and more spread apart. These are called compressions and rarefactions. An example of this type of wave is a sound wave.see more
The easiest way to solve this problem is by using differentiation.
dy/dx = (6x^2 + 4x + 2)/dx
dy/dx = 6*2*x^1 + 4*1*x^0 + 0
dy/dx = 12x + 4
When you set the derivative equal to zero you find a point on the curve where there is no change in gradient (where the curve is momentarily horizontal).
12x + 4 = 0
Solving this gives the x value for when the function has a maximum or a minimum.
x = -1/3
You can then use this x value to find the value of y at that maximum or minimum. This is done by substituting x back into the formula for y.
y = 6(-1/3)^2 + 4(-1/3) + 2 = 4/3
You could go a step further and find whether that value was a maximum or a minimum. To do this you would differentiate again.
d^2y/dx^2 = (12x + 4)/dx
d^2y/dx^2 = 12*1*x^0 + 0
d^2y/dx^2 = 12
This is greater than 0 and so the value is a minimum as the rate of change of the gradient is positive.see more
The problem with this fraction is the "i" on the denominator, it is not rational. To rationalise this fraction we use the complex conjugate.
This sounds more complicated than it is. If a + ib is a complex number, then a - ib is its conjugate. The only thing that changes is that the imaginary part of the number changes its sign.
To rationalise the fraction you multiply both top and bottom of the fraction by the the complex conjugate of the denominator.
(8 + 6i)/(6 - 2i) * (6 + 2i)/(6 + 2i)
To make this less messy in text I will solve the top and bottom separately. First the top:
(8 + 6i)*(6 + 2i) = 48 + 16i + 36i + 12i^2 = 48 + 52i - 12 = 36 + 52i
Then the bottom:
(6 - 2i)*(6 + 2i) = 36 - 12i + 12i - 4i^2 = 36 + 4 = 40
Putting these together gives:
(36 + 52i)/40
Simplified this is:
9/10 + (13/10)isee more