O If and exist in a neighborhod of (a, b) and are continuous at (a, b) then f(x, y) is differentiable at (a, b). af af and always exist. dy lim (1,9) → (a, b) O f f(x, y) is differentiable at (a, b) then a tangent plane to f(x, y) exists at (a, b). O if f(z, y) is continuous at (a, b) then it is differentiable at (a, b). O if f(z, y) is differentiable at (a, b) then it is continuous at (a, b). f(x, y) always exists.

Q: Let f(x+y)-(x)(y) for all x, y e R and suppose that fis differentiable at x = 0.Show that fis…

A:

Q: (d) (e) Determine the range f(A) where A = (a, 0] for some fixed a < 0,a # Determine the supremum…

A:

Q: Which of the following functions are continuous? 0. .(x, y) =(0,0) f(x,y) ={ 2x(x² – y²) I) (x, y) =…

A:

Q: f(z) = f(x+iy) = Z

A:

Q: Determine whether the function xy²/³ + xy ³√x⁹ + 2y³¹ 1 √3' if (x, y) = (0,0) if (x, y) = (0,0) is…

A: Let us consider a function f then f is said to be continuous at c if f(c) is defined limx→cf(x)…

Q: Let Fn be Fibonacci sequence. Prove that:

A: By the definition of the Fibonacci sequence, Fn+2=Fn+1+Fn and the Fibonacci sequence is satisfied…

Q: Prove that if f is a differentiable function such that ∇f(x0, y0) = 0, then the tangent plane at…

A: Here the objective is to prove that if f is a differentiable function such that

Q: A function F is defined on R² By: f (x, y) (x, y)# (0,0) F (x, y) = %3D (x, y) = (0,0) x²y+2xy² Let…

A:

Q: Let f be a function of R sucht hat Ә f(x, y) = f(x,y); for all (x, y) = R then prove that - y^ f(x,…

A: Since given partial differential equation is               fx=fy⟹      fx-fy=0Auxillary equation for…

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: Q\ Let F(x,y,z)=(sin(xy), e",2x+yz") then find: 1. Grad F(x,y,z) at (-1,1,-1) 2. Div F(x,y,z) at…

A:

Q: Let f(x, y) = and let g(x, y) O O rºy6 26+y6 O 0 0 6x³y³ 6x6 + 6y6 Which of the above functions are…

A: fx,y=x6y6x6+y6if x,y≠0,00if x,y=0,0 gx,y=6x3y36x6+6y6if x,y≠0,00if x,y=0,0

Q: Tutorial 1 (1.1) Let B = {r, y} and c> 0. Suppose the functionf on Bx B satisfies S(17, 2) = f(y, v)…

A:

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: Find the limit 2yz lim (x,y,z) (0,0,0) x4 + y² + z² along the curve x = 2t, y = 3t², and z = 4t².

A: I am going to solve the problem by using some simple calculus to get the required result of the…

Q: How would you remove the dis . z46 f(r) =

A:

Q: Let f(x, y) = sin (x - y) [x] + [y] ' 0, |x | + (x, y)

A:

Q: b) If the function f(x), g(x) and h(x) are continuous on [a, b] and derivable on (a, b), show that…

A: The given functions f,g and h are continous on [a,b] and differntiable on (a,b). So intermediate…

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: Suppose that f is differentiable on [a, b]. Prove that if f'(a)f'(b)< 0 then there exists a point c…

A:

Q: Let & be continuous in [a,b] and differentiable in (a,b). It's true that: a) f (a) = f(b) = 3 x €…

A: Statement (a): f(a)=f(b)  implies there exists x0​∈(a,b)  such that the tangent line to the graph of…

Q: Calculate of of and for the function f(x, y) defined as : ax' dy sin (xy) 1 + y f(x, y) =

A:

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

A:

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: Theorom:- Let f-3GDCI); XoEIthen O TfaeR then a fisdefferantible at (a P) (xo)= QP(xo) D ft…

A:

Q: Consider the function f defined by f(r,y) = y %3D lim(z,y) (0,0) f(x, y) =

A:

Q: If f', and y are continuous on [a, b] and derivable on Ja, b[, then show that there is a value c…

A:

Q: Consider the function f defined by f(x, y) = r lim(z.y)¬(3,5) f(x, y) =

A: