1. A heated plate has temperature $T(x, y) = 100 - (x^2 + 4y^2)$ at the point (x, y), where T is
measured in °C, and x and y in meters.
(A) Find $\lim_{(x,y) \to (2,1)} T(x, y)$.
(B) Find $T_x(2, 1)$.
(C) Interpret the value of $T_x(2, 1)$ in context without using 'rate of change'.
(D) Find the direction in which the temperature increases most rapidly at (2, 1).
(E) Find the greatest increase of the plate's temperature at (2, 1).
(F) Find all direction(s) in which the temperature does not change at (2, 1).
(G) How fast does the plate's temperature change at (2, 1) in the direction of $<1, 3>$?
(H) Find all direction(s) in which the temperature at (2, 1) increases at a rate of 4°C per meter if
there is any.
(I) Find all direction(s) in which the temperature decreases at (1, 1) at a rate of 9°C per meter if
there is any.
(J) Use the tangent plane to T(x, y) to approximate T(1.9, 1.2).
(K) Find all the critical point(s).
(L) Use the Second-Order Derivative Test to find all the local extrema of T(x, y).
(M) Find the global extrema of T(x, y) over R = {(x, y) | 0 ≤ y ≤ 1 - x²}.
(N) Use Lagrange multipliers to find the maximum/minimum of T(x, y) subject to x² + y² = 16.
(O) Sketch the following on the same window: contour map including level curves with at least
z = 36, 84, and 92, gradient vector field of T(x, y), VT(2, 1), and x² + y² = 16. Confirm and
explain your answer in (N) above using your sketch.
