c) By using Green's theorem evaluate $ (4xy – y°) dx + (x³ + 3xy) dy where C is a closed curve starting from the origin to point (0, 1) along -1 the y-axis, then to point G) along the semi-circle V2' v2, x = -V1- y?, then on a straight line, back to the origin.

Q: ind the equation for the tangent plane to the surface z = In(6x2 +9y2 + 1) at the point (0,0, 0).…

A:

Q: c) By using Green's theorem evaluate $ (4.cy – y*) dx + (2³ + 3xy) dy where C is a closed curve…

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Q: 2.2 Prove that Te -dz < 3 ez z2 +7+1 where C is the arc of the circle |2| = 3 from z = 3i to z = -3.

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Q: - Use the graphs of = irection in which the curve is traced as t increases. = f(t) and y = g(t) to…

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Q: Show that the curve y = x? sin 3x touches the parabolas y = x andy =-x² alternately at infinite…

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Q: Use Green's theorem to calculate §. -y² dx + xy dy where C is the vertex triangle (1,3), (4,0) and…

A: Consider the given integral. I=∮c-y2dx+xydy The green theorem is defined as, ∮cPdx+Qdy=∬R∂Q∂y−∂P∂x…

Q: (b) Find the circle of curvature at (0, 1)of the curve y = x3 + 2x² + x +1

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Q: Verify that Green's Theorem is true for S,3(y? – x) dx + (x2 - y)dy given that the curve C is formed…

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Q: Use Green's theorem to evaluate 2y3 dx – x³y2 dy, where C is the positively oriented circle of…

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Q: 1- Find all of the points where the curve r = 2 cos (-) intersects the circle of 3, radius v2…

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Q: The curve C is the circle of radius 5 contained in the plane z 5 and centered around the z-axis,…

A: Solution:-

Q: b. The curve y = cosx ,y 0, x = - and x=

A: Solution Question 26 b Given, The curve y = cosx  We have to find the area of region bounded by the…

Q: c) By using Green's theorem evaluate $ (Ary – y*) de + (* + 3ry) dy where C is a closed curve…

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Q: 22. Find an equation for the circle of curvature of the curve r(r) = (2 In )i- [t + (1/]j, esIse, at…

A: The formula for the curvature is given by  κ(t)=r⇀'(t)×r⇀''(t)r⇀'(t)3 In order to use this formula,…

Q: Verify Green's Theorem where F1=y^2-7y , F2=2xy+2x and the curve C is a circle x^2+y^2=1

A: Given: F1=y2-7y , F2=2xy+2x and C is a circle i.e., x2+y2=1 As, we have:…

Q: Find the length of the curve r(t) = (cos 3t, sin 3t, 2t3/2),0 sts 3. %3D

A:

Q: Prove that the curve y: (-,-)→ R³ is regular with factor H and determine whether the image y (t) =…

A: If γ't not equal to 0 for all t∈-π2,π2then it is regular. If γ''t=kγ't for all t∈-π2,π2 then the…

Q: Use Green's Theorem to evaluate F dr, where F = C consists of the arc of the curve y = læ – x from…

A: Given that       F = &lt; √x + 4y, 3x + √y &gt; C consists of the arc of the curve y = x - x2 from…

Q: ind the equations of the tangent line to the ellipse x² + 4y² + z² = 9, x = 2 at the point (2, -1,…

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Q: Find a parametric description for the circle x2+y2-6y=0, oriented clockwise and where t begins at 0.…

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Q: Use Green's Theorem to evaluate F. dr, where F C consists of the arc of the curve y 1x x2 - from…

A:

Q: 8.) Find the exact length of the curve y= over the interval (0.4]. 2 %3D -

A: Given,          The exact length of the curve y=x22 over the interval 0,4 .

Q: Find the exact length of the curve. y = In(sec(x)), OsxSa/4

A: Length of a curve f(x) in interval [a,b] is given by ∫ab1+(f'(x))2dx

Q: (6) Suppose the closed curve C in R? starts at the point (-1, 1), goes along the parabola y = r? to…

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Q: Show that the tangent line at a point P = (x0, y0) on the hyperbola(x/a)2−(y/b)2= 1 has equation Ax…

A: Given equation of hyperbola  x2a2-y2b2=1 and given point is P=xo,yo. Since hyperbola is passing…

Q: Find the center of the circle(s) tangent to 5x – 2y – 1 = 0 at (1, 2) with a radius of 3. Express…

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Q: 1 4. Let f(=) =- s · Show that J (z)dz = 0 , where C is a simple closed contour not passing…

A: Moreras theorem

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

A: We have to evaluate the integral ∫5xy+x2+y2dx+x2-ydy over the closed curve C. Comparing with…

Q: c) By using Green's theorem evaluate UTM £ (kry – y") dz + (z² +3ry) dy where C is a closed curve…

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Q: B. Find the length of the circle of radius r defined parametrically by x = r(v1 – sin t) and y =…

A: We need to find the length of the circle which is parametrically defined as below.

Q: Find the points on the graph of f(x, y) = 3x 2 - xy - y 2 at which thetangent plane is horizontal.

A: Given that, fx, y=3x2-xy-y2z=3x2-xy-y2

Q: 3. Find a parameterization of the curve C= r(t) = (x(t), y(t)) a. Where C is the line segment…

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Q: Prove that the curve given by the equations x=R(cost+tsint),y=R(sint−tcost) is the involute of the…

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Q: Find the equation of the tangent plane to the sphere x² + y2 + z² = 1 at the point (1/ V3, 1/V3, 1/…

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Q: Use Green's theorem to evaluate $ 2y3 dx – x3y² dy, where C is the positively oriented circle of…

A:

Q: Evaluate |*(x + 7) – 2y ds where C is the circle of radius 1 centered at (-3,0) and lying on the…

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Q: (a) Obtain the equation of the tangent plane and the condition for tangency to the conicoid Ar? +…

A: Conicoid given is- Ax2+By2+cz2=1 at the point (α,β,γ) We have to obtain the equation of the tangent…

Q: Find the points on the curve x2 + xy + y2 = 1 in the xy-plane that are nearest to and farthest from…

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Q: Find parametric equations for the semicircle x2 + y2 = a2, y &gt; 0, using as parameter the slope t…

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Q: In which direction is the curve x = -2 sin t, y = 2 cos t, for 0 ≤ t &lt; 2π, generated?

A: x = -2 sin t, y = 2 cos t Table for values of x and y: t x y 0 0 2 π 0 -2 π2 2 0 2π 0 2

Q: ind an equation for the tangent plane of z = –2x²y – 3x² + y at the point (1, –1, –2).

A:

Q: Find the exact length of the curve. y = In(sec(x)), Osxs- 6

A:

Q: (b) F. dr, C is the line segment from (1, 1) to (5, 1), F = (y, -x). 3 of 7 (c) Set up but do not…

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Q: 99. Determine the concavity of the curve x= 2t + In t, y = 21 – In t. %3D

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Q: The tangent plane to the elliptic paraboloid z = -2x - 7y? at the point (-3, -4) is

A: Consider the provided question, The equation of the tangent plane to the elliptic paraboloid at the…

Q: Find parametric equations for the arc of a circle of radius 3 from P = (0,0) to Q = (6,0). x(t) =…

A: Given :-   A circle of radius 3 from P=(0,0) to Q=(6,0)  To find:-  Find parametric equations for…

Q: The circle of radius 1 centered at (-3, 4,1) and lying in a plane parallel to the yz-axis is a)…

A: Since you have asked multiple questions here, we will solve the first one for you. To get the…

Q: Use Green's Theorem to evaluate / F · dř, where F = (Va + 6y, 3x + y9) C consists of the arc of the…

A:

Q: Find the lengthof the curve r= a sin (0/2), 0s 0S T, a>0

A: The formula for finding arc length is L = ∫0πr2+ drdθ2dθ, where r = a sin2(θ2)

Q: ind the equation of the tangent line to right branch of the hyperbola == sec t, y= tant, 2 - sts at…

A: Topic:- application of derivatives