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Linda C.

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Trigonometry

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Question:

What quadrant contains the terminal side of the angle 10pi/3?

Linda C.

Answer:

Let's find out what angles fall in each quadrant first. Think the whole quadrant is like a circle (2pi) divided by 4 equal parts, circling around counter clock wisely. In the 1st quadrant, angles between 0 to pi/2 (0 to 90 degrees) fall into it. In the 2nd quadrant, angles between pi/2 to pi (90 to 180 degrees) fall into it. In the 3rd quadrant, angles between pi to 3pi/2 (180 to 270 degrees) fall into it. In the 4th quadrant, angles between 3pi/2 to 2pi (270 to 360 degrees) fall into it. The angles repeat themselves (fall into the same quadrant) every 2pi (360 degrees). This is very easy to understand, it visually means if you ran around in circles, you go back to your starting point every 2pi (360 degrees, or a full circle). That said, you can add 2npi ( n is an integer) to the boundaries of the quadrant. It is like the angle circles n*360 degrees and falls back to its quadrant. In the 1st quadrant, angles between 0+2n(pi) to pi/2+2n(pi) fall into it. In the 2nd quadrant, angles between pi/2+2n(pi) to pi+2n(pi) fall into it. In the 3rd quadrant, angles between pi+2n(pi) to 3pi/2+2n(pi) fall into it. In the 4th quadrant, angles between 3pi/2+2n(pi) to 2pi+2n(pi) fall into it. So we just need to re-write 20pi/3 in our question into a different format and see which quadrant it falls in. 20pi/3=6pi+2pi/3=2*3*pi+2pi/3 (you can see here n=3) And 2pi/3 falls between pi/2 to pi. Therefore, 20pi/3 falls in the 2nd quadrant.

Calculus

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Question:

Find the slop of the line tangent to y(X) when X is equal to 6. y(X)=X^4-X^3+4X^2

Linda C.

Answer:

The slop of the line tangent to a function f(x) at point x is actually the derivative of the function at that point. For example, the slope of the line tangent to f(X)=2X is the derivative of f(X), df(X)/dX=2. This means function f(X)=2X has a constant slope 2, no matter the value of x. Another example, the slope of the line tangent to f(X)=X^2 is the derivative of f(X), d(X^2)/dx=2X. When x is 6, the slope is 12. So the slope of the line tangent to y(X)=x^4-X^3+4X^2 is d(X^4-X^3+4X^2)/dx=4X^3-3X^2+8X When x is 6, plug 6 into the slope, which is: 4*6^3-3*6^2+8*6=804

Algebra

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Question:

If x is the average (arithmetic mean) of 2m and 7, y is the average of 4m and 19, and z is the average of 6m and 22, what is the average of x, y, and z in terms of m?

Linda C.

Answer:

First, let's break down the problem one segment of sentence by one. 1)" If x is the average (arithmetic mean) of 2m and 7" In mathematic terms, this sentence can be written as: x=(2m+7)/2 2) "y is the average of 4m and 19" In mathematic terms, this sentence can be written as: y=(4m+19)/2 3) "z is the average of 6m and 22" In mathematic terms, this sentence can be written as: z=(6m+22)/2 The question asks "what is the average of x, y, and z in terms of m?" So what is the average of x, y, and z written in mathematic terms? It should be (x+y+z)/3 We know x, y, and z, as x=(2m+7)/2, y=(4m+19)/2 and z=(6m+22)/2. Therefore, (x+y+z)/3={(2m+7)/2+(4m+19)/2 +(6m+22)/2}/3={(12m+48)/2}/3=2m+8. Since (x+y+z)/3 is the average of x, y, and z, and it equals to 2m+8 Therefore, the average of x, y, and z in terms of m is 2m+8

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