Problem 1. (14 Points) Let y : R R be given by Y(t) = (3t – t°, 3t², 3t + t³). %3D (a) Find the velocity, the speed, and the acceleration of y. Show that y is a regular space curve. (b) Compute the unit tangent T. (c) Compute the vector y x ". (d) Compute the binormal B. (e) Compute the curvature k and the torsion T of y. Show that the curvature K and the torsion T of the curve y coincide: K(t) = T(t) for all t e R.

Q: 2. C. b. Let (t)= and (u) = At what point P do these two space curves intersect. a. Find the tangent…

A: As per rule I need to solve only first three subparts.

Q: (9.) r(t) = (1, cos t, 2 sin t)

A: As per your query the solution of question no 9 is given the next step.

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A: It is a basic questions of 3-D geometry , we have equate the equation only

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A: We have to find the equation of tangent line.

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A: Length and direction of vectors

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A: Solution by using formulaes :

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A: As asked we shall solve only part (d) and (e).We have to eliminate the variable t and write the…

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Q: 4. Find the unit tangent vector T(t) for the vector function: r(t) = .

A: To find the unit tangent vector.

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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.