Let f: R² R2, f = (f1, f2), be defined by f(0,0) = (0,0), and y³ 23 x² + y²¹ f₂(x, y). x² + y² (a) Show that f is continuous at (0,0). (b) Show that f is not differentiable at (0.0). (c) Show that f is differentiable at the point (1,0) by using the limit-definition directly. (You must explicitly give the linear operator f'(1, 0) and show that it is the (Fréchet) derivative of f at that point.) f₁(x, y) = for (x, y) (0,0). #

Q: Suppose that Q(x, y) is a function such that 1/Q(x, y) is continuous for all (x, y). Which of the…

A: Given: Suppose that Q(x, y) is a function such that 1/Q(x, y) is continuous for all (x, y). To Find:…

Q: Determine if f(x, y) is continuous (x - 1)²y² f(x, y) = (x - 1)² + y² 0 at (1,0) where if (x, y) =…

A:

Q: Show that the function f defined by (r, y) = (0,0) f(x, y) = { 22 + y (r, y) # (0,0) x + y

A:

Q: Show that the function f (z) = Z is continuous at the point z=0 but not differentiable at z=0.

A:

Q: f(r,t) = cos(3r) exp(4t) %3D fi(x,t)+ frz(x,t) = f(r,t) %3D

A:

Q: Show that the function f defined by (x, y) = (1, –1) f(x, y) = r? + y (x, y) # (1, –1) x + y is not…

A:

Q: Show that A Function with a Single Point of Discontinuity 2xy x² +1,²² 0, is continuous at every…

A:

Q: Consider the function f(x)=1/x+4. a) As x→−∞, f(x)→ b)As x→∞, f(x)→

A:

Q: Let f(x, y) = x sin ry. Find: fyy (x, y) fyz(x, y)

A: f(x,y)=xsinxyfx=sinxy+xycosxyfyx=xcosxy+x(cosxy-xysinxy)=2xcosxy-x2ysinxyfy=x2cosxyfyy=-x3sinxy

Q: Assume that f : (0, 1) → R is a differentiable function with the derivative f' increasing on (0,…

A: Given f:0,1→R is a differentiable function and derivative of f is increasing in 0,1 that is f'x>0…

Q: Assume that f(x, y) is a function given by if (r, y) # (0,0), f(x, y) 0 if (r, y) = (0, 0). Page 1…

A:

Q: { 2xy? 9x2 + y4 (x, y) # (0,0) (x, y) = (0,0) f (x, y) ·

A: Topic = Limits and  Continuity

Q: Assume that f(r, y) is a function given by if (r, y) # (0,0), f(r, y) = 0 if (r, y) = (0, 0).

A:

Q: Consider the function f: RR defined by 1 (a) (b) By using definition, explain whether f is…

A:

Q: Find domain and vange f (2y)- し、 (x'+ゴ) ) fCz, y) = Sin" (y-z) Ž (3,y)- tan

A: According to the question,  We need to specify the domain and range of the given function: (A) f(x,…

Q: What value of c makes the following function continuous at (0,0)? f(z, y) = C, x+ y³ +2, (x,y) +…

A:

Q: Let f(x, y) = x/y. Then the directional derivative of f in the direction of √2 point (1, 1) is ○ (1,…

A:

Q: If f (s Continuous then on -fex) True or felse?

A:

Q: Let f(x, y) and let g(x, y) = O O 8xy8 828+8y8 O 3xy6 3x² + 3y¹2 Which of the above functions are…

A:

Q: Let F(x , y , z) = g(y) + h(z) + 5 In(x - 2) , where g(y) and h(z) are differentiable functions of…

A: If a function is a function of more than one independent variable then the derivative of this…

Q: d) Let f be a continuous function. Then 1 [ S f(x,y) dady = f*f* f(x,y) dyda. 0

A:

Q: Assume that the functions ?,?,ℎ:ℝ→ℝf,g,h:R→R are differentiable such that  f′(a)=b, g′(b)=c,…

A: given: ℎ:ℝ→ℝf,g,h:R→R are differentiable such that  f′(a)=b, g′(b)=c, h′(c)=d, f(a)=e, g(e)=ℓ,…

Q: Let f be a differentiable function such that f(h) f(x + y) = f(x) + f(y) + 2xy and lim = 15. Find…

A:

Q: Let f be a function with continuous second derivatives, defined by: z = f (xy, x² + y") If we take u…

A: Solution:-

Q: Show that the function g(x, y) = is not continuous at (0,0). xy 2x² + y²¹ 0, (x, y) = (0,0) (x, y) =…

A: g(x,y)=(xy/(2x2+y2), (x, y) not equal to (0,0)               0                 ,(x, y) = (0,0)

Q: Let F(x, y, z) y³ 2:5 . What is the value of div F at (1, 2, 3)?

A:

Q: Assume that f: R→ R and g: R → R are infinitely differentiable functions. Prove the following…

A: We have given that ,  are both infinitely differentiable.We need to prove that , We will use…

Q: Suppose that f: R² R is a function given by y³ x² + y² f(x, y) = 0 for x = y = 0. Show that f(x, y)…

A:

Q: Suppose that f: R2R is a function given by S x²+y² f(x, y) 0 for xy = 0. Show that f(x, y) is…

A: We have to show the given function  f(x, y) is continuous  at ( 0,0)

Q: Is f(z) = z² + 6xy + y² + 2xyi -x + yi analytic nowhere?

A:

Q: 2x'y+x?y? + 3xy³ 16x4 – y4 if (x, y) # (0,0) Let f(x, y) = if (x, y) = (0,0) Determine the set of…

A:

Q: Let f(x,y)=(x^2+y^2 +1)^1/2. Find fx(0,0) and fy(0,0)(if they exist),and decide if f(x,y) is…

A: The given function f(x,y)=x2+y2+1