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- Exercise III Let (a) o = x²y +xż and (b) ó = x² + y² + z?. Then, respectively attempt to find the directional derivative at • (1,2, 1) in the direction of the vector (2i - 3j+ 4k). • (3,0, 1) in the direction of the vector (i- 3j + 2k).Vector 1 A) Prove that: V x (FxG) = (GV)F- (FV)G+ F(VG) - G(V.F). B) Find all of the second derivatives for f(x. y) = (3xy² + 2xy + x²) In: +y² Brie differente directia derivatives with stable gnh an eations. Find the direcional avati 220. vector in t direction of ere E) Evaluat the trde integral I need answer Only branch A (a) dx dy= (b) dyd = 24 GEL direct رسالت +vana is the unitMake a vector field plot of the differential equation. Find any equilibria of the differential equation and use your vector field plot to classify whether each equilibrium is stable or unstable. dN = 3N In dt N>0 Choose the correct vector field plot below. O B. Oc. OD. AdN/dt AdN/dt AdN/dt AdN/dt -8- -8-
- QI-A) Fund the position, vekocity and the speed of the vector function ( r(t) ) that given bekow at t-2. r(t) = 1i-t'iWhat is the graph of r(t)=2 costi + sintj? Calculus 3 Vector valued functions and space curves.(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) =
- QI :(A) What is the angle between the normal vectors to the curve (* + 1)j + 1³k at the points (1,1,0) and (4,2,1) 7 (1) = (21² +7 +A charged particle begins at rest at the origin. Suddenly, a force causes the particleto accelerate according to the vector function a(t) = ⟨ sin(t) , 6t , 2cos(t)⟩Find functions for the velocity, speed and position of the particle at time t(8) Subtracting the two equations, find a vector equation for the curve of 3 intersection between y= 4x² +=' and y-1=3x += for x > 0. Find 4 1 and simplify the tangential component of acceleration for your curve. 3 2 cos (2t) + cos 2t a sin? t- sin? (2r) –4 cos?t 3 2sin? (21) - sin 21 Vsin' - sin? t- sin? (2t) –4cos?t 2sin (21)- sin 21 3 sin? t + sin? (21) +4cos?t 3 2 sin (21) +sin 21 /sin?t + sin? (21) + 4 cos²t
- I need some help with verifying that these differential equations are vector spaces.Solve the initial value problem for r as a vector function of t. dr = - 40k dt? Differential equation: dr = 5i + 5j Initial conditions: r(0) = 10k and dt t=0 r(t) = (Di+ (Dj+ (DkThe position vector r describes the path of an object moving in the xy-plane. Position Vector Point r(t) = t'i + tj (4, 2) (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. v(2) = a(2) =