Question

Calculus help?

A company introduces a new product for which the number of units sold, S, is projected to be
S(t)= 200(5- (9/2+ t)) where t is the time in months.
a)Find the mean value of S(t) during the first year using 0 for the start year and 12 for the end.
b) During what month does S'(t) equal the mean value?

Answer 1

\[avg = \frac{1}{b-a} \int\limits_a^b S(t) dt\]

Answer 2

S'(t) = -200

Answer 3

is that the mean? I don't understand...

Answer 4

@dumbcow

Answer 5

@phi

Answer 6

sorry i was away...yes avg is same as mean
are you sure you wrote function correctly? it doesn't make sense for the derivative to be a constant

Answer 7

well the function is set up \[S(t)= 200 (5 - \frac{9 }{ 2+t })\]

Answer 8

ahh ok i misread it

Answer 9

so you want to set mean = derivative

\[\frac{1}{12} \int\limits_0^{12} S(t) dt = S'(t)\]

Answer 10

for integral use fact that
\[\int\limits \frac{1}{u} = \ln u\]

Answer 11

I have no idea how to solve that...........

Answer 12

ok
\[\frac{50}{3} \int\limits_0^{12} (5 - \frac{9}{2+t}) dt\]
\[= \frac{50}{3} |_0^{12} 5t -9 \ln(2+t)\]
\[=\frac{50}{3} [ (60-9 \ln(14)) - (0-9\ln(2)) ]\]
\[= 1000 - 150(\ln(14) -\ln(2))\]
\[=1000 - 150 \ln 7\]

Answer 13

thats the last answer because there are no like terms right?

Answer 14

right so that is mean (part a), now you set that equal to derivative for part b

Answer 15

the derivative is s'(t)= 1800/(t+2)^2

Answer 16

right?

Answer 17

so 1000-150In7 = 1800/(t+2)^2 ?

Answer 18

and how do I solve that?

Answer 19

use algebra to isolate "t"

first i would move (t+2)^2 to other side by multiplying

Answer 20

to help simplify you could replace"1000 -150ln7" with a constant like "k"
\[k = \frac{1800}{(t+2)^{2}}\]

Answer 21

\[(t+2)^{2} = \frac{1800}{k}\]

Answer 22

something is wrong the answer is a neg "t" value.
are you learning about the Mean Value Thm ??
i think you want the mean value of S'(t) ??

Answer 23

sorry guys just came back

Answer 24

they said S(t) not the derivative, I tried wolframalpha but it doesn't undrestand the inquiry

Answer 25

oh is that how you type it in wolframalpha?

Answer 26

so thats the answer?

Answer 27

its seems pretty big number though?,,

Answer 28

yeah I see that. ok so how do i find b

Answer 29

@jdoe0001 @jigglypuff314

Answer 30

I don't get it. the problem is wrong?

Answer 31

for convenient to other helpers http://www.wolframalpha.com/input/?i=integral++%28200%285-+%289%2F%282%2Bt%29%29%29%29+from+0+to+12

Answer 32

http://www.wolframalpha.com/input/?i=+%28200%285-+%289%2F%282%2Bt%29%29%29%27