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Pre-Calculus

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Question:

Given the polar coordinates (3, pi/2), find the rectangular coordinates.

Lash T.

Answer:

The polar coordinates (3, pi/2) mean that you have a point that is 3 distance away from the origin in the direction of pi/2. If you don't quite understand what this means yet, don't worry. Start by drawing a simple graph with x and y axis. With polar coordinates, you want to start with the direction first. In these problems, the direction means we must find the angle on the unit circle. Pi/2 is the same thing as 180/2, or 90 degrees (remember that pi = 180 in this case, not 3.14...). What do you do with 90 degrees? Start from 0 degrees, which is the right side of the x-axis on the graph that you drew. Go counter-clockwise (because 90 degrees is positive, clockwise means the angle is negative) until you reach the top half of the y-axis [note: it doesn't matter how far up or down you are on the y-axis yet, because we are first looking for the direction]. Now that we know that our direction goes upward on the y-axis, we can see how far away we must go from the origin. We look back to our polar coordinates and see that the 3 is the value we must use to determine distance. In our new direction, go 3 away from the origin. You now have a point on the y-axis at [0,3]. This is your answer! Once you can see how this works visually, you can now attempt to solve these problems without the use of a graph. This may be necessary for harder problems where it isn't so obvious where the point is exactly. If you remember SOHCAHTOA, you know that the sine of an angle is the opposite divided by the hypotenuse, and the cosine of an angle is the adjacent divided by the hypotenuse. Why is this relevant? Well, we have an angle: pi/2. We also have the number 3. We determined that 3 is the distance from the origin, and if you think about the problems you did in geometry, isn't this the same thing as the hypotenuse? So, plugging in what we are given sin(pi/2) = opposite / 3. If we multiply both sides by 3 and solve, we get 3 x 1 = opposite, or the opposite = 3. Again, thinking back to geometry, what does the opposite represent? Our y value! It is the distance of the point from the x-axis if we draw a straight line from the point between the two, and since it is above the x-axis, it is positive. We know our y-coordinate, but what about the x-coordinate? We can simply use cosine (angle) = adjacent / hypotenuse. cos (pi/2) = adjacent / 3 --> 3 x 0 = adjacent, or adjacent = 0. Again, we can visualize the adjacent value being the x-coordinate, or the distance in the x-direction from the y-axis. Since it is 0 in this case, the point will be directly on the y-axis. Given that adjacent is 0 and opposite is 3, we can see that our rectangular values are [0,3]. Done!

Calculus

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Question:

What is the limit as x approaches 0 of (cos2x - 1)/x ?

Lash T.

Answer:

The first step in finding a limit is to simply plug in the value that you are approaching. If you plug in 0 for all values of x, you see that cos2(0) -1 is 0, and the denominator is simply 0 since you plugged 0 in for x. This means you found that the limit as x approaches 0 is 0/0. Great news! This means it can most likely be solved by using l'Hopital's rule. To use l'Hopital's rule, start by looking at the numerator. Ignore the denominator for now. The numerator is cos2x - 1. You start by taking the derivative of this entire function, leaving you with -2sin2x [don't forget to use chain rule!]. Set the answer for the numerator aside, and now do the same thing with the denominator. The denominator for your original limit is just x, so the derivative of x is 1. Now you have -2sin2x for the numerator and 1 for the denominator. By l'Hopital's rule, you may now use this as your new limit. Try taking the limit as x approaches 0 of (-2sin2x)/1. Stuck? Pretend that this is your new problem, and forget the original problem that was given to you for now. What is the first step when trying to solve a limit? Plugging in! Plug 0 in for all values of x, and you see that (-2sin[2{0}])/1 ---> (-2sin[0])/1 ---> (-2[0])/1 ---> 0/1 = 0. Your answer is lim x-> 0 (cos2x - 1)/x = 0!

Algebra

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Question:

What is the "order of operations", how does it work, and why is it important?

Lash T.

Answer:

The "order of operations" is as follows: parentheses, exponents, multiplication/division (multiplication and division can be done in any order after parentheses and exponents), and addition/subtraction (similarly, these two can be done in any order after parentheses, exponents, multiplication, and division). A great way to remember this is by using the phrase "Please excuse my dear Aunt Sally" to remember PEMDAS, which is the acronym for the order of operations! When given a math problem, you must start by looking for any parentheses. If you see parentheses, solve whatever is in the parentheses. If you do not see parentheses, move on to the next step, which is looking for any exponents. Continue this process all the way through addition and subtraction, and you will find your answer! If you do not follow the "order of operations", you might get a wrong answer. You may be tempted to answer the problem from left to right, but this is often the incorrect way of solving the problem. For example, 3 + (2 x 6 - 4) should give you 11, but you may get 20 or another result if you do not apply "PEMDAS" correctly.

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