Let F be the conservative vector fieldF(x, y) = (x2 + xy2, y2 + x2y)and let C be the circle x2 + y2 = 2 of radius √2 centred at (0, 0).(a) Find a scalar potential function ϕ for F that satisfies ϕ(0, 0) = 0.(b) Use the method of Lagrange multipliers to find the maximum and minimum of ϕ from(a) on C

If F(x,y,z)𝐹(𝑥,𝑦,𝑧) is a vector field such that curl(F)≠0(𝐹)≠0 then:a.there will be a scalar field f𝑓 such that F=∇(f)𝐹=∇(𝑓).b.F𝐹 is a conservative vector field.c.F𝐹 is irrotational.d.none of the choices are true.

Let f (x, y, z) = sin(xy + z), and P be the point (0, −2, π3 ).(a) (6 pts) Compute ∇f (x, y, z).(b) (2 pts) At P , find the direction along which f obtains maximum directional derivative.(c) (4 pts) Calculate the directional derivative ∂f∂u (P ), where u is a unit vector making an angleπ6 with the gradient ∇f (P ).(d) (3 pts) The level surface f (x, y, z) = √32 defines z implicitly as a function of x and y near P .Compute ∂z∂x at P .

Letf : R^2 -> R: (x,y) -> x^4 + y^2 + 2x^2y -1Answer the following:(a) Graph f(x, 0) against x in two dimensions and graph f(0, y) against y in two dimensions.(b) Draw the countour set f(x, y) = 3(c) Calculate the directional derivative of f at the point (1, 1) in the direction of thevector (−1, 1).(d) At the point (1, 1), find the direction of steepest ascent. That is, find the vector v∗ with length 1 that causes f to increase most rapidly.(e) Find the equation of the tangent plane at the point (1, 1) and comment on the relationship between the tangent plane and the slope of steepest ascent.

A consumer spends an amount 𝑚𝑚 to buy 𝑥𝑥 units of one good at the price of 6 per unit and 𝑥𝑥 units of a different good at the price of 10 per unit. Here 𝑚𝑚 is positive and suppose that 8 < 𝑚𝑚 < 40. The consumer’s utility function is 𝑈(𝑥, y) = 𝑥y + y^2+ 2𝑥 + 2y, so that her problem is: Maximize 𝑥y+y^2 +2x+2y subject to 6𝑥+10𝑥y=𝑚 (a) Find the optimal quantities 𝑥∗ and y∗ and the Lagrange multiplier, all of them as functions of 𝑚. (b) Write down the maximum value of the utility function as a function of m. (c) Find the derivative of the above utility function for m=20 (d) What are the solutions for 𝑥∗ and y∗ if (i) 𝑚 ≤ 8? (ii) 𝑚 ≥ 40?

Find the velocity field due to two sources of strength m: one located at (0, 0, l) andthe other – at (0, 0, −l). Where is the velocity zero? Find the velocity at an arbitrary point (x, y, 0) inthe xy-plane. Where in the xy-plane is the speed greatest?Hint: The scalar potential for the two-source system isφ(x, y, z) = φ(r) = − m|r − lk| − m|r + lk|