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Active Calculus - Multivariable

Nicholas Long, Mitchel T. Keller, Steve Schlicker

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Activity 11.4.4.
As we saw in Activity 11.3.5, the wind chill \(w(v,T)\text{,}\) in degrees Fahrenheit, is a function of the wind speed, in miles per hour, and the air temperature, in degrees Fahrenheit. Some values of the wind chill are recorded in Table 11.4.12.
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Table 11.4.12. Wind chill as a function of wind speed and temperature
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\(v \backslash T\) \(-20\) \(-15\) \(-10\) \(-5\) \(0\) \(5\) \(10\)
\(10\) \(-41\) \(-35\) \(-28\) \(-22\) \(-16\) \(-10\) \(-4\)
\(15\) \(-45\) \(-39\) \(-32\) \(-26\) \(-19\) \(-13\) \(-7\)
\(20\) \(-48\) \(-42\) \(-35\) \(-29\) \(-22\) \(-15\) \(-9\)
\(25\) \(-51\) \(-44\) \(-37\) \(-31\) \(-24\) \(-17\) \(-11\)
\(30\) \(-53\) \(-46\) \(-39\) \(-33\) \(-26\) \(-19\) \(-12\)
\(35\) \(-55\) \(-48\) \(-41\) \(-34\) \(-27\) \(-21\) \(-14\)
(a)
Estimate the partial derivatives \(w_{T}(20,-15)\text{,}\) \(w_{T}(20,-10)\text{,}\) and \(w_T(20,-5)\text{.}\) Use these results to estimate the second-order partial \(w_{TT}(20, -10)\text{.}\)
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(b)
In a similar way, estimate the second-order partial \(w_{vv}(20,-10)\text{.}\)
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(c)
Estimate the partial derivatives \(w_T(20,-10)\text{,}\) \(w_T(25,-10)\text{,}\) and \(w_T(15,-10)\text{,}\) and use your results to estimate the partial \(w_{Tv}(20,-10)\text{.}\)
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(d)
In a similar way, estimate the partial derivative \(w_{vT}(20,-10)\text{.}\)
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(e)
Write a few sentences to explain what the values \(w_{TT}(20, -10)\text{,}\) \(w_{vv}(20,-10)\text{,}\) and \(w_{Tv}(20,-10)\) indicate regarding the behavior of \(w(v,T)\text{.}\)
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