Consider the curve r (t) = (sin³ (t), cos³ (t), sin² (t)) for 0 ≤ t≤ π/2. Compute the following quantities at the point t = π/4. The unit normal vector N (4) = (−, −, −). (a component in first answer box, y component in second answer box, z component in third) The unit binormal vector B (4) = (−, —, —). (a component in first answer box, y component in second answer box, z component in third) The curvature (1)

The position vector r describes the path of an object moving in the xy-plane. Position Vector Point r(t) = 2 cos ti + 2 sin tj (VZ, V2) (a) Find the velocity vector, speed, and acceleration vector of the object. v(t) = s(t) a(t) = (b) Evaluate the velocity vector and acceleration vector of the object at the given point. a(#) =

Find Components of the Acceleration For the curve defined by 7(t) = (e-t cos(t), eʻ sin(t)) find the unit tangent vector, unit normal vector, normal acceleration, and tangential acceleration at t = 3 T(플) - N(플) - 3 3 AN =

Subtracting the two equations, find a vector equation for the curve of intersection between y= 4x2+(3/4)z2 and y-1= 3x2+(1/2)z2 for x>0. Find and simplify the tangential component of acceleration for your curve.

Please solve 2b)

Find a normal vector at point P. xy2 - 4x2 + y + 8x = 7, P(4, 3) Find a tangent vector at point P.

Let r(t) = (-3t5 - 1, 2e³, 4 sin(4t)) Find the unit tangent vector T(t) at the point t = 0 T(0) -

TRANSCRIBE THE FOLLOWING TEXT IN DIGITAL FORMAT

2) Consider the vector function r(t) = (sin(t), −3t, cos (t)). Analyze the function, then sketch a graph.

(1n |t – 1], e', vî ) 1. Let 7(t) = (a) Express the vector valued function in parametric form. (b) Find the domain of the function. (c) Find the first derivative of the function. (d) Find T(2). (e) Find the vector equation of the tangent line to the curve when t=2. 2. Complete all parts: (a) Find the equation of the curve of intersection of the surfaces y = x? and z = x3 (b) What is the name of the resulting curve of intersection? (c) Find the equation for B the unit binormal vector to the curve when t= 1. Hint: Instead of using the usual formula for B note that the unit binormal vector is orthogonal to 7 '(t) and 7"(t). In fact, an alternate formula for this vector is ア'(t) × ア"(t) ア(t) ×デ"(t)| B(t) =