0, x? < y < 2x² 1, otherwise. Show that f.(0, 0) and f (0, 0) exist, but ƒ is not differentiable at 102. Let f(x, y) = (0, 0).

Q: 4. Find the linear approximation to f(x) = cos(x) near a =- [Where L(x) = f(a) + f'(a)(x – a) ]

A:

Q: f(x.y) =e^sin(y) + In(xy) ; әз of Әхд-у

A: Given: fx,y=exsiny+lnxy To find the partial derivative of ∂3f∂x∂2y Partial differentiation: Partial…

Q: Suppose that z is a differentiable function of x and y with dz (2, 5) = 108 and- dx dz (2, 5) = 9.…

A:

Q: f(z) is a twice differentiable function with f(0) = 1, f(2) = 4, f'(2) = 5, Then af"(x)dx = %3D

A:

Q: Find the local linear approximation to f(x) = Vx at x = 1 and use it to approximate both 1.1 and…

A: To know which one is better we have to compare with calculator values of sqrt(1.1)  and sqrt(0.1)

Q: At what points (x, y, z) in space are the functions in continuous? h(x, y, z) = ln (z - x2 - y2 - 1)

A:

Q: Using a local linear approximation of f(x) = v4 + x at a 0 to estimate the value of v4.3 gives

A:

Q: Use linear approximation to find y = L (x) of the function f (x) = = COS X about the point xXo = JT…

A: Use formula of  linear approximation

Q: Let f(x) = e2sinz 1. Use the local linear approximation of f at x = 0 to approximate e2 sin(0.01

A:

Q: Find and by implicit differentiation, and confirm that the - dy results obtained agree with those…

A: The solution is as follows

Q: the linear approximation for f(x,y) = sqrt (y+cos^2 x) at (0,35) is ?

A:

Q: Assuming - 6x° -y² + 7xy = 0 defines y as a differentiable function of x, use the theorem Fx to find…

A:

Q: Let v(x, y) cos(4x) sinh(4y). Find f(z)with Im( f(z)) = (x, y).

A: using the concept of cauchy - Riemann equations to find u and putting  the value of u and v (given)…

Q: if (x,y) # (0,0) 1. Let f:R2 → R be defined by f(x,y) = Find if f is 0, if (x,y) = (0,0) continuous…

A:

Q: is concave up on interval The graph of f(x)= x2 – 1 OA(-1,1) OB (0,1) oC(-1.0) U (1, 00) OP (-x,-1)U…

A: F is concave up if f’’(x) &gt; 0.

Q: 4. If f is a differentiable function of x and y with w = f(x, y), x = r cos 0 and y = r sin 0, show…

A: Given: w=fx,y, x=rcosθ, y=rsinθ If w is a multi variable function w(x,y,a,........z) thus the…

Q: find ∇f at the given point. ƒ(x, y, z) = 2z3 - 3(x2 + y2)z + tan-1 xz, (1, 1, 1)

A:

Q: 4. If f is a differentiable function of a and y with w f(x, y), x =r cos 0 and y = r sin 0, %3D show…

A: To verify the given relations between partial derivatives.

Q: Assuming 8x - y +7xy = 0 defines y as a differentiable function of x, use the theorem dx Fx to find…

A: By using partial differentiation .

Q: Find the derivative of the given function at P. (3, 3, 3) in the direction of u = 3i + 4j f (x, y,…

A:

Q: If x, y, and z are lengths of the edges of a rectangularbox, then the common length of the box’s…

A:

Q: Using the linear approximation of f (x, y) near (x, y) = (1, 0) tells us that f (0.98, 0.05) is…

A: The linear approximation of a function f(x, y) at (a, b) is L(x, y) = f(a, b) + fx(a, b)(x − a) +…

Q: Consider the function 23 – 3y if (x, y) # (0,0), f (x, y) = x² + y? if (x, y) = (0,0). Show that…

A:

Q: tion z2 – (x2 + y2)2 and let p = (0, 0, 0). ermine the set Tp(Zf ) of all tangent prs to Zf at p. Is…

A: Sol

Q: Find a point zo where the function f (z) = (z – 8 – 10i) (z – 8 – 10i) is differentiable. What is…

A:

Q: Find the linear approximation of the given function at (0, 0). f(x,y) = sqrt(y+(cos(x))^2)

A: The given function fx, y=y+cos2x The linear approximation in 2 variables is given by Lx, y-fa,…

Q: Let v(x, y) = cos(4x) sinh(4y). Find f(2)with Im(f(z)) = v(x, y).

A:

Q: The tangent line to the curve y=f(x) is horizontal at where X = Xo f f(xx) = 00 A O f(xo) = 00 .8 O…

A: The tangent line to the curve y=f(x) is horizontal at where x=x0 if

Q: xY (a) Prove that the function f(x,y) is differentiable everywhere in its do- x² + y? main.

A: Differentiablility: A function z=f(x, y) is said to be differentiable in its domain D if the partial…

Q: Show that the function no-f 0, z = 0 f(2) = { – y* z#0 x2 + y? x² + y² is not differentiable at z =…

A:

Q: -2 X Given f(x,y,2) sin (z²y), show that fxyuy = fyxy %3D %3D

A: Given:- f(x, y, z) =e-2x sin(z2y)  To prove:- fxyy = fyxy

Q: Assuming 9x° – 5y - 4xy = 0 defines y as a differentiable function of x, use the theorem dx - to…

A: Given: 9x3-5y2-4xy=0 We have to calculate dydx(1,1) by the given formula.

Q: prove that xy^2/x^2+y^2 is not differentiable at 0,0 when  xy^2/x^2+y^2 if (x,y) not equal to (0,0)…

A:

Q: Find He valve of fe2) when fg functions such that are dilferentrable %3D g(x) >o 1 for all x, while…

A:

Q: Find the linear approximation at x = 0 to each of f(x) (x + 1)?, g(x) = 1+ sin(2x) and h(x)= e2*.…

A:

Q: (i) Find the linear approximation of the function f(x, y) = 1 - xy cos πy at (1, 1) and use it to…

A: Here, the given function is: f(x,y)=1-xycos πy To find the linear approximation of the function. For…

Q: Find the best linear approximation to f(x) = ex on the interval [-1,1]

A: We have to find the best linear approximation of f(x)=ex on the interval -1,1.

Q: (a, b), f (x, 42. Show that ifJ (2, Y) is differentiable at then the function of one variable J (2,…

A: Definition (Differentiability of two variable function): A real valued function f(x,y) is said to be…

Q: Find fr and fy- f (x, y) = | 5e dt %3| fa(x, y) = fy(x, y) =

A: We apply fundamental theorem of calculas and solve it.

Q: If f(x, y, z) = In(x³ + yl² + 2®) + V (x12 + y6 + 23), then find fr, fy and f; at (1, 1, 1).

A:

Q: At what points (x, y, z) in space are the functions in continuous? h(x, y, z) = xy sin 1/z

A: Given function is  h(x, y, z) = xy sin 1z  We  have no additional conditions for x y part for any x…

Q: Let f (x,y) = sin(xy) + x² ln(y). fyx at (0,) is equal to: %3D А. 33 В. О С. 3 D. 1

A:

Q: if (2,y) # (0,0) Let f(x, y) = aª + yª Show that D1f(0,0) and if (2,y) = (0,0) D2f(0,0) exist but f…

A:

Q: if z=0 ne function f(z)= )² if z+0 is differentiable at z= 0 LL

A:

Q: At what points (x,y,z) in space are the functions continuous? a. h(x,y,z) = In (2z2 – x² – 3y³ – 4)…

A: we want to find the domain of given  function

Q: Find the linear approximation L(x) to y = f(x) near x = a for the function. f(x) = sin(x), a = 2…

A:

Q: prove that x^2y/(x^2+y^2) is not differentiable at 0,0 when  x^2y/(x^2+y^2) if (x,y) not equal to…

A: f(h,k)-f(0,0) =hfx(0,0) +kfy(0,0) +h2+k2∅(h,k) h2kh2+k2-0 =hA+kB+h2+k2ϕh,k ...(1) A =fx(0,0)…

Q: implicitly dx

A: Solved below

Q: if (x, y) # (0,0) (1 if (x,y) (0,0) x+y (a) Prove that the function f(x, y) =2x+y is not continuous…

A: