Geom. Applications of Calc. (1 Viewer)

[pLuvia]

^ o yeaa.. ahaha..oops



answer says minima?

So it is my bad, my maths is rotting very fast

 

[-pari-]

^

(2) Find the turning point of the curve y = 3x^4 + 1 & determine its nature.

So y'=12x^3

y''=36x^2

when y'' > 0 it means its concave. So it never becomes concave down...

y"< 0 means its concave down

yeah, but
y" = 0 means a point of inflexion
ie, y" on the LHS of the point should be different to the y" on the RHS of the point (in terms of bigger/smaller than zero)

right?

but in this case,
despite y" = 0
y">0 on BOTH sides of the turning point.

how does that work??

 

[bos1234]

^



yeah, but
y" = 0 means a point of inflexion
ie, y" on the LHS of the point should be different to the y" on the RHS of the point (in terms of bigger/smaller than zero)

right?

but in this case,
despite y" = 0
y">0 on BOTH sides of the turning point.

how does that work??

Point of inflexion means CONCAVITY

For example

y = x^3
y'=3x^2
y''=6x

At turning point,
dy/dx = 0
3x^2 = 0
x=0

at point of inflexion
y''=0

therefore, x=0

Therefore there is a horizontal point of inflexion at (0,0)

when x<0 y''<0
when x=0 y''=0
when x>0 y > 0

So the concavity is changing about the origin. So on the negative side of the x-axis it is concave down. As it is approaching (0,) its starting to become concave up. Then after 0, 0 its concave up

----------------

In your question
<TABLE cellSpacing=0 cellPadding=3 width="100%" border=0><TBODY><TR><TD class=alt2 style="BORDER-RIGHT: 1px inset; BORDER-TOP: 1px inset; BORDER-LEFT: 1px inset; BORDER-BOTTOM: 1px inset">(2) Find the turning point of the curve y = 3x^4 + 1 & determine its nature.

So y'=12x^3

y''=36x^2

when y'' > 0 it means its concave. So it never becomes concave down...

y"< 0 means its concave down</TD></TR></TBODY></TABLE>
This means that the concavity doesnt change. It remians concave up.

POINT OF INFLEXION REFERS TO CONCAVITY.

WHETHER THE CURVE CHANGES FROM CONCAVE UP TO CONCAVE DOWN AND IF IT DOES AT WHICH POINT DOES THIS HAPPEN

ask again if you dont understand it

 

[-pari-]

k, wait...so one thing,

when y" = 0, what does this mean??

(a)does it mean that the point in question is a point of inflexion?

(b) or does it just mean that at the point in question, there is "no concavity"...at that point it is flat...horizontal....
and THEN on either side of that point, whatever the concavity is will determine what KIND of point it is.

 

[bos1234]

k, wait...so one thing,

when y" = 0, what does this mean??

(a)does it mean that the point in question is a point of inflexion?

(b) or does it just mean that at the point in question, there is "no concavity"...at that point it is flat...horizontal....
and THEN on either side of that point, whatever the concavity is will determine what KIND of point it is.

(b) is right

if y'=(5,?)
and y''=(5,?)
then it means its a horizontal point of inflexion

 

[-pari-]

^ ooo right. k, i've been thinkin (a) ...that explains it lol

thanks

 

[bos1234]

kk cool

im gonna do some stuyding myself

cya and goodluck with the rest of ur studies:wave:

 

[-pari-]

(1) Sketch: y = x^2/(x-2)

domain: x=/= 2
range: i got y=/= 0,

i made x the subject of the equation, that put y on the denominator on the RHS, so y can't equal zero.

but! one of the turning points are (0,0).

what'd i do wrong when finding the range?

2) the forumla for surface area of a cylinder is given by SA = 2(pi)r (r + h)
the volume = 54 (pi) metres^3

surface area = 2(pi)r^2 + [108∏/r]
Find the radius that gives the minimum surface area.

after getting the equation, and differentiating it, i equate y' to zero.
but....
i get (pi)r^3 = 27...

now the answer is 3....
so if i ignore the (pi) it all works out. but am i allowed to ignore the (pi)??

 

[-pari-]

bump

am sure someone knows?

 

[nathan71088]

Sorry, just to make sure i understand you, what are you saying the dom,ain and range are in part 1?

 

[nathan71088]

if S.A. = 2(pi)r^2+(108(pi))?r then:

differentiating S.A. and making it = 0
d (S.A.) = 4(pi)r - (108 (pi))/(r^2) = 0
d (r)

then taking onto the other side to make everything positive:

4(pi)r = (108 (pi))/(r^2)

Divide everything by 4:

(pi)r = 27 (pi)/(r^2)

times both sides by r^2:

(pi)r^3 = 27(pi)

divide both sides by (pi):

r^3 = 27

then get 3rd root:

r = 3

you must have forgotten a pi somewhere