VCE Methods Question Thread!

[soccerboi]

When you solve |sin(x)+3cos(x)|=3 for [-pi,pi] on CAS, do you get four solutions? It should be 5 solutions if you graph them and count the intersection points.

But i keep getting 4 solutions with the solve command and i'm sure i've typed it correctly and included the domain. The solution at -pi doesn't show even though i specified -pi in the domain. Does anyone know what's wrong?

[Homer]

{x=0,x=-3.14159265358979,x=3.14159265358979,x=-2.49809154479651,x=0.643501108793284} These are the values I get.

[soccerboi]

i'm still missing the value x=-3.14... thats weird... and i've tried it with my other CAS. and also missing the x=-3.14...

[Greatness]

i'm still missing the value x=-3.14... thats weird... and i've tried it with my other CAS. and also missing the x=-3.14...

Yeah I didn't get the x=-3.14 either, if you don't approximate when you hit enter you get 4 exact solutions so they're your solutions, I'm thinking that -3.14..... is slightly larger than -pi hence it's not showing up.

[rife168]

i'm still missing the value x=-3.14... thats weird... and i've tried it with my other CAS. and also missing the x=-3.14...

Yeah I didn't get the x=-3.14 either, if you don't approximate when you hit enter you get 4 exact solutions so they're your solutions, I'm thinking that -3.14..... is slightly larger than -pi hence it's not showing up.

Wolfram Alpha says that it is exactly
Link

[Greatness]

Hmmm that's weird, what do the solutions say? If it's 5 then something is wrong with our calculators........ lol

[BubbleWrapMan]

is a solution but calculators don't necessarily know that it's exactly that. Depending on the algorithm they use to arrive at solutions (I would guess a numerical method was used in this instance), they might be slightly off, and if it's even slightly less than it, of course, won't show up as a solution. Generally there's always some slight amount of error when dealing with circular functions on a calculator, so if your calculator contradicts your common sense then it's probably because of a slight inaccuracy in the calculator.

Does it work if you set the lower bound to -3.15?

[Homer]

make sure the domain implied is inclusive. This is what I typed in solve(abs(3*cos(x)+sin(x))=3|-π≤x≤π,x) and I got 5 solutions

[dinosaur93]

1. How do you find the value of a?

2. Find Pr(X>0.5)

3. Pr (X>2 | X>0.5)

[Lasercookie]

1. How do you find the value of a?

If that's a probability density function (assuming it is from the rest of the question), the area from x = 0 to x = a will be equal to 1. You'll have two integrals that add up to 1 there.



You don't have to evaluate integrals, you can see that |x-1| will form a triangle, and 0.1 will form a rectangle.

Triangle:

If the area of the triangle is 1/2, to add up to 1 the rectangle must be 1/2 too.

Solve for

[b^3]

1. Remember that the total area under a probability density function must add to 1.

2. So we need to find the area under the probability density function for X>0.5, that is we split it up into its sections, integrate from 0.5 to 1 for the first section and form 1 to a (which is 6) for the second section.

(I just used the areas using triangles and such instead of integrating since its easier to do)
3. Same deal except we now have conditional probability.


EDIT: Beaten by laseredd

[Homer]

a=6 since the area under abs(x-1)=.5, hence area under 0.1 between "a" and 1 must equal 0.5.

Pr(x>5)= 1/8 +.5 = 5/8

Pr(X>2)= 2/5 hence (2/5)/(5/8)= 16/25

[dinosaur93]

awesome, tnx guys...

1. Remember that the total area under a probability density function must add to 1.

2. So we need to find the area under the probability density function for X>0.5, that is we split it up into its sections, integrate from 0.5 to 1 for the first section and form 1 to a (which is 6) for the second section.

(I just used the areas using triangles and such instead of integrating since its easier to do)
3. Same deal except we now have conditional probability.


EDIT: Beaten by laseredd

some pretty detailed answer right here ^^

sorry, but I dont get how you got0.5 for part d

[Homer]

http://www.mathgoodies.com/lessons/vol6/images/conditional_formula2.gif

[bonappler]

This was from a TSSM exam and I'm not sure about the solution.