Question

what is a sign graph and how do i draw one for 3x^2-4x+2 for F'(x)

Answer 1

sign graphs are annoying. it is easier to find the zeros of \[3x^2-4x+2\] if any and then recall that the graph of \[y=3x^2-4x+2\] is a parabola that faces upwards, so it will be positive outside the zeros and negative between them (from the picture).

Answer 2

unless i am not paying attention it looks like this does not have any zeros.
\[b^2-4ac=(-4)^-4\times 3\times 2 = 16-24= -8\] so no real zeros. means the parabola is always above the x axis and so always positive. if this is a derivative of something, then that something must always be increasing. check my work because i might have made an arithmetic mistake.

Answer 3

Sign graphs are an excellent tool. For example in this case, find derivative f'(x) = 6x^2-4. Find critical points 3x^2-2=0
3x^2=2
x^2=2/3
x=sq rt (2/3)
So you evaluate from negative infinity to critical point and from critical point to positive infinity. Plug in any number between negative infinity to critical point to find if it is positive or negative within integral.

Answer 4

i suppose. but if \[3x^2-2=0\]
\[3x^2=2\]
\[x^2=\frac{3}{2}\]
then there are two zeros,
\[x=\sqrt{\frac{6}{2}}\] and
\[x=-\sqrt{\frac{6}{2}}\]
don't need to test anything, since we know \[y=3x^2-2\] is a parabola facing up, negative between zeros and positive outside them. guess it is just a matter of taste.

Answer 5

a sign graph i believe is the numberline marked out for + and - regions right?

Answer 6

<...............(c)..............>
- +
slope is decreasing when (-); amd increaseing when (+)

Answer 7

and (c) is the zero of f'(x)

Answer 8

yes, usually you stack the factors up
x-1 - - - - - - - - - - - - - - - - - - - - - - + + + + + + +
x+3 - - - - - - + + + + + + + + + + + + + + + + + +

_________-3______________1__________

+ - +

looks something like the above

Answer 9

i perfer the stacking; makes it simpler :)

Answer 10

Yes, it is bookkeeping, where you can list behavior of f(x), f'(x) and f''(x). It is not a big deal, but for someone learning calculus, this subject is overwhelming and it is a good way to read the graph.

Answer 11

the second derivatives sigh chart tells you concavity and inflection...right?

Answer 12

are there any intriguing details for higher derivatives?

Answer 13

i will not argue that point, but it is also good to know in advance what a quadratic looks like, what a cubic looks like etc, so you don't have to reinvent the wheel each time.
yes second derivative gives concavity.

Answer 14

third derivative is sometimes called "jerk" as it is the rate of change of the acceleration (assuming your original function is the position function, so first derivative is velocity and second is acceleration)

Answer 15

hunh... give medal changed to good answer :) more descriptive i spose of its intent lol

Answer 16

its good to know, i think, that 1st derivative zeros can give false readings; in the case where an inflection point has a 0 slope to it