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(1) The graph consists of two sections. For x < 0, y = -x + 3x - 4 For x >/= 0, y = x + 3x - 4 (2) Sketch the graph of y = 1/(x^2 - 1), see that y > 0 or y < -1. .: range is {x: x > 0} U {x: x < -1}. (3) Similar to (2), graph the function to find out. (4) y = |x|/x^2 can be written as...

No such point P can exist in the real x-y space because AB = sqrt[(3+5)^2 + (-1-4)^2] = sqrt(89). .: PA^2 + PB^2 > 4. :) :) :wave:

What has been translated?

Symmetric equations, e.g. x - 1 = y + 2 = z - 3 :) :) :wave:

:) :) :wave:

:) :) :wave:

Wrong substitutions 1/2 [(1+0.04) + 2 (1/4 + 1/9 + 1/16)] :) :) :wave:

Continue from my last post. [(1-sqrt(3)) + (1+sqrt(3)i]/2sqrt(2) = cos(7pi/12) + isin(7pi/12) .: (1-sqrt(3))/2sqrt(2) + (1+sqrt(3))i/2sqrt(2) = cos(7pi/12) + isin(7pi/12) .: cos(7pi/12) = (1-sqrt(3))/2sqrt(2) and sin(7pi/12) = (1+sqrt(3))/2sqrt(2) Note: 2sqrt(2) = sqrt(8) For...

Work out z in polar form to be 2sqrt(2) cis(7pi/12). .: (1-sqrt(3)) + (1+sqrt(3)i = 2sqrt(2) cis(7pi/12) .: [(1-sqrt(3)) + (1+sqrt(3)i]/2sqrt(2) = cis(7pi/12) .: [(1-sqrt(3)) + (1+sqrt(3)i]/2sqrt(2) = cos(7pi/12) + isin(7pi/12) Compare the two sides to obtain answers. :) :) :wave:

What is in 4a) ? For 6) Change 1- i , -1 + sqrt3 i to polar form. Then multiply and divide in polar form.

Thanks for your kind words

:) :) :wave:

:) :) :wave:

(1) |x-3|/(3-x) = x, .: |x-3| = x(3-x) and x =/= 3. Two possibilies: x-3 > 0 or x-3 < 0. If x-3 > 0, then x-3 = x(3-x) and x > 3. x-3 = 3x - x^2 x^2 -2x - 3 = 0 (x-3)(x+1)=0 .: x = 3...

Let's do the last one first. x^(3/2) = 1/8 , .: x = (1/8)^(2/3) = (1/2^3)^(2/3) =(2^(-3))^(2/3) =2^(-2) =(1/2)^2 =1/4...

Consider e.g. a = -108, do you agree that -108 < 0. Now what is |-108|? Of course |-108| = 108 = -(-108), i.e. |a| = -a if a < 0. :) :) :wave:

How do you derive these formulae? Is it a good idea for sec students to learn the derivations?

Divide the numerator by the denominator, then integrate. 2) ò..... dx together denote integration. Sometimes the integrand may have variables other than x, ò....dx means to integrate with respect to x only. :) :) :wave:

Try to make a simple one like: http://hyperphysics.phy-astr.gsu.edu...indmot.html#c1 Warning! Do not connect directly to the mains supply. Use school lab power pack AC. Make sure electromagnet has many windings in order not to short circuit the power pack. :) :) :wave:

How did you arrive at the figure 99.999999%? 1 miss out of 100000000. Close enough is not good enough for a pure mathematician. Keeep trying. :) :) :wave: