Question

Since January 1, 1960, the population of Slim Chance has been described by the formula P=44000(0.96)^t, where P is the population of the city t years after the start of 1960. At what rate was the population changing on January 1, 1974?

rate =
people/yr

Answer 1

@mathmale @Directrix @Kainui @satellite73

Answer 2

Given a differentiable function, what's the usual method for coming up with a formula for the rate of change?

Answer 3

slope

Answer 4

(y2-y1)/(x2-x1)

Answer 5

take the derivative, plug in 14

Answer 6

that's for the slope of a straight line connecting 2 points. Not what we need to do here.

Again, imagine I give you a function f(t) which is differentiable. starting with f(t), how would you find a formula for the rate of change of this function f(t) with respect to t?

Answer 7

derivative maybe?

Answer 8

Why "maybe?"

Answer 9

To benefit most from calculus, try to learn WHY you are finding results named "derivative," "maximum," "optimize" and the like.

Answer 10

You are actually given a function of time: P=44000(0.96)^t.
What's it for?

Now supposing you found dP/dt. What is the resulting function of t? What are some possible interpretations of its meaning?

Answer 11

I took the derivative of P=44000(.96)^t and plugged in 14 and i got the rate of change

Answer 12

Sounds good. The derivative of P(t) does represent the time rate of change of P(t).
Next question: Why substitute 14 into the derivative?

Answer 13

because that is how many years took place between the initial year and initial population to the answer they are looking for

Answer 14

Supposing that satellite73 had not suggested plugging in 14, what would you have done to find the appropriate t value?

Answer 15

You sure about that? I'm not trying to give you a hard time so much as I am ensuring that you understand exactly where info such as that '14' comes from and what the initial value was.

Answer 16

well i know that 14 years is the time it took between 60 and 74

Answer 17

Very good. That's what I was probing for. Now let's check out that initial population, which you claim was 4,400. Would you defend that claim, please?

Answer 18

my bad 44000 not 4400

Answer 19

Cool. You deserve praise for having stuck with this trouble-shooting.

Answer 20

:D

Answer 21

P(t) represents the pop of Slim Chance, with t representing the number of years after 1960.
Finding the derivative, with respect to time, of P(t)

Answer 22

ok

Answer 23

gives you a formula for the rate of change of the pop with respect to time.
Then, if you happen to be interested in how

Answer 24

fast the pop is changing after 14 years, you subst. 14 for t, and then you've got it made.

Answer 25

One more question: What are the proper units of measurment

Answer 26

for the time rate of change of the population, P(t)?

Answer 27

people per/year

Answer 28

oh um

Answer 29

And have you found how many people per year represents the time rate of change in 1974?

Answer 30

yeah I did

Answer 31

Great. Any questions?

Answer 32

No thank you so much :D

Answer 33

My great pleasure, best wishes to you. Bye!