Exercise 26

In the following questions, we determine and apply the linearization for several different functions.

1. Find the linearization $L(x,y)$ for the function $f$ defined by $f(x,y) = \cos(x)(2e^{2y}+e^{-2y})$ at the point $(x_0,y_0) = (0,0)$. Use the linearization to estimate the value of $f(0.1, 0.2)$. Compare your estimate to the actual value of $f(0.1, 0.2)$.

2. The Heat Index, $I$, (measured in apparent degrees F) is a function of the actual temperature $T$ outside (in degrees F) and the relative humidity $H$ (measured as a percentage). A portion of the table which gives values for this function, $I=I(T,H)$, is provided in Table.

   Heat index.

   T $\downarrow \backslash$ H $\rightarrow$	$70$	$75$	$80$	$85$
   $90$	$106$	$109$	$112$	$115$
   $92$	$112$	$115$	$119$	$123$
   $94$	$118$	$122$	$127$	$132$
   $96$	$125$	$130$	$135$	$141$

   Suppose you are given that $I_T(94,75) = 3.75$ and $I_H(94,75) = 0.9$. Use this given information and one other value from the table to estimate the value of $I(93.1,77)$ using the linearization at $(94,75)$. Using proper terminology and notation, explain your work and thinking.

3. Just as we can find a local linearization for a differentiable function of two variables, we can do so for functions of three or more variables. By extending the concept of the local linearization from two to three variables, find the linearization of the function $h(x,y,z) = e^{2x}(y+z^2)$ at the point $(x_0,y_0,z_0) = (0, 1, -2)$. Then, use the linearization to estimate the value of $h(-0.1, 0.9, -1.8)$.