Question

Please help with this two part question (Calculus)^^

Answer 1

Answer:

1) -0.016 pounds per square inch per cubic inch.

2) $\displaystyle V'(P)=-\frac{800}{P^2}$

Step-by-step explanation:

We are given the equation $PV=800$.

Part A)

We want to determine the average rate of change of P as V increases from 200 cubic inches to 250 cubic inches.

To find the average rate of change between two points, we find the slope between them.

Rewrite the given equation as a function of V:

$\displaystyle P(V)=\frac{800}{V}$

Hence, the average rate of change for V = 200 and V = 250 is:

$\displaystyle \begin{aligned} m &= \frac{P(250) - P(200)}{250 - 200} \\ \\ & = \frac{3.2 - 4}{250 - 200} \\ \\ & = -0.016\end{aligned}$

Therefore, the average rate of change is -0.016 pounds per square inch per cubic inch.

Part B)

We want to express V as a function of P. This can be done through simple division:

$\displaystyle V(P)=\frac{800}{P}$

We want to show that the instantaneous change of V with respect to P is inversely proportional to the square of P. So, let's take the derivative of both sides with respect to P:  

$\displaystyle \frac{d}{dP}\left[V(P)\right]=\frac{d}{dP}\left[\frac{800}{P}\right]$

Differentiate. Note that 1/P is equivalent to P⁻¹. This allows for a simple Power Rule:

$\displaystyle \begin{aligned} V'(P) & = 800\frac{d}{dP}\left[ P^{-1}\right] \\ \\ & = -800(P^{-2}) \\ \\ & = -\frac{800}{P^2}\end{aligned}$

Therefore, the instantaneous change of V is indeed inversely proportional to the square of P.