Activity 11.3.5.
The wind chill, as frequently reported, is a measure of how cold it feels outside when the wind is blowing. In the table below, the wind chill \(w\) is a function of the wind speed \(v\) and the ambient air temperature \(T\text{.}\) Both \(w\) and \(T\) are measured in degrees Fahrenheit, while \(v\) is measured in miles per hour. This means that \(w\) can be expressed in the form \(w = w(v, T)\text{.}\)
| \(v \backslash T\) | \(-20\) | \(-15\) | \(-10\) | \(-5\) | \(0\) | \(5\) | \(10\) |
| \(10\) | \(-41\) | \(-35\) | \(-28\) | \(-22\) | \(-16\) | \(-10\) | \(-4\) |
| \(15\) | \(-45\) | \(-39\) | \(-32\) | \(-26\) | \(-19\) | \(-13\) | \(-7\) |
| \(20\) | \(-48\) | \(-42\) | \(-35\) | \(-29\) | \(-22\) | \(-15\) | \(-9\) |
| \(25\) | \(-51\) | \(-44\) | \(-37\) | \(-31\) | \(-24\) | \(-17\) | \(-11\) |
| \(30\) | \(-53\) | \(-46\) | \(-39\) | \(-33\) | \(-26\) | \(-19\) | \(-12\) |
| \(35\) | \(-55\) | \(-48\) | \(-41\) | \(-34\) | \(-27\) | \(-21\) | \(-14\) |
(a)
When computing the partial derivative with respect to wind speed \(w_v\text{,}\) which variable is held constant and which variable changes? Do you think \(w_v\) is positive, negative, or zero at \((v,T)=(20,-10)\text{?}\) Write a couple of sentences to explain your reasoning.
(b)
When computing the partial derivative with respect to temperature \(w_T\text{,}\) which variable is held constant and which variable changes? Do you think \(w_T\) is positive, negative, or zero at \((v,T)=(20,-10)\text{?}\) Write a couple of sentences to explain your reasoning.
(c)
Recall that we can estimate a partial derivative of a single variable function \(f\) using the central difference \(\frac{f(x+h)-f(x-h)}{2h}\) for small values of \(h\text{.}\) A partial derivative is a derivative of an appropriate trace, so by considering a difference quotient for a trace of \(w\text{,}\) estimate the partial derivative \(w_v(20,-10)\text{.}\) Include units on your answer and write a sentence explaining the meaning of the value of \(w_v(20,-10)\text{.}\)
(d)
Recall from single-variable calculus that for a function \(f\text{,}\) \(f(a+h) \approx f(a) + hf'(a)\text{.}\) Use this idea and your result above to estimate the wind chill \(w(18, -10)\text{.}\)
(e)
Estimate the partial derivative \(w_T(20,-10)\text{.}\) Include units on your answer and write a sentence explaining the meaning of the value of \(w_T(20,-10)\text{.}\)
(f)
Use your result above to estimate the wind chill \(w(20, -12)\text{.}\)
(g)
Consider how you might combine your previous results to estimate the wind chill \(w(18, -12)\text{.}\) Explain your process.
