NAME: (print!) Section: E-Mail address: MATH 251 (22,23,24 ) [Fall 2020], Dr. Z. , Exam 2, Monday, Nov. 23, 2020, 8:40-10:40am

WRITE YOUR FINAL ANSWERS BELOW

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 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).

Sign the following declaration: I Hereby declare that all the work was done by myself. I was allowed to use Maple, calculators, the book, and all the material in the web-page of this class but not other resources on the internet. I only spent (at most) 90 minutes on doing the exam. The last 30 minutes were spent in checking and double-checking the answers. I also understand that I may be subject to a random short chat to verify that I actually did it all by myself. Signed:

1. (10 pts.) Compute the line integral ∫_{C} yz dx + xz dy + xy dz ,

over the path

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

Explain! The type of the answers is: ans.

2. (10 points) Change the order of integration. Explain! ∫_{0}^{9} ∫_{√\overline{x}}^{3} F (x, y) dy dx

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

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

8. (10 points) Compute the line integral ∫_{C} f ds where

f(x, y, z) = xyz

and C is

r(t) = ⟨t, −2t, 3t⟩ , for 0 ≤ t ≤ 1

The type of the answer(s) is: ans.

9. (10 points) Compute the vector-field surface integral ∫ ∫_{S} F.dS for the given oriend surface

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

with downward pointing normal.

The types of the answers are (i) (ii)

ans. (i) magnitude :  Newtons  ; (ii) unit direction:

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.