1. Show that the curve a(t) = (t,t² + 1, t – 1) is regular.

Q: a. Show that n(t) = -g'(t)i + f(t)j and - n(t)=g'(t)i-f'(t)j are both normal to the curve r(t) =…

A: Using n(t) = - g'(t)i + f(t)j and - n(t) = g'(t)i – f'(t)j are both normal to the curve r(t) =…

Q: Let C be the closed curve indicated in the diagram. Use Green's Theorem (2,1) (o,i) (x² + y?, y³ +…

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Q: Prove the dot product rule. If C₁ (t) and C₂ (t) are differentiable curves in R" then: d [c₁ (t) ·…

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Q: 4. Find the length of the curve 7(t) = (t2,,t³,t) for 0st<4

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Q: Sketch the plane curve. r(t) = 2ti – tj, [0, 2] y y 2. 1 1 -1 1 4 -1 1 4 5 -1 y -1 2 3 4 1 2 3 4 5…

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Q: r(t) = (t2 - 1, t2 + 1, t + 1). Find the point where the line tangent to the curve at the point (3,…

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Q: A curve is given in the parametric form x = [(2x - 1)/ (t + 1)], y = [t / (t+ 1)], z = t + 2. Find…

A: According to the given information, it is required to determine the torsion of the given curve.

Q: r(t)=(-6–2t)i+tj– 2tk, 0<ts Evaluate where C is the curve given by C

A: Solve the following

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

Q: Consider the curve r (t) = (t, tº, tº) ; t > 0 Does the point (1, 1, 1) belong to the curve and if…

A: Please refer the attached image for complete solution.

Q: Find T, N, and к for the plane curve r(t) = 9t i + 9 In (cos t) j, - T(t)=(☐)i+(☐)j N(t) = ( ☐ ) i +…

A: Step 1:

Q: 1. Parametrize the curve-segment C(t) of the parabola y = x² from (1, 1) to (2,4) then find [x² dx +…

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Q: 1. Let F(x, y) = (1 — y, ½x³ +e³²), and let C with the position vector r be the portion of the…

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Q: Give the length of a curve given parametrically by x = x(t), y = y(t), and z = z(t) for a ≤ t ≤ b in…

A: given parametrically by x = x(t), y = y(t), and z = z(t) for a ≤ t ≤ b in the vector form.

Q: ) Find the length of the curve determined by 7 (t) = 2t7 + V3t27 + t

A: To find the length of the curve

Q: Let C2 be the curve defined by R(t) = (t² +t + 3, 4t – 5, 2+ t – t2), t e [0, 1]. %3D | (x + y+ 2)…

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Q: Compute the length of the curve r(t) = (-4t, 5t — 3, -2t + 3) over the interval 0 ≤ t ≤9 L =

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Q: Q2) Find the vectors „N of the curve r(1) = ) at the point (1.1)

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Q: Let : [0, 1] CR→ C be a parameterized curve. What does represent in each case? (a) (t) = i + it (b)…

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Q: If L is the tangent line to the curve 7 (t)= (2t-1, t, t-2) at (3, 4, 2) and the point where L…

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