Question

Calculus
The rate of change of the price of a vintage 1963 Chevy Corvette Coupe, in dollars per month, is modeled by the function P(t)=500cos(pi/t) for ) 12 years ago

Answer 1

\[P(t)=500\cos \left(\begin{matrix}\pi \\ t\end{matrix}\right)\]
for \[0\le t \le 12\]

Answer 2

The price is increasing at \(t=3\) if \(P'(3)>0\) and decreasing if \(P'(3)<0\).

Answer 3

can you show me how to find the derivative to the equation/

Answer 4

\[P(t)=500\cos\frac{\pi}{t}\]
Apply the chain rule. The "outside" function is \(\cos\), and the "inside" function is \(\dfrac{\pi}{t}\):
\[P'(t)=500\left[\frac{d}{dt}\cos\right]\frac{\pi}{t}\cdot\frac{d}{dt}\left[\frac{\pi}{t}\right]\]
You won't find this notation in a textbook, so I should explain. Normally, you would write \(f(t)=500\cos t\) and \(g(t)=\dfrac{\pi}{t}\). The given function is then a composition, \(f(g(t))\). So, by the chain rule, the derivative is
\[f'(g(t))\cdot g'(t)=500\left(-\sin\frac{\pi}{t}\right)\cdot \left(-\frac{\pi}{t^2}\right)\]

Answer 5

So when I plug in t=3 would I use sin to find \[(-\frac{ \pi}{ t^2})\]

Answer 6

No.

The expression

500(−sin[π/t])⋅(−π/[t^2)])

has three separate factors, including the -500 and the sine term. Evaluate the third term separately, by letting t = 3.

Answer 7

Then multiply the three (all numeric) factors together.