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Exercice 2 (5pts) Let f given by f(x, y) Isinyif (x, y) (0,0) and f(0,0) 0...
Exercice 2 (5pts) Let f given by f(x, y) Isinyif (x, y) (0,0) and f(0,0) 0 1V224 1. Is f continuous at (0,0). 2. Compute the partial derivatives of f at any (x, y) E R2. Are the partial derivatives continuous (0,0). at (0,0) (0,0) and 3. Compute the second derivatives 4. Compute the linear approzimant of f at (0,0).
Exercice 2 (5pts) Let f given by f(x, y) Isinyif (x, y) (0,0) and f(0,0) 0 1V224 1. Is f...
Problem 1: Let F(, y,) be a function given by F(, y, z) (r2+y)e. Let S be the surface in R given ...
Problem 1: Let F(, y,) be a function given by F(, y, z) (r2+y)e. Let S be the surface in R given by the equation Fr, y, 2) 2. (a) Find an equation of the tangent plane to the surface S at the point p(-1,1,0) (b)Find the directional derivative -1,1,0) of F(,y,2) in the direction of the unit vector u = (ui, t», t's) at the point p(-1,1,0) - In what direction is this derivative maximal? In what direction is...
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).
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...
3 This problem is to prove the following in the precise fashion described in class: Let O C R2 be...
Please write carefully! I just need part a and c done.
Thank you. Will rate.
3 This problem is to prove the following in the precise fashion described in class: Let O C R2 be open and let f: 0+ R have continuous partial derivatives of order three. If (ro, o) O a local maximum value at (To, Va) (that is, there exist r > 0 such that B. (reo) O and (a) Multivariable Taylor Polynomial: Suppose that f has...
Question 8 (15 marks) Consider the function f: R2 R2 given by 1 (, y)(0,0) f(r,y) (a) Consider the surface z f(x, y). (...
Question 8 (15 marks) Consider the function f: R2 R2 given by 1 (, y)(0,0) f(r,y) (a) Consider the surface z f(x, y). (i Determine the level curves for the surface when z on the same diagram in the r-y plane. 1 and 2, Sketch the level curves (i) Determine the cross-sectional curves of the surface in the r-z plane and in the y- plane. Sketch the two cross-sectional curves (iii) Sketch the surface. (b) For the point (r, y)...
the function of two real variables defined below: 1 –9x + 2y“ (x, y) + (0,0),...
the function of two real variables defined below: 1 –9x + 2y“ (x, y) + (0,0), f(x, y) = { 6x + 3y 10 (x, y) = (0,0). Use the limit definition of partial derivatives to compute the following partial derivatives. Enter "DNE" if the derivative does not exist. fx(0,0) = DNE fy(0,0) = 0
DUE DATE: 23 MARCH 2020 1 1. Let f(x,y) = (x, y) + (0,0) 0. (x,...
DUE DATE: 23 MARCH 2020 1 1. Let f(x,y) = (x, y) + (0,0) 0. (x, y) = (0,0) evaluate lim(x,y)=(4,3) [5] 2r + 8y 2. Show that lim does not exist. [10] (*.w)-(2,-1) 2.ry + 2 3. Find the first and second partial derivatives of f(x,y) = tan-'(x + 2y). [16] 4. If z is implicitly defined as a function of x and y by I?+y2 + 2 = 1, show az Əz that +y=z [14] ar ду 5....
2. Let if r and y are not both 0 f(x, y) = 0 if (x,...
2. Let if r and y are not both 0 f(x, y) = 0 if (x, y) = (0,0) (a) Show that and we both exist at the origin are are zero (b) Let v = (v1, v2) be a unit vector with vị and v2 both not zero. Prove that V (f) at the origin exists, and compute it directly from the definition. Does the formula Vu(f) = (Vf). ✓ hold at the origin? (c) Is f differentiable at...
Question 2 (20 points): Consider the functions f(x, y)-xe y sin y and g(x, y)-ys 1. Show f is differentiable in its domain 2. Compute the partial derivatives of g at (0,0) 3. Show that g is not d...
Question 2 (20 points): Consider the functions f(x, y)-xe y sin y and g(x, y)-ys 1. Show f is differentiable in its domain 2. Compute the partial derivatives of g at (0,0) 3. Show that g is not differentiable at (0,0) 4. You are told that there is a function F : R2 → R with partial derivatives F(x,y) = x2 +4y and Fy(x, y 3x - y. Should you believe it? Explain why. (Hint: use Clairaut's theorem)
Question 2...
Exercice 2 (5pts) Let f given by f(x, y) Isinyif (x, y) (0,0) and f(0,0) 0 1V224 1. Is f continuous at (0,0). 2. Compute the partial derivatives of f at any (x, y) E R2. Are the partial derivatives continuous (0,0). at (0,0) (0,0) and 3. Compute the second derivatives 4. Compute the linear approzimant of f at (0,0).
Exercice 2 (5pts) Let f given by f(x, y) Isinyif (x, y) (0,0) and f(0,0) 0 1V224 1. Is f...
Problem 1: Let F(, y,) be a function given by F(, y, z) (r2+y)e. Let S be the surface in R given by the equation Fr, y, 2) 2. (a) Find an equation of the tangent plane to the surface S at the point p(-1,1,0) (b)Find the directional derivative -1,1,0) of F(,y,2) in the direction of the unit vector u = (ui, t», t's) at the point p(-1,1,0) - In what direction is this derivative maximal? In what direction is...
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).
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...
Please write carefully! I just need part a and c done.
Thank you. Will rate.
3 This problem is to prove the following in the precise fashion described in class: Let O C R2 be open and let f: 0+ R have continuous partial derivatives of order three. If (ro, o) O a local maximum value at (To, Va) (that is, there exist r > 0 such that B. (reo) O and (a) Multivariable Taylor Polynomial: Suppose that f has...
Question 8 (15 marks) Consider the function f: R2 R2 given by 1 (, y)(0,0) f(r,y) (a) Consider the surface z f(x, y). (i Determine the level curves for the surface when z on the same diagram in the r-y plane. 1 and 2, Sketch the level curves (i) Determine the cross-sectional curves of the surface in the r-z plane and in the y- plane. Sketch the two cross-sectional curves (iii) Sketch the surface. (b) For the point (r, y)...
the function of two real variables defined below: 1 –9x + 2y“ (x, y) + (0,0), f(x, y) = { 6x + 3y 10 (x, y) = (0,0). Use the limit definition of partial derivatives to compute the following partial derivatives. Enter "DNE" if the derivative does not exist. fx(0,0) = DNE fy(0,0) = 0
DUE DATE: 23 MARCH 2020 1 1. Let f(x,y) = (x, y) + (0,0) 0. (x, y) = (0,0) evaluate lim(x,y)=(4,3) [5] 2r + 8y 2. Show that lim does not exist. [10] (*.w)-(2,-1) 2.ry + 2 3. Find the first and second partial derivatives of f(x,y) = tan-'(x + 2y). [16] 4. If z is implicitly defined as a function of x and y by I?+y2 + 2 = 1, show az Əz that +y=z [14] ar ду 5....
2. Let if r and y are not both 0 f(x, y) = 0 if (x, y) = (0,0) (a) Show that and we both exist at the origin are are zero (b) Let v = (v1, v2) be a unit vector with vị and v2 both not zero. Prove that V (f) at the origin exists, and compute it directly from the definition. Does the formula Vu(f) = (Vf). ✓ hold at the origin? (c) Is f differentiable at...
Question 2 (20 points): Consider the functions f(x, y)-xe y sin y and g(x, y)-ys 1. Show f is differentiable in its domain 2. Compute the partial derivatives of g at (0,0) 3. Show that g is not differentiable at (0,0) 4. You are told that there is a function F : R2 → R with partial derivatives F(x,y) = x2 +4y and Fy(x, y 3x - y. Should you believe it? Explain why. (Hint: use Clairaut's theorem)
Question 2...
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