I am a student at Durham University doing a master’s degree in Aeronautical Engineering. I’m in my final year and I’ve thoroughly enjoyed my time here. I really enjoy maths and physics and I’m hopefully quite good at both by now. I’ve taught people aged 12-17 before so I’ve had experience dealing with people of different ages. I’m approachable and easy going, so don’t be afraid to ask questions or contact me at any point. Tutoring: What I teach in each session is up to you and I can go into as much or little detail as you like in each subject. The more information you give me beforehand the better I can prepare and tailor your experience. When teaching I’ll be talking you through the problem at hand and noting down any necessary or helpful diagrams and analogies. If you are unsure of any step I will take the time to reinforce your knowledge of that particular area. If you have any questions drop me a message through MyTutor and if you think I’d make a good tutor for you, please arrange a ‘Meet the tutor session’ with me. Thanks, HarryI am a student at Durham University doing a master’s degree in Aeronautical Engineering. I’m in my final year and I’ve thoroughly enjoyed my time here. I really enjoy maths and physics and I’m hopefully quite good at both by now. I’ve taught people aged 12-17 before so I’ve had experience dealing with people of different ages. I’m approachable and easy going, so don’t be afraid to ask questions or contact me at any point. Tutoring: What I teach in each session is up to you and I can go into as much or little detail as you like in each subject. The more information you give me beforehand the better I can prepare and tailor your experience. When teaching I’ll be talking you through the problem at hand and noting down any necessary or helpful diagrams and analogies. If you are unsure of any step I will take the time to reinforce your knowledge of that particular area. If you have any questions drop me a message through MyTutor and if you think I’d make a good tutor for you, please arrange a ‘Meet the tutor session’ with me. Thanks, Harry

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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.

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.

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.

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.

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)i**

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)i**