Activity 11.4.5

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 $T$. 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 $w = w(v, T)$.

1. When computing the partial derivative with respect to wind speed $w_v$, which variable is held constant and which variable changes? Do you think $w_v$ is positive, negative, or zero at $(v,T)=(20,-10)$? Write a couple of sentences to explain your reasoning.

2. When computing the partial derivative with respect to temperature $w_T$, which variable is held constant and which variable changes? Do you think $w_T$ is positive, negative, or zero at $(v,T)=(20,-10)$? Write a couple of sentences to explain your reasoning.

3. 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 $h$. A partial derivative is a derivative of an appropriate trace, so by considering a difference quotient for a trace of $w$, estimate the partial derivative $w_v(20,-10)$. Include units on your answer and write a sentence explaining the meaning of the value of $w_v(20,-10)$.

4. Recall from single-variable calculus that for a function $f$, $f(a+h) \approx f(a) + hf'(a)$. Use this idea and your result above to estimate the wind chill $w(18, -10)$.

5. Estimate the partial derivative $w_T(20,-10)$. Include units on your answer and write a sentence explaining the meaning of the value of $w_T(20,-10)$.

6. Use your result above to estimate the wind chill $w(20, -12)$.

7. Consider how you might combine your previous results to estimate the wind chill $w(18, -12)$. Explain your process.