I can't get the right answer for these two, mainly because they include trigonometry and lns.
For the first:
(a)
we want $\displaystyle S(12)$
$\displaystyle S(12) = 72 - 15 \ln (12 + 1) = 72 - 15 \ln (13) \approx 33.53$
(b)
we want $\displaystyle S'(4)$
Now $\displaystyle S'(t) = - \frac {15}{t + 1}$
$\displaystyle \Rightarrow S'(4) = - \frac {15}{5} = -3$
Do you know how to find the derivative of ln ?
(c)
I assume the original score is to be taken when $\displaystyle t = 0$
This means the original score is 72
we want to know when does the average score becomes less than $\displaystyle \frac {3}{4} 72 = 54$
So we must solve for $\displaystyle S(t) < 54$
we want $\displaystyle t$ such that:
$\displaystyle S(t) = 72 - 15 \ln(t + 1) < 54$
$\displaystyle \Rightarrow -15 \ln(t + 1) < -18$
$\displaystyle \Rightarrow ln(t + 1) > \frac {6}{5}$
$\displaystyle \Rightarrow e^{ \frac {6}{5}} > t + 1$
$\displaystyle \Rightarrow t > e^{ \frac {6}{5}} - 1$
The second
The rate of change of voltage is $\displaystyle v'(t)$
Now, $\displaystyle v'(t) = -2 \sin(t) - 2 \sin(2t)$ ........By the Chain rule
we want $\displaystyle t$ such that $\displaystyle v'(t) = 0$
That is, we want to solve $\displaystyle -2 \sin(t) - 2 \sin(2t) = 0$
$\displaystyle \Rightarrow \sin(t) + \sin(2t) = 0$
$\displaystyle \Rightarrow \sin(t) + 2 \sin(t) \cos(t) = 0$
$\displaystyle \Rightarrow \sin(t) (1 + 2 \cos(t)) = 0$
$\displaystyle \Rightarrow \sin(t) = 0 \mbox { or} 1 + 2 \cos(t) = 0$
$\displaystyle \Rightarrow t = 0 \mbox { or} \cos(t) = - \frac {1}{2}$
$\displaystyle \Rightarrow t = 0 \mbox { , } \pi \mbox { , } 2 \pi \mbox { or } t = \frac {2 \pi}{3} \mbox { , } \frac {4 \pi}{3} $
EDIT 1: added $\displaystyle \pi $ and $\displaystyle 2 \pi $ to the solutions
EDIT 2: Thanks for looking out behemoth100...would you believe that I intentionally left the answer incomplete to see if SportfreundeKeaneKent would catch it?
Ok well given that its only 7am in the morning this may be a stupid comment but...
Isnt sin(t) = 0 at 0 pi and 2pi? The interval is 0<t<2pi (except they are equal to or less than signs).
So isn't your answer incomplete?
If I have made a mistake on BASIC DIFFERENTIATION AND TRIG THEN I SHOULD BE SHOT!!! I'm scared agaist going up against the almighty Jhevon