Graphing derivatives , and derivative's characteristics
http://screencast.com/t/GxfCh0DY
So basically this exercise is 100% to be in our final exam today at Calculus , as stated by our teacher so I am trying to master it

- Christos

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- reemii

and how's it going until now?

- Christos

For example On (a) do I have to find roots for only f(x) or f'(x) and f''(x) ?

- Christos

it says find roots of f

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## More answers

- reemii

the roots of f:
the x's such that f(x)=0
they are (graphically) the x's where the graph crosses the x-axis

- Christos

ooh first = 3/2

- Christos

second,third = (-6 +- sqrt(72))/2

- Christos

Which brings me to another problem of mine on how to actually point this number on the graph

- reemii

not sure. in the form \((x+3)^2\) you should not compute the square, actually they already made the factorization for you. you can just read the answer: second and third roots are equal: x=-3.

- reemii

(it's (-6 ± sqrt(0)) / 2 )

- Christos

ah yea that's true

- Christos

so we have one root for 3/2 and one for -3

- Christos

for b I just find f(0) of the function ?

- reemii

yes

- Christos

-27

- reemii

yes again ;)

- Christos

So the intervals (-inf,-9) increase
(-8,0) decrease
(0,inf) increase
At least thats what I got

- reemii

the derivative of the function is \(6x(x+3)\). the intervals are (-inf,-3), (-3,0), (0,inf)

- Christos

arent we using multiplication rule for the derivative

- reemii

yes.
whether the values you found are right or wrong, at least verify that the intervals don't leave gaps.
(your intervals left a gap between -9 and -8)

- reemii

or was it a typo?

- Christos

it wasnt a typo but I guess you are right on that one

- reemii

directly using the mult. rule:
\(((2x-3)(x+3)^2)' = 2(x+3)^2 + (2x-3)2(x+3) = 2(x+3) \times [(x+3)+(2x-3)]\).

- reemii

oops. last one is
\[ 2(x+3)\times [(x+3) + (2x-3)] = 2(x+3)[3x] = 6x(x+3)\]

- Christos

hm let me redo it

- Christos

2(x+3)^2 +4x^2 + 12x - 4x -12 up until now all correct?

- reemii

i don't think so.
do you apply the rule on \((2x-3)(x^2+6x+9)\) ? what's your starting point?

- Christos

Yes I apply the rule however i start with the product rule first before applying it

- reemii

show the first steps plz

- Christos

2(x+3)^2+2(x+3(2x-3)

- Christos

2(x^2+6x+9)+2(x+3)(2x-3)

- Christos

those are correct?

- reemii

yes, you made a mistake when computing the product \(2(x+3)(2x-3)\). this is equal to \(4x^2+6x-18\).

- Christos

Why mistake what do you multiply first with what

- Christos

ok ok I got it now for concave up/down (-inf,-1) down (-1,inf) up

- Christos

infection points x=-1

- Christos

and I am stuck at (f)

- reemii

your derivation is correctly done.
then it's only multiplication:
\[2(x^2+6x+9) + 2(x+3)(2x-3)\\
\quad = 2x^2+12x+18 + 2(2x^2-3x+6x-9) \\
\quad = 2x^2+12x+18 + 4x^2 + 6x - 18\\
\quad = 6x^2 + 18x \\
\quad = 6x(x+3)\]-> roots are -3 and 0.
-> (-inf, -3), (-3,0), (0,inf) are the intervals.

- Christos

Yea I see now are my next moves correctly done??

- reemii

you didn't obtain the correct expression for f' (you got the wrong roots -> wrong intervals). the inflexion point is at -3/2.

- Christos

you mean for f''(x) I got it wrong?

- Christos

I damn you are right

- reemii

\(f'\) is unfortunately wrong, so you could not get the right \(f''\).

- Christos

(-inf, -3/2) down (-3/2, inf) up I get it now

- Christos

right?

- reemii

ouch. no too fast.
\(f'(x) = 6x(x+3)\) with roots -3 and 0. -> (-inf,-3) up, (-3,0) down, (0,+inf) up.

- reemii

\(f''(x) = (6(x^2+3x))' = 6(2x+3)\) with root x=-3/2. -> inflexion point at -3/2.

- Christos

Bro I mean concave up down :D I moved a step

- Christos

I understood the previews one

- Christos

I am just stuck at (f) could you help me out surpass it?

- reemii

oh ok. (about concave up down)
relative extrema means you have to study what happens at each x such that f'(x)=0,
at -3 and at 0 in our case.

- Christos

only for f'(x) and f''(x) ?

- reemii

the relative extrema (local/global min or max) are only possibly located at the roots of \(f'(x)\). Look at \(x=-3\) and say if it's a min, max, or none of these (inflexion point). Do the same with \(x=0\). You can use \(f''\) to tell you about the shape of the curve at that particular point.

- Christos

is not a min nor a max at -3 0

- reemii

you mean (-3,0)?
it is a local maximum. Reason: \(f''(-3)<0\). (or just \(f\) is increasing before -3 and decreasing after -3.).

- Christos

bro you told me to check at f'(x) so which one do I check? do I check both f'(x) and f''(x) ? Do I only use the roots from the first derivative?

- reemii

1) to answer the question you have to do some job at each of the points that solve \(f'(x)=0\). We know that these points are -3 and 0.
2.1) what happens at -3 ?
2.2) what happens at 0 ?
To answer 2.1 and 2.2, you can use any means you want. For example, a quick way is to use \(f''\).
\(f''(-3)>0\) means it's a MIN.
\(f''(-3)<0\) means it's a MAX.
\(f''(-3)=0\) means it's an inflexion point.

- Christos

so I dont use -3/2 at all? I can only just use the roots of the first derivative for the second?

- Christos

@reemii

- reemii

What we know is that -3/2 is an inflexion point.
But it happens (often?) that it is not an extrema. Drawing:
|dw:1369665423749:dw|

- reemii

so you don't need to look at it at all in this question (f).
You would have to look at it if it was solution of \(f'(x)=0\). But it is NOT.

- Christos

Ok and the last and hardest part for me can you help me analise/make the graph?? Would it be possible??

- reemii

\(x=-3\) -> \(f''(-3) = 6(2*(-3)+3) < 0\) -> local max
\(x=0\) -> \(f''(0) = 6(0 + 3)>0\) -> local min.
Ok

- reemii

1) Always start with : put points of \(f\) that are easy: the roots, the intercept.
2) draw the missing part using the info in \(f'\) and \(f''\).

- Christos

Kinda can you show me what you do to graph it? I have like 5 minutes before the exams that's the most I can get atm

- reemii

first step: a few points
|dw:1369666161790:dw|

- reemii

then join using use \(f''\) at the extrema (-3 and 0)
|dw:1369666278751:dw|

- reemii

join in a healthy way (without forgetting hte inflexion point)
|dw:1369666362664:dw|

- reemii

use that order:
1)fixed points,
2)concavity at extrema
3) join. (pay attention to inflexion point, but that is less important than the rest)

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