Question

Some derivative graph problems. Pics attached

Answer 1

the 2 graphs on top were provided

Answer 2

the bottom 4 are my ideas

Answer 3

the r value is the growth rate

Answer 4

bottom one the derivative

Answer 5

can anyone check my work please? :)

Answer 6

update: r is the per capita growth rate

Answer 7

I am not sure I know what per capita growth rate means, but the derivative graphs I can say is absolutely correct.

Answer 8

let me see if I can describe it mathematically

Answer 9

From looking at the solid line on the first graph of r I was going to guess it meant it's the ratio of N to dN/dt but I can't really tell by looking at the second graph from that guess.

Answer 10

hmm

Answer 11

what would you expect?

Answer 12

I suppose a bit

Answer 13

flatter initially?

Answer 14

until the midpoint maybe?

Answer 15

I guess since the graph is going down for the second graph it would be r=N/N' but it's really sorta hard to say for sure, it is only an approximate guess.

Answer 16

well I know this r value needs to hit 0 at the end, if that helps

Answer 17

I should have said r=N'/N but I still don't know what per capita means. What is capita? That's what we're dividing by haha.

Answer 18

ok, I think you may be right about the ratio of N to dN/dt describing the relation

Answer 19

lol

Answer 20

My guess is Growth Per Capita means N' / N. So that means N' is growth, / is per, and N is Capita lol.

Answer 21

per capita=for each individual

Answer 22

if it's positive, there are more people. If it's negative, there are less people

Answer 23

as compared to the total

Answer 24

idk if I'm making sense, long night lol

Answer 25

I think you're making sense, so can we say N=people?

Answer 26

yep, population

Answer 27

reviewing your points, I think I agree with the ratio of N to dN/dt =r

Answer 28

so r= (N x deltaN)/delta T ?

Answer 29

It seems that way based on notes

Answer 30

no!

Answer 31

finally, got it

Answer 32

deltaN/deltaT = r N

Answer 33

Ahh so then N'/N=r is true! =D

Answer 34

yep

Answer 35

Ok, well are we done here or what remains I kinda got lost.

Answer 36

checking the graphs

Answer 37

second row

Answer 38

you said the one for logistic growth (on the right) seemed strange

Answer 39

so I'm thinking flat until the midpoint, then curving down?

Answer 40

Yeah that sounds good to me.

Answer 41

ok, and for the left one on the second row, for exponential growth? It's constant, right? Thanks by the way :)

Answer 42

ok, I'm pretty sure it's right. Thanks again.

Answer 43

Yeah, just a simple consequence of the fact: \[\Large N=e^{ax} \\ \Large \frac{dN}{dt} =ae^{ax} \\ \Large \frac{N'}{N}=a\]

Answer 44

erm nevermind the mixing of the x's and t's you know what I mean lol.

Answer 45

lol, yeah, e^x magic