Prove that f(x, y) = x sin y is differentiable everywhere.

Q: Assuming that z is a differentiable function of x and y, Let F(x.y.2) = In(x³y) – e" -tan-z. Find F,…

A: We can find the answer as below.

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: Get z = f(x,y), x = es cos t, y= es sin t 2 + əx. 2 2 Show that it's + at. as

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Q: Determine if the following are homogeneous or non-homogeneous functions f (x, y) = cos = cOS x + y f…

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Q: 11. Verify that the conclusion of Clairaut's Theorem holds for the function: f(x, y) = e*Y sin(y)

A: The given function is: fx,y=exy siny To verify that the conclusion of Clairaut's theorem hold for…

Q: Find a non-zero function h for which F(x, y) = h(x)(x sin y + y cos y)i + h(x)(x cosy – y sin y)j is…

A: Given vector field is, Fx,y→=hxxsiny+ycosyi+hxxcosy-ysinyj.

Q: verify that wxy = wyx . w = x sin y + y sin x + xy

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Q: Calculate fxxyz for f(x, y, z) = y sin(x + z).

A: For a function in several variables, partial derivative with respect to one variable say x can be…

Q: Let { sin(r-y) |r|+ |y| 0 (x, y) = (0,0) f(x, y) = Is f continues at the origin? Why?

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Q: If f(x) • cos ( z ) t'dt then f'(x) =

A: We can find the answer as below

Q: u(x, y) = e¯ª (cos x cos y cosh y - sin x sin y sinh y)

A: We have to find the analytic function for which the real part is u(x,y)=e-x(cosx cosy coshy-sinx…

Q: Use the Cauchy-Riemann Equations to show that f(z) = z Im(z) is only differentiable at z =0 and find…

A: Cauchy-Riemann Equations: Let f=u+iv be a complex valued function , then the Cauchy-Riemann…

Q: Let f(x) = sin? (x). Define F(x) | f(t) dt. What is F" -) ?

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Q: c = sec t, and y =

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Q: (a) If f(x,y) = e" siny – e sinx then verify that fæy = fyx. %3D -

A: Given that  f(x,y)=e^xsiny-e^ysinx To verify the fxy=fyx

Q: c) Let f(r, y) = Vlry| Prove that f(r, y) is not differentiable at point (0, 0).

A: Given a function : f(x,y)=xy We need to show that f(x ,y) is not differentiable at (0 ,0). Let's…

Q: Find the linear approximation of the function at the given point: f(x, y) = 2 sin x – 3 cos y at (0,…

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Q: Where is the function f(x, y) = e' arctan (VI+In(x+y)

A: In this question, the concept of continuity is applied. Given function is fx,y=etan-11+lnx+y

Q: If f is differentiable everywhere and f(1)=f(2)=π, then ∫12 f'(t)dt=0   True or false?

A: True because,

Q: 11. Verify that the conclusion of Clairaut's Theorem holds for the function: f(x, y) = exy sin(y)

A: The objective is to verify the conclusion of clairaut's theorm.

Q: Show that f is continuous at (0, 7) where a cos y + y(e" – 1) if r + 0 f(x, y) = ay if æ = 0.

A: In the given question we have to show that the given function f(x,y) is continuous at the point…

Q: If F(x)= J. x) = [", "+!dt then F'(x) dt , then F'(x)= /3 sint

A: If Fx=∫π3xt2+1sintdt, then evaluate the value of F'x.

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

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

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Q: (a) f(x) = e" centered at r = 0.

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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)…

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Q: Verify that the conclusion of Clairaut's Theorem holds for the function: f (x, y) = e*y sin(y)

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

A: using CR equation we get....

Q: Apply the first part of the fundamental theorem to determine the value of F'(x) if: e) F (x) = S…

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Q: Let f(x, y) = y cos (xy), then f.(x, y) = fy(x, y).

A:

Q: c) Let f(x, y,z) = cos (xyz)sin(x²y=

A: The given function is f(x,y,z)=cos2(xyz)sin(x2y2z2) ..............(1) we have to find;  fx=∂f∂x=?and…

Q: Determine the set of points at which the function is continuous. cos(ry) er – y² ' b) f(x,y) =

A: The given function is: f(x,y)=cos(xy)ex-y2

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=f(x,y) x=rcosθ and y=rsinθ

Q: Show that f(z) = x³+i(1−y)³ is differentiable only at x = 0, y = 1 and f'(z) = 0.

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Q: b) f(x): 1 + sec2

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Q: Verify whether the function f(z) = e^x (cos y + isin y) satisfies Cauchy- Riemann equations or not.

A: We know that cauchy-Riemann equations  ux = vy  and uy = -vx Here  u = excosy and v = exsiny

Q: xY 2. (a) Prove that the function f(x, y) is differentiable everywhere in its do- %3D x2 + y? main.

A: Solution is given below:

Q: Let U (x, y) = e-(y sin x Find a real-valued function V (x, y) such that f(z) = U + iV is analytic.…

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Q: If F(x)=[ cos(t² +1)dt , calculate using the fundamental theorem of calculus derived version.

A: Given : F(x)=∫-xxcos(t2+1)dt Aim: to calculate using the fundamental theorem of calculus ,the…

Q: Find the antiderivative F(x) of f(x) = 8(sec x)(sec x – 4 tan x).

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Q: F(x) = [,**dt COS x 1 %3D 19

A: Answer is given below

Q: Verify the linear approximation at ( 0, 0) for e^x cos(xy)

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Q: Suppose f(x) = x² + sin æ and F is an antiderivative of f that satisfies F(T) = 0. Then F(x) =

A: Solution..

Q: Find an antiderivative F(x) of the function f(x)= 7sinx / 12+cosx such that F(0)=1.

A: Now,

Q: The gravitational force exerted by the planet Earth on a unit mass at a distance r from the center…

A: To find the continuity of F(r).

Q: Use the Second Fundamental Theorem of Calculus to find F'(x). F(x) = 5t cos(t) dt %3D F'(x) =

A: Simple use the second fundamental theorem of calculas   go Through the solution..

Q: Suppose f(x) = x2 + sin x and F is an antiderivative of f that satisfies F(T) = 0. %3D Then F(x) =

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Q: Let f (x, y) = x+ y Evaluate . f (x,y)ds when x sin t, y = sin t

A: Explanation of the solution is given below...

Q: f(x) = f(x) +² dt where g(x) = [cosx[1 + sin t²]dt, find f'(-)

A: It is known that: ∫g(t)h(t)f(x)dx=h'(t)f(h(t))-g'(t)f(g(t)). We need to calculate f'π2 using the…

Q: Show that the function f(x,y) =x² cos² y+ ysin² x+xe" , satisfies the 2 COS Lipschitz condition in…

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Family of Curves

A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.

XZ Plane

In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.  

Euclidean Geometry

Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.

Lines and Angles

In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.