Activity 11.5.4 — Active Calculus Multivariable

As we saw in Activity, the wind chill $w(v,T)$ , 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 the table below.

$v \backslash T$	$-20$	$-15$	$-10$	$-5$	$0$	$5$	$10$
$10$	$-41$	$-35$	$-28$	$-22$	$-16$	$-10$	$-4$
$15$	$-45$	$-39$	$-32$	$-26$	$-19$	$-13$	$-7$
$20$	$-48$	$-42$	$-35$	$-29$	$-22$	$-15$	$-9$
$25$	$-51$	$-44$	$-37$	$-31$	$-24$	$-17$	$-11$
$30$	$-53$	$-46$	$-39$	$-33$	$-26$	$-19$	$-12$
$35$	$-55$	$-48$	$-41$	$-34$	$-27$	$-21$	$-14$

Estimate the partial derivatives $w_{T}(20,-15)$ , $w_{T}(20,-10)$ , and $w_T(20,-5)$ . Use these results to estimate the second-order partial $w_{TT}(20, -10)$ .
In a similar way, estimate the second-order partial $w_{vv}(20,-10)$ .
Estimate the partial derivatives $w_T(20,-10)$ , $w_T(25,-10)$ , and $w_T(15,-10)$ , and use your results to estimate the partial $w_{Tv}(20,-10)$ .
In a similar way, estimate the partial derivative $w_{vT}(20,-10)$ .
Write a few sentences to explain what the values $w_{TT}(20, -10)$ , $w_{vv}(20,-10)$ , and $w_{Tv}(20,-10)$ indicate regarding the behavior of $w(v,T)$ .

Long, Keller & Schlicker, Active Calculus Multivariable, §11.5 Second-Order Partial Derivatives, Activity 11.5.4

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