Question 1. Find the tangent plane for each surface at the given point. 1) x + z

2 2

 xyz = 4, point P(3, 2, 1). 2) z = sin x cos y, point Q Question 2. The pressure of a mole of an ideal gas is related to its volume and temperature by the equation

temperature in kelvins. 1) Write the total differential of the function P at the point, where the volume is 4 L and the temperature is 300 K.

2) Use this differential to find the approximate change in pressure, when the volume increases from 4 L to 4.3 L and the temperature decreases from 300 K to 295 K.

Question 3. Find      4  1  , , .  4 2 P  8 T , where P is pressure in kilopascals, V is volume in litres and T is V    and z z  t s if z = x ln y and x = 2 s t

Question 4. Suppose z = z(x, y) is given implicitly by the equation  and z  x

Question 5. Find the direction and rate of the fastest increase of the function f (x, y) = e  x y

Question 6. Find the directional derivative of the function f (x, y, z) = z sin (2x + 3y) at point P(3, 2, 1) in the direction of vector v = < 2, 3, 6 >. Question 7. Find the points and values of local minima and maxima of the function f (x, y) = x  z .  y  2  3 x y at point P(1, 1). + 3xy 2 3  15 x  12 y.