Let D be the triangle in the xy plane with vertices at (-2, 2), (2, 0), and (5, 5). Describe the boundary aD as a piecewise smooth curve, oriented counterclockwise. (Use t as a parameter. Begin the curve at point (-2, 2).) tE [0, 1) f(t) : tE [1, 2) tE [2, 3)

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Q: Let z = g(x, y) f2(3, 6) = 5. f(3 cos(ry), y + e=y) provided that f(3,6) = 3, f1(3,6) = 2, i) Find…

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Q: Let C be a smooth curve traced by Ř(t) with R(1) = (2, –1, 1) and R' (1) = (1, –2, –1). Let g(t)…

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Q: Sketch the plane curve. r(t) = 4ti – tj, [0, 4] %3! y y 4 4 2 X 10 15 10 15 -2 -2 y y 4 4 2 10 15 10…

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Q: Find T, N, and k for the space curve r(t) = - 2e cos t)i- (2e sin t) j+ 2k.

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Q: Evaluate | F. dR, where F(x, y, z) = (x, –2, y) and C is the curve Ř(t) = (2t, 3t, –t²), 0 <t < 2.

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Q: Let P be the tangent plane to the surface z=x2+2y2 at the point Q=(-1,-1,3). Which of the following…

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Q: A possible way for the parameterization of the curve C corresponds to: (9t – t² A) r(t) = , t, z(t)…

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Q: Which of the following is the equation of the tangent plane to the surface z = 3x2 +x-y²+y at the…

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Q: F = (x² – y²,x, 2yz); C is the boundary of the plane z = 6 – 2x – y in the first octant.

A: Given, F=x2-y2,x,2yz Find the line integral ∮CF·dr in the following. Here C is the boundary of the…

Q: Sketch the plane curve r(t) = t3i + t2j and find its length over the given interval [0, 1] .

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Q: Let P be the tangent plane to the surface z=2x2+y2 at the point Q=(-1,-1,3). Which of the following…

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Q: Find the equation of the tangent plane to *+ y? at the point (2, –3, 10) the surface z = 4 O x + 4y…

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Q: Find T(t), N(t), aT , and aN at the given time t for the plane curve r(t) r(t)=(t-t3)i+2t2j, t=1

A: The solution is given as follows

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Q: Let C be the curve in space given by the following parameterization. r(t) = tỉ + t²j + 9k, 0 < t s…

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Q: Find an equation of the tangent plane to the surface z = 5y – 4x? at the point (4, 2, -44).

A: Given: Equation of surface is z=5y2-4x2 and a point -4, 2, -44. To find: Equation of tangent plane…

Q: Sketch the plane curve. r(t) = 4ti - tj, [0, 4] y y 4 4 2 10 15 10 15 o-4 o-4 y y 4 X 5 10 15 10 15…

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Q: Let z=f(x,y) = x²+y³. Find the slope of the line tangent to this surface at the point (-1,1) and…

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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: The value of [F. dī where F= x?y² i + y j and C is the curve y? = 4x in the XY-plane from (0, 0) to…

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Q: Find an equation of the plane tangent to the surface z = x2 + y2 at (2, 3, 13). [Hint: Consider the…

A: Note: Here, f(x,y,z) = x2 + y2 - z and the point is (2, 3, 13).

Q: Given the curve y = x´ and the lines y = 0 and y = 2 – x wwww Find the ordinate of the centroid Y חו…

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Q: Find the length of the curve r(t) = (√2t)i + (√2t)j + (1 - t2)k from (0, 0, 1) to (√2, √2, 0)

A: given, r(t)=(2t)i +2tj+1-t2kwe have to find the length of the  given curve from (0,0,1) to 2,2,0

Q: Let Ci be the rectangle with vertices (0,0), (2,0), (0,1), and (2,1) oriented counterclockwise, and…

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Q: Sketch the space curve r(t) = ⟨4t, −cos t, sin t⟩ and find its length over the given interval [0,…

A: Consider the given space curve, rt=4t,-cost,sint use the graphing utility to graph the curve,

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Q: 12. Let T be the line tangent to the space curve represented by r(t) = (t³ + e=t )i+ (ln(t) + 2t )j+…

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Q: Find the equation w=T(x,y,z) of the tangent hyperplane at p. F(x,y,z)=3x²-2y²+xz², p=(1,2,-1)

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Q: Use Stoke’s theorem to evaluate ꬹϲ F.dr, if F(x,y,z)=(-xy,)I –(xz)j+(2yz)k and C is the closed…

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Q: Let C be the boundary curve counterclockwise of the semidisk æ? + y? 0. Then fc(-2ydx + 3xdy) 5dydx…

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Q: Let F(x. y. z) = (3x y: - 3y) i (x'z - 3x) j + (x'y+ 2:) k Evaluate , F dr. where C is the curve…

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Q: Sketch the plane curve. r(t) = 2ti-tj, [0, 2] y -1 y 2 1 Find its length over the given interval. 2…

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Q: Let P be the tangent plane to the surface z=x2+2y2 at the point Q=(-1,-1,3). Which of the following…

A: Which of the following points also lies in P

Q: Sketch the plane curve r(t) = t2i + 2tk, and find its length over the given interval [0, 3]

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