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- How would I write a vector function r(t) that follows the below guidelines2. Calculate the gradient vector Vf of the function f (x, y) = x² – x + y - x²y - 2y2 at the point (2,1) and sketch it on the attached contour plot (you can save the picture, open in photo editor and use drawing tools). Explain in one paragraph (about 200-300 words) the meaning of the gradient vector Vf(2,1), negative gradient vector -Vf(2,1).If r(t) = u(t) x v(t), where u and v are the vector functions given below, find r'(2). u(2) = (6, 3, -6), u'(2) = (4, 0, 2), v(t) = (t, t², t³)
- #3. Consider the function if (x, y)# (0,0), if (x, y)= (0,0). f (x, y)= x2+y2 Calculate the directional derivative of f at the point (0,0) in the direction of the vector (1,3).(22) in the direction of (1,2). Then, Suppose f(x, y) (a) ▼ f(x, y) = = (b) ▼ ƒ(6, π) = = sin (c) ƒu (6, ñ) = Du ƒ(6, ñ) = and u is the unit vectorYou are given the derivative of a vector function r in the component form is -(e,3e",-2t , dt ,3e",-2t). You are also given that r(0) = 21-j+k. r(0) = 2i -j+k J Determine the vector function r (t) in the form r(t)= (x(t),y(t), z(t)} An efficient notation for the vector equation of a straight line in 3D(or 2D) is given by ((t)- a+ bt where t is any real number, a is the position vector from the origin to a point on the line and b b) Write the vector equation of the tangent line to the curve C generated by r(t) at the point (2, -1, 1) using the above form a
- Suppose f(x, y) = √/tan(x) + y and u is the unit vector in the direction of (1, 1). Then, (a) ▼ f(x, y) = (b) ▼ ƒ(−0.8, 10) = (c) fu (−0.8, 10) = Du f(−0.8, 10) =Find r'(t), r(t), and r'(to) for the given value of to. r(t) = (et, e²t), to = 0 r'(t) = r(t) = r'(to) = Sketch the curve represented by the vector-valued function, and sketch the vectors r(t) and r'(to). y 3₁ -1 y 3 1 -1 r (1, 1) 1 r' 2 3 X -1 2 -1 (0, 1) r 1 r' 2 3 X -1 y 3 2 1 r (1, 0) 1 r' 2 3 X -1 y 3 2 1 -1 r (1, 1) 1 P¹ 2 3 XWhat is the velocity vector?
- True or False: Given a function of two variables f(x, y), the vector Vf(a, b) is perpendicular to the graph z = f(x, y) at the point with (x, y) = (a, b). True FalseFind r'(t), r(t), and r'(to) for the given value of to. r(t) = (et, e²t), to = 0 r'(t) = r(t) = r'(to) = Sketch the curve represented by the vector-valued function, and sketch the vectors r(to) and r'(to). y 3r O -1 eBook 2 (0, 1) r 1 r' 2 X 3 -1 لانا y 3 2 1 r (1, 0) 1 2 X 3 -1 y 2 1 r (1, 1) 1 2 X 3 -1 y 3 2 1 -1 (1, 1) 1 2 3 XThe position vector r describes the path of an object moving in the xy-plane. Position Vector Point r(t) = ti + (-t2 + 8)j (1, 7) (a) Find the velocity vector v(t), speed s(t), and acceleration vector a(t) of the object. v(t) s(t) a(t) (b) Evaluate the velocity vector and acceleration vector of the object at the given point. v(1) = a(1)