BEC'S methods questions

[bec]

yeah i get what you mean about quotient/product rule as a general thing for keeping it in the right form, that does help

would i need to be able to do part (d) of this one without a calculator?:

where k is a real number.
a) Find f'(x) (x-k)(3x-k)
b) Find the coordinates of the points at which the graph of f has zero gradient (k,0) and ()
c) Show that the equation of the straight line joining the points where the graph of f has zero gradient is
d) Hence show that the point of intersection of f and the equation of the straight line found in part c occurs at the midpoint between the two points of zero gradient.

Also, I've seen Qs like this a couple of times before - what's the equation in (c) all about? does that apply to a general case in some way?

[Mao]

this applies to all cubics, given it is a "curve" (as opposed to saddle), the straight line joining the two turning points will pass through the point of inflection as well (the centre, midpoint, whatever you want to call it)

for part d, you have 2 choices:

1. solve

2. find the midpoint of and , then substitute into and to show that it is a point of intersection of these two graphs.
the second option shouldnt need calculator =)

[bec]

In multiple choice, what's the quickest way of working this out:



a)

b)

c)

d)

I lose a lot of time on Qs like this because even though i can do them, i can't efficiently get them into the right form...any tips?

[mark_alec]

In multiple choice, what's the quickest way of working this out:

See that or use the program DERIV on a TI-83+ or better.

[bec]

i do cas so i've got the ti-89, but my problem was that (in this question and in others) i can get the answer on my calc, but it's in a different form to the options.
it's interesting though, when i input it as

like you suggested, it did give the right form.

so is there any way of seeing, from the answers, what form i need to put something into the calculator? or do i just need to do these Qs the old-fashioned way and figure it out by hand?

[mark_alec]

so is there any way of seeing, from the answers, what form i need to put something into the calculator? or do i just need to do these Qs the old-fashioned way and figure it out by hand?

Can't say there is any shortcut to recognising the forms of questions besides being familiar with the question so that it is second nature as to what the answer is.

That being said, I would not worry too much if you need to spend a minute or two on a multiple choice question, since there will be others that are a simple calculation and do not require any manual work.

[Collin Li]

I lose a lot of time on Qs like this because even though i can do them, i can't efficiently get them into the right form...any tips?

By glancing at the options available, it is apparent that your denominator should have , except for option E. This is because the second term as simplified will differentiate into a term.

Once again, you should seek to obtain a common denominator by multiplying your other term by , unless the answer is E.

[bec]

Find the equation of the normal to the curve with equation that is parallel to the line with equation

The solution in the text says it's
I get

The equation I got is for a line that is PARALLEL to the equation they give in the question, but NORMAL to the curve at x=5....that's the way I interepreted the question.

Am I misinterpreting what the question is asking, or are the solutions wrong?

[Collin Li]

Yeah, you're supposed to find the equation of the normal to the curve, such that the normal will be parallel to .

Working backwards:
- This means you are looking for a normal with the same gradient as .
- This means you are looking for a normal that has a gradient of .
- That means you are solving for when is orthogonal (perpendicular) to , i.e.: .

Your gradient seems correct. I haven't verified the equation with calculations, but the solutions are wrong.

[bec]

oh good, that's what i thought - thanks!

is this solution wrong too, or am I:
The top of a ladder 5 metres long is sliding down a vertical wall at a rate of 3 metres per second. Find the rate at which the base of the ladder is sliding away from the wall, in metres per second, when the base of the ladder is 4m away from the wall.

I got (where b=distance between base of ladder and the wall, t=time)

The answer in the book is 4 .
I think this is the soln you'd get when h=4, but the question asks us for when b = 4 doesn't it?

[Mao]

Find the equation of the normal to the curve with equation that is parallel to the line with equation

The solution in the text says it's
I get

The equation I got is for a line that is PARALLEL to the equation they give in the question, but NORMAL to the curve at x=5....that's the way I interepreted the question.

Am I misinterpreting what the question is asking, or are the solutions wrong?

your interpretation is correct, and the book is wrong.

the gradient for is within (its the top part of a hyperbola, part of the specialist course)
that is, y cannot be parallel to , so it must be meant for the normal, with a gradient of

[bec]

thanks mao

how do i do this one:
The sum of two numbers is 14. Find the numbers if their product is a maximum.
I can do it just because i know that 7*7 would be the biggest of all the possible combinations, but I don't know how to do it methodically....or with a cas calculator

[Mao]

oh good, that's what i thought - thanks!

is this solution wrong too, or am I:
The top of a ladder 5 metres long is sliding down a vertical wall at a rate of 3 metres per second. Find the rate at which the base of the ladder is sliding away from the wall, in metres per second, when the base of the ladder is 4m away from the wall.

I got (where b=distance between base of ladder and the wall, t=time)

The answer in the book is 4 .
I think this is the soln you'd get when h=4, but the question asks us for when b = 4 doesn't it?

and yes, you are right.

the book wrongly substituded b=4 into h.

[Mao]

thanks mao

how do i do this one:
The sum of two numbers is 14. Find the numbers if their product is a maximum.
I can do it just because i know that 7*7 would be the biggest of all the possible combinations, but I don't know how to do it methodically....or with a cas calculator

let be one number
the other number would be

and their product is

now you just gotta maximise it =)
(maximum turning point)

[bec]

hahah ok yes obvious

i've got a more general question: how do you sketch "unfamiliar" graphs, in the non-calc section?
eg.

here's the way i do it (and still often get wrong):
1. find x and y-ints
2. find stationary points, work out if max/min/spoi
3. look for obvious restrictions in domain (eg in the one above, )
4. work out if they are "holes" or asymptotes by subbing in points slightly above and slightly below the x-value that doesn't exist

it's mostly asymptotes that get me - is there a more efficient way of "testing" for asymptotes than my step 3/4?

thanks