NAME: (print!) Section: E-Mail address: MATH 251 (22,23,24 ), Dr. Z. Second Chance Club for Exam 2 Worksheet Due Friday, Dec. 13, 2020, 8:00pm. Email an attachment called scc2FirstLast.pdf to DrZcalc3@gmail.com Subject: scc2 Make sure that you have the posted solutions:

http://www.math.rutgers.edu/~zeilberg/calc3NNN/mt2S.pdf

and understand each problem. In this worksheet, you are supposed to state what was your error for each of the exam questions (if you made an error) and say how to avoid it in the future. Then, regardless of whether you got it right or wrong do the similar problems given Exam Problem 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! : Here is what I did wrong (if applicable): Problem 1a. Compute the line integral

∫_{C} xe^{xyz} dx + ye^{xyz} dy + ze^{xyz} dz ,

over the path

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

Explain! Problem 1b. Compute the line integral

∫_{C} (4x^{3}y^{2} + 1) dx + (2x^{4}y + 1) dy ,

over the path

r(t) = ⟨sin t^{2}, cos t^{2}⟩ , 0 ≤ t ≤ \sqrt{π/2} .

Explain!

: Exam Problem 2. (10 points) By changing the order of integration, if necessary, evaluate the double-integral

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

Here is what I did wrong (if applicable): Problem 2a: Changing the order of integration

∫_{0}^{1} ∫_{0}^{e^{x}} f(x, y)dy dx

Problem 2b: Changing the order of integration

∫_{0}^{π} ∫_{0}^{sin x} f(x, y)dy dx

Problem 2c: Changing the order of integration

∫_{0}^{1} ∫_{e^{y}}^{e} f(x, y)dx dy Exam Problem 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. Here is what I did wrong (if applicable):

Problem 3a. Find the equation of the tangent plane at the point (1, 2, 4) to the surface given parametrically by x(u, v) = u^{2} , y(u, x) = uv , z(u, v) = v^{2} , −∞ < u < ∞ , −∞ < v < ∞ . Express you answer in explicit form, i.e in the format z = ax + by + c.

Problem 3b. Find the equation of the tangent plane at the point (−1, −1, 2) to the surface given parametrically by x(u, v) = u^{3} , y(u, x) = v^{3} , z(u, v) = −2uv , −∞ < u < ∞ , −∞ < v < ∞ . Express you answer in explicit form, i.e in the format z = ax + by + c.

Exam Problem 4. 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 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.

Here is what I did wrong (if applicable):

Problem 4a Let f(x, y, z) = sin(x + y^{2} + z^{3}), and let

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

Let C be the curve

r(t) = ⟨t, t^{2}, t^{3}⟩ , 0 ≤ t ≤ 3 .

Find the value of the line-integral

∫_{C} F.dr .

Explain! Just giving the answer will give you no credit. Problem 4b Let f(x, y) = e^{cosx+3 sin y}, and let

F = ⟨\frac{∂f}{∂x} , \frac{∂f}{∂y} ⟩

Let C be the curve

r(t) = ⟨sin 2t, cos t⟩0 ≤ t ≤ π .

Find the value of the line-integral

∫_{C} F.dr .

Explain! Just giving the answer will give you no credit. Exam Problem 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} .

Here is what I did wrong (if applicable):

Problem 5a Evaluate the triple integral ∫_{R} (x + y)(x^{2} + y^{2} + z^{2})^{2} dx dy dz ,

where R is the region in 3D space given by

{(x, y, z) | x^{2} + y^{2} + z^{2} ≤ 1 , x ≥ 0, y < 0, z < 0} .

Problem 5b Evaluate the triple integral ∫_{R} z(x^{2} + y^{2} + z^{2}) dx dy dz ,

where R is the region in 3D space given by

{(x, y, z) | x^{2} + y^{2} + z^{2} ≤ 1 , y < 0} .

Problem 5c Evaluate the triple integral ∫_{R} (z − x) dx dy dz ,

where R is the region in 3D space given by

{(x, y, z) | x^{2} + y^{2} + z^{2} ≤ 2} .

Exam Problem 6. Evaluate the double integral

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

Here is what I did wrong (if applicable):

Problem 6a Convert the integral to polar coordinates, do not evaluate.

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

Problem 6b Convert the integral to polar coordinates, do not evaluate.

∫_{0}^{4} ∫_{−\sqrt{16−x^{2}}}^{0}(x^{2} + y) dy dx

Problem 6c Convert the integral to polar coordinates, do not evaluate.

∫_{−\sqrt{2}/2}^{\sqrt{2}/2} ∫_{x}^{√\overline{1−x^{2}}}(x^{3} + y^{2}) dy dx

Exam Problem 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} ,

Here is what I did wrong (if applicable):

Problem 7a: Decide whether the following limit exists. If it does, find it, if not, explain!

lim_{(x,y)→(1,3)} \frac{x − 1}{y − 3} ,

Problem 7b: Decide whether the following limit exists. If it does, find it, if not, explain!

lim_{(x,y,z)→(0,0,0)} \frac{x + y + 2z}{2x + y + z} ,

Exam Problem 8. 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) Here is what I did wrong (if applicable): Problem 8a Compute the line integral ∫_{C} f ds where

f(x, y, z) = xy^{2} + yz^{2} + z

and C is the line segment starting at (0, 0, 0) and ending at (1, 1, −1) Problem 8b Compute the line integral ∫_{C} f ds where

f(x, y) = x + y

and C is the upper circle {(x, y) : x^{2} + y^{2} = 1, y > 0}. Exam Problem 9. 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. Here is what I did wrong (if applicable): Problem 9a Compute the vector-field surface integral ∫ ∫_{S} F.dS if F is

F = ⟨x + z, y + z, −x⟩ ,

and S is the oriented surface z = 9 − x^{2} − y^{2} , x < 0, y < 0, z ≥ 0 with upward pointing normal. Problem 9b Compute the vector-field surface integral ∫ ∫_{S} F.dS if F is

F = ⟨x + z, y + z, −x⟩ ,

and S is the oriented surface

z = 9 − x^{2} − y^{2} , 0 < x < 1, 0 < y < 1, z ≥ 0

with downward pointing normal. Exam Problem 10. Find the point on the plane x + 2y + 3z = 18 where the function f(x, y, z) = xyz is as large as possible. Here is what I did wrong (if applicable): Problem 10a Find the maximum value of the function f(x, y, z) = xyz on the plane 2x + y + z = 4 Problem 10b Find the point on the plane 2x + y + z = 4 where f(x, y, z) = xy^{2}z is as large as possible. (You can use Maple)