y? - 2xy x + y2 if (x, y) + (0,0) 7. Given the piecewise function: f(x,y) 0 if (x, y) = (0,0) a) Show that: limf(x,y) does not exist. *(x,y) (0,0) b) Find: fy(0,0). c) Where is f continuous? Where is f differentiable? Explain.
3. F =< y2 + x-3, 2xy + e? -y + 2, ye? +2z -4> (1) Prove or disprove that F is conservative. (ii) If F is conservative find the potential function f.
f(x, y) = x2 + y2 + 2xy + 6. 1- Find all the local extremas and 2) does the function f have an absolute max or min on R2
2. Let f:R2 + R be defined by gry, if (x, y) + (0,0) f(x,y) := { x2 + y2 + 1 0 if (x, y) = (0,0). Show that OL (0, y) = 0 for all y E R and f(x,0) = x for all x E R. Prove that bebu (0,0) + (0,0).
(1 point) Consider the function defined by ?(?,?)=??(9?2+5?2)?2+?2F(x,y)=xy(9x2+5y2)x2+y2 except at (?,?)=(0,0)(x,y)=(0,0) where ?(0,0)=0F(0,0)=0. Then we have ∂∂?∂?∂?(0,0)=∂∂y∂F∂x(0,0)= ∂∂?∂?∂?(0,0)=∂∂x∂F∂y(0,0)= Note that the answers are different. The existence and continuity of all second partials in a region around a point guarantees the equality of the two mixed second derivatives at the point. In the above case, continuity fails at (0,0)(0,0). (1 point) Consider the function defined by F(x, y) = xy(9x2 + 5y2) x2 + y2 except at (x, y) = (0,0)...
2. A) Calculate the work done by the field } = (x² - y2,-2xy) when moving an object from the origin to the point (1, 2) along the path C: x = t?, y = 2t. B) Use a Theorem from 16.3 to determine whether or not F = (x2 - y2,-2xy) is a conservative vector field. C) Deduce the work done by the field } = (x2 - y2,-2xy) moving an object from the point (1, 2) to the...
e 09, 201 (6) 2 points An equation for the level curve of f(z, y) = In(z+y) that passes through the point (0, e2) is A. z + y = e2 B. I+y e C. z+y 3. D. None of the above (7) 2 points The gradient of f(z,y, z) = ep at the point (-1,-1,2) is A. (2e2,e2,2e2). B. (-e,-e,2e2). C. (-2e2,-2e2, e) D. (-2e2,-e,-e) (8) 2 points Let f be a function defined and continuous, with continuous first...
1. Find lim(x,y)=(1,1) x2-y2 2xy 2. Show that lim(x,y)-(0,0) 21 z does not exist 3. Show that lim(x,y)=(0,0) z?”, does not exist 4. Find lim(x,y)=(0,0) eye if it exists, or show that the limit does not exist
8.) (minimum along lines does not mean minimum) Define f: R2 and, if (a, y)0, R by f(0,0) (a) Prove that f is continuous at (0,0). Hint: show that 4r4y2 < (z4 + y2)2. (b) Let & be an arbitrary line through the origin. Prove that the restriction of f [0, π) and t E R. (c) Show that f does not have a local minimum at (0,0). Hint: consider f(1,12). to ( has a strict local minimum at (0,0)....
(1 point) Consider the function defined by F(x, y) = x2 + y2 except at (r, y) - (0, 0) where F(0,0)0 Then we have (0,0) = (0,0) = ax dy Note that the answers are different. The existence and continuity of all second partials in a region around a point guarantees the equality of the two mixed second derivatives at the point. In the above case, continuity fails at (0,0) Note: You can earn partial credit on this problem...