Mathematics Question Thread

[Ellie__]

Hey guys,

Can someone please explain how to answer locus questions?? I'm so confused on what they are

Ah also, how do I complete the square???

Thankyou!!

Mod Edit [Aaron]: Posts merged. You are able to edit posts you have already made.

[anotherworld2b]

I was wondering how do you know when to use substitution when you are asked to antidifferentiate exponentials and trigonometry? Would it be when there is product rule involved?

For those questions, you have to use substitution. For g), you know that the derivative of e^-x is -e^-x.
So use u=1+e^-x, du/dx = -e^-x
So you end up with
You can see that there's a function, and the derivative of that function in each question. So let u = the number to a power / more complicated part then find the derivative of u, and use that to substitute in to antidiff with respect to u.

For a)
You can divide it up into 5 sections, I'll call them 1,2,3,4,5.
Section 3 and 4 cancel out as they're equal in area and opposite signs
Area 1 = positive 1/4 * π * 1² = π/4
Area 2 = negative 1/2 *(1*1) = -1/2
Area 5 = positive 1/4 * π * 1² = π/4
Area 1 + Area 2 + Area 5 = π/2 - 1/2

Edit: this is for b)
Area from 0-1 is positive, areas from 1-2 and 2-3 cancel out as one's positive and one's negative and they're equal, and area from 3-4 is negative
Area 1 = positive (1-1/4*π)
Area 4 = negative (1/4*π)
Area 1 + Area 4 = 1-1/4*π-1/4*π = 1-π/2

[jamonwindeyer]

Can i have help with these questions please?

Sure! For your first one, expand:



Then just integrate each piece second one, split it into two terms:



Then integrate those pieces when you can expand (ie, the power is a square) or otherwise separate into smaller pieces, do this! Almost always it will yield an easier integration!

It could be because it is 2:30am, but I don't see a way to do that last one easily without just expanding that last bracket? It's a cubic, which isn't horrendous, you'll end up with 6 terms - You'll then be able to multiply all of them by the \(\frac{1}{x^2}\), group like terms then integrate each piece

[jamonwindeyer]

Hey guys,

Can someone please explain how to answer locus questions?? I'm so confused on what they are

Ah also, how do I complete the square???

Thankyou!!

Check the bottom half of this guide! There is a quick summary/explanation of Locus - And feel free to pop any specific examples that are troubling you here and we can work through them for you!

Completing the Square, let's do an example.

Solve \(2x^2-5x+3=0\)

Note: We could just factorise this normally, but I'll show you what it looks like with completing the square. The process is identical

Step 1 - Remove the coefficient of \(x^2\). This is just the way we normally do it - Keeping it there isn't much harder at all but I'll just do what your textbook probably does.

So dividing by two:\(\implies x^2-\frac{5}{2}x+\frac{3}{2}=0\)

Step 2 - Move the constant to the other side.



Step 3 - Split the middle term in half.



Step 4 - Take the coefficient of one of those halves, and square it. In this case, \(\left(\frac{-5}{4}\right)^2=\frac{25}{16}\). Add this number to BOTH sides of the equation (we can't do just one, gotta keep the sides the same!)



Now look at the LHS - We have something in the form \(a^2+2ab+b^2\) - Exactly what we need to do a perfect square factorisation. So this becomes:



So that's the method - Take the middle number, halve it, and square it - Then add that to your quadratic. It will turn it into a complete square. As to WHY this works, the best explanation is visual - This is probably the best/most famous explanation for 2U students

[jamonwindeyer]

I was wondering how do you know when to use substitution when you are asked to antidifferentiate exponentials and trigonometry? Would it be when there is product rule involved?

If you can't do it with your basic rules, substitution is almost always the solution (there is also integration by parts but if you don't learn it, obviously it won't be that )

[1937jk]

So I'm having trouble figuring out what this question is asking and how to solve it:
Find d/dx (lnx/x)

[asd987]

Hi, is there anyway to solve this question without drawing the actual cos graph and y=+- 1?

[asd987]

So I'm having trouble figuring out what this question is asking and how to solve it:
Find d/dx (lnx/x)

Hi, you would have to use quotient rule
u=ln , u'=1/x
v=x  , v' = 1

so (1 - lnx)/x^2

[jakesilove]

So I'm having trouble figuring out what this question is asking and how to solve it:
Find d/dx (lnx/x)

Yep, the answer above is absolutely correct! Just differentiate the function by using the quotient rule

[jakesilove]

Hi, is there anyway to solve this question without drawing the actual cos graph and y=+- 1?

Hey! You could solve the equation for cos(x)=+-1, however honestly it is much easier to just sketch the curve.

[jamonwindeyer]

Hey! You could solve the equation for cos(x)=+-1, however honestly it is much easier to just sketch the curve.

Ditto - Like, solving that equation is just a thing you learn, just the same as the exact ratios! But the graphs are the best way to jog your memory of them (the \(\sin\) and \(\cos\) curves equal to 1 or 0 are called the boundary angles in a lot of textbooks)

[katnisschung]

hi hi!

need to check if my answer is right

bill wants to put a small rectangular vege garden
in his backyard using two existing walls (x,y) as part of the
border. He has 8m of garden edging for the other two sides

Find the dimensions of the garden bed that will give the greatest area

Thanks!

i got 4m by 4m (presuming that p=16m because x+y=8 then the other side also has
to be 8 as its a rectangle)

[jamonwindeyer]

hi hi!

need to check if my answer is right

bill wants to put a small rectangular vege garden
in his backyard using two existing walls (x,y) as part of the
border. He has 8m of garden edging for the other two sides

Find the dimensions of the garden bed that will give the greatest area

Thanks!

i got 4m by 4m (presuming that p=16m because x+y=8 then the other side also has
to be 8 as its a rectangle)

That's correct

[katnisschung]

hello again!

no answers so i want to check..

A piece of wire 10m long is broken into two parts
which are bent into the shape of a rectangle as shown.
Find the dimensions x and y that make the total area
a minimum

basically the rectangle is x,y for its dimensions
and the square is x for its length

somehow i got 1.25 for both x and y which can't be correct
becos one is a rectangle?? 

[jamonwindeyer]

hello again!

no answers so i want to check..

A piece of wire 10m long is broken into two parts
which are bent into the shape of a rectangle as shown.
Find the dimensions x and y that make the total area
a minimum

basically the rectangle is x,y for its dimensions
and the square is x for its length

somehow i got 1.25 for both x and y which can't be correct
becos one is a rectangle?? 

Can't really tell without the diagram, I'd expect 2.5 maybe? Perhaps upload pics of your working so we can glance it