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- 4. ) Find the directional derivative of the function f(x, y) = sin²(xy) at the point P (2,7) in the direction of the unit vector ū = −²¹+] √5Let f(w)f(w)be a function of vector ww, i.e. f(w)=1/(1+e−wTx). Determine the first derivative and matrix of second derivatives of ffwith respect to w ?Define two vector functions 7(t) = 8 sin(t)i + 6 cos(t)+ (t - 2)k ü(t) = 6 sin(t)i + 8 cos(t)j + (t – 2)k Compute 7 (t) ü(t)
- f(x, 3) In(x2 + Yof function P(3, 4) at the point of = (9, 12) Directional derivative in the direction of the vector to you.Find r’(t), r(t), and r’(t) for the given value of tå. r' (t) r(to) r'(to) = = = = r(t) = 3 cos(t)i + 3 sin(t)j, to T 2 Sketch the curve represented by the vector-valued function and sketch the vectors r(t) and r'(t). = r(t) starts at (x, y) = (0, 0) and ends at (x, y) = r'(to) starts at the terminal point of r(t) at (x, y) = and ends at (x, y) =Define two vector functions F(t) = 2 sin(t)i + 8 cos(t)j + t°k ü(t) = 8 sin(t)i + 2 cos(t)j + (t² – 2)k Compute 7 (t) · ü(t) = ||
- Sketch the curve with the vector equation r(t) = cos(t)i − cos(t)j + sin(t)k. Show the direction of increasing t with an arrow drawn on your curve.Find a equation vector and the equation of the tangent line at the point P0 where t=0.2 on the graph of the vector function r(t)=e2ti+(t2-t)j+(ln(t))k2.31 Let A and B be vector functions of position vector x with continuous first and second derivatives, and let F and G be scalar functions of position x with continuous first and second derivatives. Show that: (a) V. (V x A) = 0. (b) ▼x (VF) = 0. (c) V (VF x VG) = 0. (d) V. (FA) = A·VF+ FV.A.