Question

Question 1 (4 marks) Given the function f(x,y,z) = z ln x + x^2 y z^3 - x e^{xy} Calculate ?f/?x and ?^2 f / ?z?x. Question 2 (4 marks) Find the equation of the tangent plane of the surface given by z = 1 + x^2 y + ln(x + y), which passes through the point (2, -1, -3). Question 3 (4 marks) During certain vertical pressurisation, the base radius R of a cylinder increases at 2 cm/s and its height h is decreasing at a rate of 5 cm/s. Find the rate of change of the volume of the cylinder when R = 30 cm and h = 1 m. (Hint: the volume of a cylinder is V = ?R^2 h) Question 4 (4 marks) Find the rate of change of f(x,y,z) = z - x e^y cos z + 1 at the point (1,0,0) in the direction of the vector u = i - 2j + 4k. Question 5 (4 marks) Use the method of Lagrange multipliers to find the maximum and the minimum values of the function f(x,y) = x^2 + 2y^2 + 2y subject to the constraint x^2 + 3y^2 = 12.

          Question 1 (4 marks)
Given the function
f(x,y,z) = z ln x + x^2 y z^3 - x e^{xy}
Calculate ?f/?x and ?^2 f / ?z?x.

Question 2 (4 marks)
Find the equation of the tangent plane of the surface given by
z = 1 + x^2 y + ln(x + y),
which passes through the point (2, -1, -3).

Question 3 (4 marks)
During certain vertical pressurisation, the base radius R of a cylinder increases at 2 cm/s and
its height h is decreasing at a rate of 5 cm/s. Find the rate of change of the volume of the cylinder
when R = 30 cm and h = 1 m.
(Hint: the volume of a cylinder is V = ?R^2 h)

Question 4 (4 marks)
Find the rate of change of
f(x,y,z) = z - x e^y cos z + 1
at the point (1,0,0) in the direction of the vector u = i - 2j + 4k.

Question 5 (4 marks)
Use the method of Lagrange multipliers to find the maximum and the minimum values of the function
f(x,y) = x^2 + 2y^2 + 2y
subject to the constraint x^2 + 3y^2 = 12.

Question 1 (4 marks)
Given the function
f(x,y,z) = z ln x + x^2 y z^3 - x e^xy
Calculate ?f/?x and ?^2 f / ?z?x.

Question 2 (4 marks)
Find the equation of the tangent plane of the surface given by
z = 1 + x^2 y + ln(x + y),
which passes through the point (2, -1, -3).

Question 3 (4 marks)
During certain vertical pressurisation, the base radius R of a cylinder increases at 2 cm/s and
its height h is decreasing at a rate of 5 cm/s. Find the rate of change of the volume of the cylinder
when R = 30 cm and h = 1 m.
(Hint: the volume of a cylinder is V = ?R^2 h)

Question 4 (4 marks)
Find the rate of change of
f(x,y,z) = z - x e^y cos z + 1
at the point (1,0,0) in the direction of the vector u = i - 2j + 4k.

Question 5 (4 marks)
Use the method of Lagrange multipliers to find the maximum and the minimum values of the function
f(x,y) = x^2 + 2y^2 + 2y
subject to the constraint x^2 + 3y^2 = 12.

Added by Heather G.

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