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 F(t) = < -1² + 5, t4 + t, In a. Determine the coordinates of the point on the curve traced out…

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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: Find F-dX where C is the curve X(r) = r°I+( 2r+ 1J, 0<I55, and Fxy) = rI+yJ.

A: 9

Q: x-1 y-2 z-1 2 -6 7 1 F(t) = ( 21 + ¹ )i + ¹/ (m² − 3)} j + t√21³-1k at t=1 1 4 (c) show that the…

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Q: Let C be the parametric curve defined by æ(t) = t? – 2, | y(t) = t3 – 3t + 1. Determine if C is…

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Q: Let C be the closed curve oriented in the counterclockwise direction, consisting of the line segment…

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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: Let C be the curve defined by r(t) [6 cos(t) – 5 cos (t), 6 sin(t) – 5 cos(t) sin(t), cos(t)] for 0…

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Q: Evaluate S. (2xy – x²)dx + (x + y²) dy such that c is a curve bounded by y? – x , y – x² ?

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Q: Let C be the parametric curve defined by x(t) = t² – 2, y(t) = t³ – 3t + 1. Determine if C is…

A: Given that,A parametric curve C is defined byx(t)=t2-2y(t)=t3-3t+1The concavity of the parametric…

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: Let (t)=(-5t2+4,t³-+4,2+³) 7 Find the line (L) tangent to (t) at the point (-1,0,2) L: (x,y,₁2)= +t

A: The given problem is to find the line tangent to the given parametric curve at the given point…

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: 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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Q: Let C be the parametric curve defined by                         x(t) = t2 - 2;…

A: The given curve C is xt=t2-2yt=t3-3t+1. We need to find that the C is concave up or concave down at…