NAME: (print!) Section: E-Mail address: MATH 251 (22,23,24 ) [Fall 2020], Dr. Z. , Makeup for Exam 1, Sunday, Oct. 25, 2020, 9:00-11:00am 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.) Find a parametric representation, in terms of the parameter, t, of the line of intersection of the planes y = 0 and z = 0. What is the usual name for that line?

The type of the answers is:

ans.

2. (10 points) Use any method to find \frac{∂^{2}h}{∂r^{2}} at (q, r) = (1, 1) where h(u, v) = u^{5} v^{6}, u = q^{2} r, v = q r^{2}

The type of the answer is:

ans.

3. (10 points) Find the point on the line y = x + 6 (in the xy-plane) where the rate of change of the function f(x, y) = x^{2} + y^{2} in the direction < 1, 1 > is zero.

The type of the answer is:

ans.

4. (10 points) Find an equation of the tangent plane to the following surface at the given point

z = x^{2} + y^{3} , (1, 2, 9) .

The type of the answer is:

ans.

5. (10 points) Compute f_{xyy}(1, 1) (aka \frac{∂^{3}f}{∂x∂^{2}y} (1, 1), if f(x, y) = ln(x^{2} + y). The type of the answer is: ans.

6. (10 points) Use Linearization to apprximate \sqrt{(3.001)^{2} + (3.999)^{2}}. Part of the challenge is to find the appropriate function f(x, y) and the appropriate ‘nice’ point (x_{0}, y_{0}).

The type of the answer is:

ans.

7. (10 points) Decide whether the following limit exists. If it does find it. If it does not Explain why not?

lim_{(x,y,z)→(1,1,1)} \frac{x + 2y + 3z − 6}{3x + 2y + z − 6}

Hint: To prove that a limit exists at a designated point you first try to plug it in. If there are no issues you are done, and the value is the limit. Failing this, you try to use algebra to simplify and then plug it in, if possible. Failing this you try to prove that it does not exist. One way of doing it is to pick any two different lines through the point in question and show that if you approach the designated point via these lines you get different limits (in the sense of single-variable calculus).

8. (10 points) Find the minimum location(s) (i.e the point(s) where the function is the smallest) of the function

f(x, y, z) = x + 2y − z

in the closed pyramid whose vertices are

(1, 1, 0) , (−1, 1, 0), (1, −1, 0), (−1, −1, 0), (0, 0, 1)

The type of the answer(s) is:

ans.

9. (10 points) A certain particle of mass 2 kilograms has position function, expressed in meters, where t is time, expressed in seconts,

r(t) = ⟨ e^{t}, 2 cos t, 2 cos t ⟩ ,

(i) Find the magnitude of the force acting on it at time t = 0. Explain! (ii) Find the unit direction. Reminder from physics: F = ma, i.e. the force equals the mass times the accelaration. The type of the answer(s) is:

ans. magnitude :  Newtons  ; unit direction:

10. (10 points) A certain function f(x, y, z) depends on three variables called x, y and z. At a certain time the rate of change of this function with respect to time, happens to be 140. It is also known, that at that same time

• The rate of change of f(x, y, z) with respect to x is 1

• The rate of change of f(x, y, z) with respect to y is 2

• The rate of change of f(x, y, z) with respect to z is 3

• The rate of change of y with respect to time is two times The rate of change of x with respect to time.

• The rate of change of z with respect to time is three times The rate of change of x with respect to time. What is the rate of change of x with respect to time at that instant?

The type of the answer is:

ans.