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Let f:R2→R be defined by f(x,y) =|xy|e−(x2+y2). Evaluate ∫R2f, if it exists Let f : R2...
Let f:R2→R be defined by f(x,y) =|xy|e−(x2+y2). Evaluate
∫R2f, if it exists
Let f : R2 + R be defined by f(L,y) = [tyle=(3++y?). Evaluate Sir2 f, if it exists
(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)...
(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...
2. Consider the function f : R2 → R defined below. r3уг_ if (x,y) (0,0) f(x,y)...
2. Consider the function f : R2 → R defined below. r3уг_ if (x,y) (0,0) f(x,y) = if (x, y) (0, 0) (a) Prove that f is continuous at (0,0) (b) Calculate the partial derivatives (0,0) and (0,0) directly from the definition of partial derivatives. (c) Prove that f is not differentiable at (0,0).
4. Let F(x,y) - PiQj be a smooth plane vector field defined for (x,y) f (0,0), and F - dr for int...
4. Let F(x,y) - PiQj be a smooth plane vector field defined for (x,y) f (0,0), and F - dr for integer j, and all suppose Q - Py for (z, y) (0,0). In the following L-JF dr for integer j, and all G are positively oriented circles. Suppose h = π where G is the circle x2 + y2-1. (a) Find 12 for G : (x-2)2 + y-1. Explain briefly. (b) Find Is for Cs: ( -2)y 9. Explain...
Let F = (P,Q) be the vector field defined by -x+y . P(x,y) = 22, (x,...
Let F = (P,Q) be the vector field defined by -x+y . P(x,y) = 22, (x, y) + (0,0) 0, (x, y) = (0,0) Q(x,y) = (x, y) + (0,0) x2+y2; 10,(x, y) = (0,0). (a) Show that F is a gradient vector field in R2 \ {y = 0}. (b) Letting D = {2:2020 + y2020 < 1}, compute the line integral Sap P dx + Q dy in the counter-clockwise direction. (c) Does your calculation in part (b)...
if (r.y) (0,0), 0,f (, y) (0, 0) 2. Consider f : IR2 -R defined by f(r,y)-+ (a) Show by explicit computation that the directional derivative exists at (x, y)- (0,0) for all oi rections u є R2 with 1...
if (r.y) (0,0), 0,f (, y) (0, 0) 2. Consider f : IR2 -R defined by f(r,y)-+ (a) Show by explicit computation that the directional derivative exists at (x, y)- (0,0) for all oi rections u є R2 with 1 11-1, but that its value %(0.0) (Vf(0,0).u), fr at least one sucli u. (b) Show that the partial derivatives of f are not continuous at (0,0)
if (r.y) (0,0), 0,f (, y) (0, 0) 2. Consider f : IR2 -R...
(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...
(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)...
Let f : [0, 1] x [0, 1] + R be defined by f(x, y) =...
Let f : [0, 1] x [0, 1] + R be defined by f(x, y) = {1 if y = 23, 0 if y + x2 Show that f is integrable on (0, 1] x [0, 1]. You may take the previous problem as given
8.) (minimum along lines does not mean minimum) Define f: R2 and, if (a, y)0, R by f(0,0) (a) Pro...
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)....
Let f : [0, 1] x [0, 1] → R be defined by f(x,y) - 1...
Let f : [0, 1] x [0, 1] → R be defined by f(x,y) - 1 if y=%, 0 if y#x2 Show that f is integrable on [0,1] [0,1]. You may take the previous problem as given
Let f:R2→R be defined by f(x,y) =|xy|e−(x2+y2). Evaluate
∫R2f, if it exists
Let f : R2 + R be defined by f(L,y) = [tyle=(3++y?). Evaluate Sir2 f, if it exists
(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...
2. Consider the function f : R2 → R defined below. r3уг_ if (x,y) (0,0) f(x,y) = if (x, y) (0, 0) (a) Prove that f is continuous at (0,0) (b) Calculate the partial derivatives (0,0) and (0,0) directly from the definition of partial derivatives. (c) Prove that f is not differentiable at (0,0).
4. Let F(x,y) - PiQj be a smooth plane vector field defined for (x,y) f (0,0), and F - dr for integer j, and all suppose Q - Py for (z, y) (0,0). In the following L-JF dr for integer j, and all G are positively oriented circles. Suppose h = π where G is the circle x2 + y2-1. (a) Find 12 for G : (x-2)2 + y-1. Explain briefly. (b) Find Is for Cs: ( -2)y 9. Explain...
Let F = (P,Q) be the vector field defined by -x+y . P(x,y) = 22, (x, y) + (0,0) 0, (x, y) = (0,0) Q(x,y) = (x, y) + (0,0) x2+y2; 10,(x, y) = (0,0). (a) Show that F is a gradient vector field in R2 \ {y = 0}. (b) Letting D = {2:2020 + y2020 < 1}, compute the line integral Sap P dx + Q dy in the counter-clockwise direction. (c) Does your calculation in part (b)...
if (r.y) (0,0), 0,f (, y) (0, 0) 2. Consider f : IR2 -R defined by f(r,y)-+ (a) Show by explicit computation that the directional derivative exists at (x, y)- (0,0) for all oi rections u є R2 with 1 11-1, but that its value %(0.0) (Vf(0,0).u), fr at least one sucli u. (b) Show that the partial derivatives of f are not continuous at (0,0)
if (r.y) (0,0), 0,f (, y) (0, 0) 2. Consider f : IR2 -R...
(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)...
Let f : [0, 1] x [0, 1] + R be defined by f(x, y) = {1 if y = 23, 0 if y + x2 Show that f is integrable on (0, 1] x [0, 1]. You may take the previous problem as given
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)....
Let f : [0, 1] x [0, 1] → R be defined by f(x,y) - 1 if y=%, 0 if y#x2 Show that f is integrable on [0,1] [0,1]. You may take the previous problem as given
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