Practice 20 questions on MAT2322 - Calculus III for Engineers at Carleton University. Free AI-generated quiz on uNotes — track your score, retake anytime.
1For a function f(x, y) = x ln(x - 2y), what is the equation of the tangent plane at the point (3, 1, 0)?
2Which of the following conditions must be met for a vector field F = P i + Q j + R k to be considered conservative in a simply connected region?
3The double integral of f(x, y) over a region from x=0 to x=2 and y=0 to y=4-x^2 is given. If we change the order of integration to treat it as a Type II region, what are the new limits?
4Calculate the directional derivative of f(x, y) = xe^y at the point (2, 0) in the direction of the vector u = (i + 2j)/sqrt(5).
5According to Green's Theorem, for a vector field F = P i + Q j and a region D bounded by a counter-clockwise curve C, the line integral of F along C is equal to the double integral over D of which expression?
6When calculating the total mass of a solid E with density delta(x, y, z) = x + y in the first octant bounded by z = x^2 + y^2 and z = 8 - x^2 - y^2, which coordinate system is most appropriate for simplifying the integral?
7Identify the critical points of the function f(x, y) = (y^2 + x^2)e^(y^2 - x^2) as described in the course materials.
8What is the result of the flux integral of F = -x i - y j - z k through the disk S defined by x^2 + y^2 <= 9 at z = 4, oriented upwards?
9A vector field F has a divergence of 3. According to the Divergence Theorem, what is the flux of F through a sphere of radius 'a' centered at the origin?
10Stokes' Theorem relates a line integral over a closed curve C to a surface integral over a surface S bounded by C. Which expressions are correctly linked by this theorem?
