cherylim23's methods question thread :)

[itolduso]

yes, velocity at t=2 is undefined

[brightsky]

yes, velocity at t=2 is undefined

Yes, that's right. My bad.

[TrueTears]

Let y = f(x) be the curve (assuming it's a function of a single variable)



The rest should be trivial right? Just integrate and then some substitutions.

[brightsky]

2x + 9y = 3 ---> 2x = 3 - 9y ---> x = (3 - 9y)/2

Sub into curve, (3 - 9y)y/2 + y + 2 = 0 -->y = -4/9 or y = 1, sub back in you get x = 7/2 or x = -3

So your points P and Q are (7/2, -4/9) and (-3, 1)

Now just implicit diff: y + xy' + y' = 0, xy' + y' = -y, y'(x + 1) = -x'y, y' = -y/(x+1)

Then sub and your done.

[Andiio]

..The gradient of a curve varies directly as x? o_o Does that just mean that m = x?

[aznxD]

..The gradient of a curve varies directly as x? o_o Does that just mean that m = x?

No direct variation is where if where k is a constant

So let the curve of the graph be f(x)
Therefore the gradient of the curve
Then as TT mentioned, you just integrate f'(x) and sub in the coordinate values to find the equation of the curve

[|ll|lll|]

I don't know if this makes any sense...
but we all know how linear graphs are always one-to-one functions, and quadratic graphs are always many-to-one functions for the domain R, however, cubic graphs may be one-to-one functions or many-to-one functions across the domain R as well.

So, for all cubic graphs with an inflection point, they are definitely one-to-one functions. Otherwise, vice versa. Is this true for all cases?

[pi]

Your understanding is correct. Cubic graphs can either be one-to-one (inflection form only) or many-to-one (general shape), and its a point a lot of people would overlook (or not care about).

I don't know if this makes any sense...

...

Is this true for all cases?

Don't really understand that last question. If you want to know if all cubics in the form f(x)=ax^3 (on Cartesian plane) are one-to-one, that is correct. This property is true for all f(x)-ax^n, where n is an odd natural number (they all have inflection points then).

[|ll|lll|]

Thanks Rohitpi. What I meant was that for all cubic graphs with an inflection point, they'd definitely be one-to-one functions yes?
I guess you pretty much answered it

Also, we know that all cubic graphs will at least have one solution, rational or irrational. Does this lead to: all cubic graphs with an inflection point would have one and only one solution?

[pi]

Thanks Rohitpi. What I meant was that for all cubic graphs with an inflection point, they'd definitely be one-to-one functions yes?
I guess you pretty much answered it

Also, we know that all cubic graphs will at least have one solution, rational or irrational. Does this lead to: all cubic graphs with an inflection point would have one and only one solution?

All cubic graphs have 3 solutions: one real and two complex/imaginary, two real and one complex/imaginary or all three real (dunno if all three can be complex/imaginary, but who cares?). A graph with an inflection point has 3 (real) solutions, but only one distinct solution. There is an important difference there.

Learnt that here:

...

• has 3 real solutions . But 2 distinct real solutions

...

^Check out his tips, they are really good (MM resources thread and scroll down go here)

[pi]

Haha, I was referring to one and only one REAL solution. Sorry if I didn't make myself clear enough.
And nope, it's not possible for a cubic graph to have three complex solutions

Thought three complex solutions would be weird (thanks for the clarification)... On the solution, yep, one real distinct solution for an inflection cubic, but remember that are really three real solutions (just one -one real one- is repeated three times) <--- wording is important. I never thought of that (the 'distinct stuff') until I read kyzoo's tips... Don't think its taught too well in textbooks

[brightsky]

Haha, I was referring to one and only one REAL solution. Sorry if I didn't make myself clear enough.
And nope, it's not possible for a cubic graph to have three complex solutions

Thought three complex solutions would be weird (thanks for the clarification)... On the solution, yep, one real distinct solution for an inflection cubic, but remember that are really three real solutions (just one -one real one- is repeated three times) <--- wording is important. I never thought of that (the 'distinct stuff') until I read kyzoo's tips... Don't think its taught too well in textbooks

It's actually referred to in Essentials, not sure about MQ. It's the fundamental theorem of algebra.

[pi]

It's actually referred to in Essentials, not sure about MQ. It's the fundamental theorem of algebra.

Haha, I stand corrected. Sorry cherylim23... (bad info there). And my school uses MQ...

[|ll|lll|]

Thought three complex solutions would be weird (thanks for the clarification)... On the solution, yep, one real distinct solution for an inflection cubic, but remember that are really three real solutions (just one -one real one- is repeated three times) <--- wording is important. I never thought of that (the 'distinct stuff') until I read kyzoo's tips... Don't think its taught too well in textbooks

Yes, think about it this way. When the discriminant is zero, there are actually two solutions, but two repeated solutions that's all.

Hence arises the confusion when the question states, for what values of (insert variable) would the equation have two solutions.
I don't know whether to let discriminant > 0 or discriminant >= 0

[pi]

Yes, think about it this way. When the discriminant is zero, there are actually two solutions, but two repeated solutions that's all.

Hence arises the confusion when the question states, for what values of (insert variable) would the equation have two solutions.
I don't know whether to let discriminant > 0 or discriminant >= 0

Assuming quadratics (as cubics also have discriminants, but big ones...), two solutions would be dis > 0.