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This is the Exam of Advanced Calculus which includes Line, Parametric Equations, Plane, Distance, Plane, Line, Surface, Tangent Plane, Parallel, Vectors etc. Key important points are: Hyperboloid, Sketch, All Points, Tangent Plane, Parallel, Plane, Remain Constant, at Horizontal, Temperature, Distances
Typology: Exams
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Marks
(a) (4 marks) Sketch the hyperboloid z^2 = 4x^2 + y^2 − 1.
(b) (8 marks) Find all points on the hyperboloid z^2 = 4x^2 + y^2 − 1 where the tangent plane is parallel to the plane 2x − y + z = 0.
[8] 2. A bug walks on a flat horizontal metal plate where the temperature is given by a C^1 function T (x, y). At a certain time, she is at the point (1, 0) in the plane (all distances are measured in meters). If she walks north (in the direction of (0, 1)) from that point, the temperature will increase at a rate of 3 degrees per meter. If she walks southeast (in the direction of (1, −1)), the temperature will remain constant. In what direction should the bug walk so as to cool off as quickly as possible, and what will be the rate of change of temperature (in degrees per meter) in that direction?
[12] 4. The plane x − y + 2z = 6 intersects the paraboloid z = x^2 + y^2 in an ellipse. Find the points on this ellipse that are closest to and farthest from the origin.
[12] 5. Evaluate the following integrals:
(a) (6 marks)
D 2 xdA, where^ D^ is the triangle in the^ xy-plane with vertices (0,^ 0), (1,^ 0), (3, 1);
(b) (6 marks)
D ydA, where^ D^ is the region in the^ xy-plane given by^ D^ =^ {(x, y) :^0 ≤ y ≤ x, x^2 + y^2 ≤ 9 }. (Hint: use polar coordinates.)
[12] 7. Evaluate each limit or prove that it does not exist.
(a) (6 marks) lim (x,y)→(0,0)
x^3 − y^3 x^2 + y^2
(b) (6 marks) lim (x,y)→(0,0)
x^2 − y^4 x^2 + y^4
[12] 8. Let
f (x, y) =
x^2 y x^2 + y^2
if (x, y) 6 = (0, 0) 0 if (x, y) = (0, 0)
(a) (4 marks) Find
∂f ∂x
(0, 0) and
∂f ∂y
(b) (8 marks) Prove that f is not differentiable at (0, 0).
The End
