2. Sketch the level surface of the function w = f(x, y, z) = 25 – x² – (z – 2)² for w = 9.

Q: Evaluate Se (5xy + x² + y²) dx + (x² - y)dy by using Green's theorem, where C is a closed curve that…

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Q: (a) Find an equation of the tangent plane to the surface of z = ye(3ry-y²) at (-1, –3, –3). (b)…

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Q: The horizontal tangent plane of the surface z = 2x² – xy – y² + 6x – 6y – 2 is given by the equation…

A: Consider the given equation. z=2x2-xy-y2+6x-6y-2 Find the derivative with respect to x and y.…

Q: Find an equation of the tangent plane to the surface g(x, y)=x²-y² at the point (7, 4, 33). O…

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Q: Find the equation of the tangent plane to the surface 4xe^yz+xz^2−7x^3y= 5 at(1,0,1).

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Q: Find the point(s) on the surface z = 3 − x2 − y2 + 6y, at which the tangent plane is horizontal.

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Q: Find the general equation of the tangent plane to the surface x2 + y2 −2xy −x + 3yz = −4 at the…

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Q: of the tangent plane to the surface z=e-4x/17ln(1y) at the point (-3, 4, 2.808).

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Q: Find an equation of the tangent plane to the surface z = x2 + y2 + 2 at the given point (1, 3, 12)

A: Given surface, So

Q: Given the surface z = In(-2x² + 5y² – xy). Find the directional derivative of the function f at…

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Q: Use Green's theorem to evaluate S. (5xy + x² + y²) dx + (x² – y)dy where C is a closed curve that is…

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Q: Find the point(s) on the surface z = x2 − xy + y2 − 2x − 2y, at which the tangent plane is…

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Q: Find the positively oriented simple closed curve C for which the value of the line integral C(y^3…

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Q: Find an equation of the tangent plane to the surface z = x2 + y2 + 3 at the given point( 2, 1, 8).

A: Known fact: Let F(x,y,z) be the surface which is differentiable at P(x0, y0, z0). then the equation…

Q: Find the equation for the tangent plane to the surface x-9xyz +y= - 11z at the point (-1,-1,-1). O…

A: The given surface is x2-9xyz+y2=-11z2 and the point is -1,-1,-1.

Q: Find an equation of the tangent plane to the surface at the given point z= 5x - xy3+y2,     (1,1,5)

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Q: Find an equation for the level surface of the function f(xy.z) = x+ ey+z that passes through the…

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Q: Find a formula for a functionf(x, y, z) whose level surface f= 3 is a cylinder of radius 5 centered…

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Q: 48. Let f(x, y, z) = xyz + 3. Find an equation of the level surface that passes through the point…

A: Given that: f(x, y, z)=xyz+3

Q: Find the absolute maximum value of the function f(x, y) = 3x² + 2y² – 4y+1 | on the disk which has…

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Q: 3. z = (c

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Q: Find the positively oriented simple closed curve C for which the value of the line integral     (y3…

A: The line integral is  ∫C(y3 − y) dx − 9x3dy The expression for Green's theorem is  ∫CP(x,y) dx…

Q: 9) Find a point on the surface z= 8 – 3x² – 2y² at which the tangent plane is perpendicular to the…

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Q: Justify why the level curves very close to the origin (0,0) for the function f(z, y) = (1-z)(1 – )…

A: As per the instruction, we will be solving the first question. For (1), We are given the function as…

Q: Q. A-Find the gradient (Vf) for the following level surface at the given point: f(x, y, = x2 + y² -…

A: Given: f(x,y,z)= x2+y2-2z2+zlnx  Gradient,∇f=i∂f∂x+j∂f∂y+k∂f∂z

Q: Consider the closed curve in the xy-plane given by x² + 2x + y* + 4y = 5. dy -(x+1) (a) Show that dx…

A: Given data:   The expression for the given function is x2+2x+y4+4y=5.   (a) Differentiate the above…

Q: Let C be the curve on the xy-plane that is the projection of a horizontal cross-section of the…

A: Given surface is,yz=ln(x+z) zxy=1x+z(1+zx)y-1x+zzx=1x+zzx=1y(x+z)-1 Take the derivative with respect…

Q: 2. Let F(x,y) = (cos(x – 2y), –2 cos(x – 2y)). Find curves C and C2 so | that, F. dr = 0 F. dr = 1

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Q: Find the area of the surface generated when the curve f(x) = (2/3)x^(3/2) - (1/2)x^(1/2) ; [1,9] is…

A: Given:

Q: Find the equation of the tangent plane to the surface 2 = r - 2r'y+ 2r y at the point where x = 1…

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Q: Find the point(s) on the surface z = 3x2 + 2y2 − 3x + 4y − 5, at which the tangent plane is…

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Q: The surface of a hill above the xy-plane is described implicitly by 3x² + y² + (z + 1)² = 2, z ≥ 0.…

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Q: Find the equation of tangent plane to the surface x² + 2y? + 3z2 = 21 which is parallel to the plane…

A: Topic- equation of tangent to surface

Q: 9) Find a point on the surface z = 8 – 3x2 – 2y² at which the tangent plane is perpendicular to the…

A: Given, Surface, z=8-3x2-2y2 at which the tangent plane is perpendicular to the line…

Q: Suppose we have a surface f (x,y)=x²y+cos (4x) · Which of the following is the gradient of f(x,y) at…

A: First I have found the partial derivatives of f(x,y)  with respect to 'x' and 'y' at point P(π/2,…

Q: Given a surface f(x,y) = -3x² + (3)y². Find the derivative in the direction of v = at the point (5,…

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Q: If (a,B,y) is a point at which the surface x²+y²-z²–2x+104=0 has a horizontal tangent plane, then…

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Q: If 0 ≤ f (x) ≤ g(x) for x in the interval [a, b], can the surface obtained by rotating the graph of…

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Q: 5) Consider the surfaces S1 : ryz = 10 and S2 : z =r² + y and the %3D point Po = (1,2,5).

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Q: 4. Find the equation of the tangent plane to the level surface x4 + yz – yz² + 5x = 0 at 3

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Q: Find the surface area of the rotating object from the curve y = x^2 with the constraint 0 ≤ x ≤ 1…

A: The formula for the surface area of the rotating object from the curve y=f(x) with the constraint…

Q: Find the surface area of the rotating object from the curve y = x^2 with the constraint 0< x <1…

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Q: 5. Are there any points on the graph of x2 – y² – 22 = 1 where the tangent plane is parallel to the…

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Q: Find an equation of the plane tangent to the surface f(x, y) = xy² – 5x at the poin (1, 2, –1).

A: Consider the provided question, Hello. Since your question has multiple parts, we will solve first…

Q: Find the equation of the tangent plane to the surface z = xy at the point (1, 2, 2) and to the…

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Q: Find the equation of the tangent plane to the surface z =e/ In (2y) at the point (2, 4,1.643). Note:…

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Q: For the surface f(x, y, z) = 2*yz, find: The derivative at the point (1,2,1) in the direction of v =…

A: To find the derivative  of the given function f(x,y,z) at point (1, 2,1) in the direction of vector…

Q: 8. Consider the surface S of equation z2-2x? + y²-x= 0 and let P (2, 3, 1) be a point at S. Suppose…

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Q: Find the equation of the tangent plane to the surface 4. =1 at the point 16 9. (1,-3,2) O a.…

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Arc Length

Arc length can be thought of as the distance you would travel if you walked along the path of a curve. Arc length is used in a wide range of real applications. We might be interested in knowing how far a rocket travels if it is launched along a parabolic path. Alternatively, if a curve on a map represents a road, we might want to know how far we need to drive to get to our destination. The distance between two points along a curve is known as arc length. 

Line Integral

A line integral is one of the important topics that are discussed in the calculus syllabus. When we have a function that we want to integrate, and we evaluate the function alongside a curve, we define it as a line integral. Evaluation of a function along a curve is very important in mathematics. Usually, by a line integral, we compute the area of the function along the curve. This integral is also known as curvilinear, curve, or path integral in short. If line integrals are to be calculated in the complex plane, then the term contour integral can be used as well.     

Triple Integral

Examples: