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- Determine the interval (s) on which the vector-valued function r(t)=ti+(sqrtt+1j)+(t^2+1)k continous A [-infinity, -1) B [-infinity, infinity) C (-1, infinity) D [-1, infinity)consider the vector-valued function r(t) = <2t3 – 15t2 + 36t, ln(4t – t^2)> a) State the domain of r(t) b) Identify where r(t) is not smooth over its domain. c) Identify any places where r(t) has vertical tangent lines. d) Identify any places where r(t) has horizontal tangent lines.Evaluate along the curve y=x2 from (-1,1) to (2,4). First find the vector valued function r(t) defining the curve.
- Find the domain of the vector-valued functions. a). Domain: r(t)=⟨t^2,tan t,ln t⟩ b). Domain: r(t)=⟨csc(t),(1)/(\sqrt{t-3 }),ln(t−2)⟩Find the domain of the vector function r(t) = } Domain: {tFind the directional derivative of the function f(z, y) = In(z* + y*) at the point (3, – 1) in the direction of the vector (- 2, 1)3. Find the directional derivative of f(x, y, z) =x*y-yz' +z at the point (1, -2,0) in the direction of the vector v =2i +j- 2k.Find the directional derivative of f(x, y, z) Prev x'y at the point (-5, 1, 1) in the direction of the vector v = (1, -3, 4).5. Let g(u) be a differentiable function, and let f(x, y) = g(x2 + y²). (a) Show that yfa = x fy. (b) Find the direction of maximal increase of f at the point (1,1) in terms of g'. (You do not need to provide this as a unit vector.)= xe" + ye² + ze* at the point Find the directional derivative of the function f(x, y, z) (1, 1, 1) in the direction of the vector v = 8i – 6k.10) Find a unit vector in the direction that f (x, y, z)= 2x² +6xy² – xz' increases most rapidly at the point (-3,–4,4) and find the rate of increase in that direction. |