Find the tangent plane for each surface - write the total

Question

Question 1. Find the tangent plane for each surface at the given point.

1) x² + z² = xyz = 4, point P(3, 2, 1).

2) z = sin x cos y, point Q(Π/4, Π/4, 1/2)

Question 2. The pressure of a mole of an ideal gas is related to its volume and temperature by the equation P = 8T/V, where P is pressure in kilopascals, V is volume in litres and T is temperature in kelvins.

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

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

Question 3. Find ∂z/∂s and ∂z/∂t if z = x²ln y and x = s/t, y = s² + t².

Question 4. Suppose z = z(x, y) is given implicitly by the equation xy/z = (x + y) ln z. Find ∂z/∂x and ∂z/∂y.

Question 5. Find the direction and rate of the fastest increase of the function

f (x, y) = e^x+y (2x + 3y) at point P(-1, 1).

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) = x³ + 3xy² - 15x - 12y.