NAME: (print!) Section: E-Mail address: MATH 251 (22,23,24 ) [Fall 2020], Dr. Z. , Final Exam , Tue., Dec. 15, 2020, 12:00noon-3:30pm Email the completed test, renamed as fianl2FirstLast.pdf to DrZcalc3@gmail.com no later than 3:30pm, (or, in case of conflict, three and half hours after the start).

WRITE YOUR FINAL ANSWERS BELOW

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

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.

Sign the following declaration:

I Hereby declare that all the work was done by myself. I was allowed to use Maple (unless specifically told not to), 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:

Possibly useful facts from school Geometry (that you are welcome to use) : (i) The area of a circle radius r is ⊂ r^{2}. (ii) The circumference of a circle radius r is 2πr (iii) The parametric equation of an ellipse with axes a b and parallel to the x and y axes respectively is x = a cos θ, y = b cos θ, 0 < θ < 2π. (iv) The area of an ellipse with axes a and b is πab (v) The volume and surface area of a sphere radius R are \frac{4}{3}πR^{3} and 4πR^{2} respectively (vi) The volume of a cone is the area of the base times the height over 3. (vii) The volume of a pyramid is the area of the base times the height over 3.

Formula that you may (or may not) need

If the surface S is given in explicit notation z = g(x, y), above the region of the xy-plane , D, then

∫ ∫_{S} F · dS =

∫ ∫_{D} (−P \frac{∂g}{∂x} − Q\frac{∂g}{∂y} + R) dA .

1. (12 pts.) Without using Maple (or any software) Compute the line integral

∫_{C} (cose^{sin x} + 5y) dx + sine^{cos y} + 11x dy ,

over the path consisiting of the five line segments (in that order)

(1, 0) → (−1, 0) → (−1, 1) → (0, 2) → (1, 1) → (1, 0) .

Explain! ans.

2. (12 points) Change the order of integration ∫_{1}^{2} ∫_{1}^{2 e^{3x}} f(x, y) dx dy

ans.

3. (12 points) Find the equation of the tangent plane at the point (\frac{π/6}{,}(\frac{π/6}{,}(\frac{π/6}{)} to the surface given parameterically by

2 sin(x + y) + 4 sin(x + z) + 8 sin(y + z) = 7

Express you answer in explicit form, i.e in the format z = ax + by + c.

The type of the answer is:

ans. z = 4. (16 points) Let a, b, c be three vectors such that

a × b = 2i + 3j − k , b × c = 4i − j + 3k , a × c = −4i + 3j − 2k , What is

(a + b + c) × (2a − b + 3c)×

ans.

5. (12 points) Find the three angles of the triangle ABC where A = (0, 0, 0) , B = (1, 2, −1) , C = (2, −1, 1) ,

ans. The angle at A is: radians ; The angle at B is: radians ; The angle at C is: radians ; 6. (12 points) Find the directional derivative of

f(x, y, z) = x^{3} + y^{3} + z^{3} + xyz ,

at the point (1, 1, 1) in a direction pointing to the point (2, 3, 4) The type of the answer is: ans.

7. (12 points) Find \frac{∂g}{∂u} at (u, v) = (0, 1) where g(x, y) = 3x^{2} − 3y^{2} x = e^{u} cos v and y = e^{u} sin v.

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

8. (12 points) Without using Maple (or any other software), compute the vector-field surface integral ∫_{S} F.dS if

F = ⟨3x + cos(y^{3} + yz) , −2y + e^{x+z^{2}} , 5z + sin(xy^{3} + e^{x})⟩ >

and S is closed surface (x, y, z) : x^{2} + y^{2} + z^{2} = 4, x ≥ 0, y ≥ 0, z ≥ 0∪(x, y, 0) : x^{2} + y^{2} = 4, x ≥ 0, y ≥ 0∪(x, 0, z) : x^{2} + z^{2} = 4, x ≥ 0, z ≥ 0∪(0, y, z) : y^{2} + z^{2} = 4, y ≥ 0, z ≥ 0 (In other words it is the boundary of the region {(x, y, z) : x^{2}+y^{2}+x^{2} < 1 , x, y, z > 0}).

ans.

9. (12 points) Compute the vector-field surface integral ∫ ∫_{S} F.dS if F is F = ⟨3z, 2x, y + z⟩ ,

and S is the oriented surface

z = 2x + 3y , 0 < x < 1, 0 < y < 1, z ≥ 0

with upward pointing normal. ans.

10. (12 points) Without using Maple or software, find the critical point(s) of f(x, y) = 4x − y^{2} − ln(2x + y) and decide for each whether it is a local maximum, local minimum, or saddle point. Explain.

ans.

11. (12 points) Without using Maple or software, using a Linearization around the point (1, 1, 2), approximate f(1.001, 0.999, 2.01) if

f(x, y, z) = \sqrt{2x^{2} + 3y^{2} + z^{2}} .

ans.

12. (12 points) Without using Maple (or any other software) and by using polar coordinates (no credit for doing it directly) find

∫_{0}^{\sqrt{2}/2} ∫_{0}^{x} x dy dx + ∫_{\sqrt{2}/2}^{1} ∫_{0}^{√\overline{1−x^{2}}} x dy dx .

ans.

13. (12 points) Convert ∫_{0}^{4} ∫_{0}^{√\overline{4−z^{2}}} ∫_{−\sqrt{4−z^{2}−y^{2}}}^{0} x^{2} y z dx dy dz ,

to spherical coordinates. Do not evaluate. ans.

14. (12 points) Find the curvature of r(t) = ⟨5, 3 sin t, 3 cos t⟩

at the point where t = \frac{π}{3} . ans.

15. (12 points) Set-up a double integral, but do not compute, for the surface area of the surface given parameterically by

r(u, v) = langleu^{2}, uv, v^{2}⟩, 0 < u < v < 1}

ans.

16. (12 points) Let f(x, y, z) = xy^{2}z^{3} and g(x, y, z) = x+y^{2}+z^{3} compute grad(f).grad(g) at the point (1, 1, 1).

ans.

17. (8 points) Decide whether the following limit exists. If it does ,find it. If not explain why it does not exist.

lim_{(x,y,z,w)→(0,0,0,0)} \frac{x + y)^{2} − (z + w)^{2}}{x + y − z − w} .

ans.