Solutions and Commentary on MATH 251 (22,23,24 ) [Fall 2020], Dr. Z. , Exam 2, Monday, Nov. 23, 2020, 8:40-10:40am

Types: Number, Function of variable(s), 2D vector of numbers, 3D vector of numbers, 2D vector of functions, 3D vector of functions, equation of a plane, parametric equation of a line, equation of a line, equation of a surface, equation of a line, DNE (does not exist), parametric equation of surface, double integral of an abstract function.

1. (10 pts.) Compute the line integral

∫_{C} yz dx + (xz + z) dy + (xy + y + 1) dz ,

over the path

r(t) = ⟨e^{t^{3}}, t^{2}e^{t^{4}}, te^{t^{7}}⟩ , 0 ≤ t ≤ 1 .

Explain!

The type of the answers is: Number.

ans. e + e^{2} + e^{3}

Comment: This was meant to be done the clever way. People who did it directly, and then used Maple to evaluate it, and got the right answer, still got full credit, but in the Final I may specifiy ”do not use Maple” (or any other software). Also I gave full credit for the answer in decimals. The perfect answer should be in terms of e. In the final exam I will insist on it (but will mention it explicitly). Sol. Whenever the vector field is conservative, then it is much easier to use the Fundamental Theorem of Line Integrals. Taking the curl(F) gives < 0, 0, 0 > (you do it!) hence there is a potential function f(x, y, z) such that grad(f) =< yz, xz+z, xy+y+1 >. i.e

f_{x} = yz , f_{y} = xz + z , f_{z} = xy + y + 1 .

From f_{x} = yz we get f(x, y, z) = xyz + ϕ(y, z) where ϕ(y, z) is to be determined. Using f_{z} = xz + z, we get xz + ϕ_{y}(y, z) = xz + z, hence ϕ_{y}(y, z) = z. Intergating with resepct to y, we get ϕ(y, z) = yz + ψ(z) where ψ(z) is to be determined. We currently have f(x, y, z) = xyz + yz + ψ(z). Using f_{z} = xy + y + 1, we get xy + y + ψ′(z) = xy + y + 1 so ψ′(z) = 1 and so ψ(z) = z + C (but C is not important for this problem) so Crucial First Step: The potential function if f(x, y, z) = xyz + yz + z. Comment: At this step it is a good idea to check That grad(f) is indeed < yz, xz + z, xy + y + 1 >. Do it! Now we almost done. By the Fundamental theorem of Line Integrals the value is f(End) − f(Start). The starting point is r(0) = (1, 0, 0), the ending point is r(1) = (e, e, e), so the final answer is

f(e, e, e) − f(1, 0, 0) = e^{3} + e^{2} + e − 0 = e^{3} + e^{2} + e .

2. (10 points) By changing the order of integration, if necessary, evaluate the doubleintegral

∫_{0}^{5} ∫_{(y/5)^{1/3}}^{1} sin x^{4} dx dy

The type of the answer is:

ans.

3. (10 points) Find the equation of the tangent plane at the point (1, 1, 1) to the surface given parametrically by x(u, v) = u^{3}v , y(u, x) = uv , z(u, v) = uv^{3} , −∞ < u < ∞ , −∞ < v < ∞ . Express you answer in explicit form, i.e in the format z = ax + by + c. The type of the answer is: ans. z = 4. (10 points) Let f(x, y, z) = e^{cos x^{2} + sin xyz+cos xz}, and let

F = ⟨\frac{∂f}{∂x} , \frac{∂f}{∂y} , \frac{∂f}{∂z} ⟩

Let C be the closed curve

r(t) = ⟨cos t, t, sin t⟩ , 0 ≤ t ≤ 2π .

Find the value of the line-integral

∫_{C} F.dr .

Explain! Just giving the answer will give you no credit. The type of the answer is: ans.

5. (10 points) Evaluate the triple integral ∫_{R} (x^{2} + y^{2} + z^{2})^{3} dx dy dz ,

where R is the region in 3D space given by {(x, y, z) | x^{2} + y^{2} + z^{2} ≤ 1 , x, y, z ≥ 0} .

The type of the answer is: ans.

6. (10 points) Evaluate the double integral

∫_{−3}^{0} ∫_{0}^{√\overline{9−x^{2}}}(x^{2} + y^{2})^{2} dy dx

The type of the answer is: ans.

7. (10 points altogether) Decide whether the following limits exist. If it does find them. If it does not Explain why not?

(a) (2 points) lim_{(x,y)→(π/2,π/2)} \frac{cos x + sin x}{x + y} , (b) (2 points) lim_{(x,y)→(0,0)} \frac{x^{2} − y^{2}}{x − y} ,

(c) (2 points) lim_{(x,y)→(0,0)} \frac{x − y}{x^{2} − y^{2}} , (d) (4 points) lim_{(x,y)→(1,1)} \frac{x + y − 2}{2x + y − 3} ,

ans. (a)  (b)  (c)  (d)

8. (10 points) Compute the line integral ∫_{C} f ds where f(x, y, z) = xyz

and C is the line segment starting at (0, 0, 0) and ending at (1, 2, −3) The type of the answer(s) is: ans.

9. (10 points) Compute the vector-field surface integral ∫ ∫_{S} F.dS if F is F = ⟨z, z, x⟩ ,

and S is the oriented surface

z = 9 − x^{2} − y^{2} , x ≥ 0, y ≥ 0, z ≥ 0

with downward pointing normal. The type of the answer is

ans.

10. (10 points) Find the point on the plane x + 2y + 3z = 18 where the function f(x, y, z) = xyz is as large as possible.

The type of the answer is:

ans.