Review from class notes the notion of a surface. Now consider F: R³ → R, a differentiable (smooth) function. Thus, the partial derivatives of F, OF OF OF FT F = , บ მე and F₂ ду = Əz exist and are continuous on all of R³. We say a point p = (P1, P2, P3) Є R³ is a critical point of F if Fr(p) Fy(p) Fz(p) = 0. Otherwise p is said to be a regular point of F. For any cЄR, the level set F-1(c) is a regular level set if F-1(c) 0 and consists entirely of regular points. Regular Level Set Theorem. Let F: R³ → R be a differentiable function. Then every regular level set of F is a surface in R³. Note: The Regular Level Set Theorem does not say that non-regular level sets of F are never surfaces! It only says that regular level sets always are. In each case below decide if the set is (or is not) a surface in R³. Give a one-line explanation of your answer in each case. The Regular Level Set Theorem may be helpful in deciding that a set is a surface. It will not help you in arguing that a set is not a surface. (In the latter situation, just state where the problem points are. No further explanation is necessary.) (i.) C = {(x, y, z) Є R³ : x² — y² + z² = 0}; - (ii.) O = {(x, y, z) Є R³ : x² + y² + z² = 0}; (iii.) F-¹(c) where F(x, y, z) = x²+ y²+z² and c> 0; (iv.) X = Graph f where f(x, y) = x² - y²; (v.) Y the solution set to the equation zxy = 0; (vi.) PUP₁ where P₁ = {(x, y, z) = R³: x = 0} and Po = {(x, y, z) = R³ : y = 0}

Where are the relative extrema (local maximum and minimum) on the curve? y = 3x² – x A. (0, 0) only В. (1,2) only C. (2, 4) only D. (4. -16) only E. (0, 0) and (2, 4) O A O E

Evaluate ʃC F dr where C is the parabola y = x2 from (0, 0) to (2, 4) and F (x, y) =< −ysinx, cosx >

13. Find an equation for the tangent plane of the graph of f at the point (xo, yo, f(xo, yo)) for: (a) ƒ: R²2 → R, (x, y) → x − y +2, (xo, yo) = (1, 1) (b) (c) ƒ: R² → R, (x, y) ↔ x² + 4y², (xo, yo) = (2, -1) f: ƒ: R² R, (x, y) → xy, → (xo, yo) = (-1, −1) (d) f(x, y) = log (x + y) + x cos y + arctan(x + y), (xo, yo) = (1, 0) -2 (e) f(x, y) = √√√x² + y², (f) f(x, y) = xy, (xo, yo) = (1, 1) (xo, yo) = (2, 1)

Pls help ASAP. Show all work and calculations.

(a) Evaluatef(2,1).(b) Estimatefx(2,1) andfy(2,1) using the contour plot.(c) Determine the linearizationL(x, y) at the point (2,1) SEE PHOTO FOR SET UP

Do the linear stability analysis for all the fixed points.

Use the definition of partial derivatives to find the equation of the line in space, tangent to the intersection curve between the surface: \[f(x, y) = \frac{\sqrt{1 - xy}}}{1 - \sqrt{x}} - \sqrt{y}}\] and the surface x=0 at the point(0,4).