• October 2nd 2008, 10:09 PM
JimDavid
I don't get how you suppose to use differentiation to solve these 3 problems, please help! I been trying to do it for hours, ahhh!

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1. Recall that the area A of a circle with radius r is pi r^2 and that the circumference C is 2 pi r. Notice that dA/dr=C. Explain in terms of geometry why the instantaneous rate of change of the area with respect to the radius should equal the circumference.
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2.An apple farmer currently has 156 trees yielding an average of 12 bushels of apples per tree. He is expanding his farm at a rate of 13 trees per year, while improved husbandry is improving his average annual yield by 1.5 bushels per tree. What is the current (instantaneous) rate of increase of his total annual production of apples? Answer in appropriate units of measure.

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3.If gas in a cylinder is maintained at a constant temperature T, the pressure P is related to the volume V by a formula of the form:

P= (nRT/V-nb)-(an^2/V^2)

in which a, b, n, and R are constants. Find dP/dV.
• October 3rd 2008, 11:49 AM
Sean12345
Hi JimDavid,

For the last one we have

$P=\left(\frac{nRT}{V-nb}\right)-\left(\frac{an^2}{V^2}\right)$ , where $a$ , $b$ , $n$ and $R$ are constants.

Note the phrase "constant temperature $T$" thus $T$ is also constant. Hence re-write the equation as follows an differentiate as normal.

$P=nRT(V-nb)^{-1}-an^2V^{-2}$

$
\frac{dP}{dV}=-nRT(V-nb)^{-2}+an^2V^{-3}=\left(\frac{-nRT}{(V-nb)^2}\right)+\left(\frac{an^2}{V^3}\right)
$