(20 points) Evaluate the following line integral along the positively oriented curve C consisting of the line segment from (0,0) to (4,0), followed by the arc of the circle x² + y² = = 4x from (4,0) to (2, 2), and the line segment from (2, 2) back to (0, 0): § (cos(2x + 1) − yx²) dx + (xy² + sin(xy)) dy .

Let Q = 4x2 + 6xy+ 4y? . The points on the curve Q = 7 which are closest to the origin are: %3D %3D OA(-눈-)and () (주·주) pue OB. (-V7, V7) and (v7, -V7) Oc. (0, ±1) OD. (tV7,0)

Find the length of the arc of the curve x2=(y-2)3 from P(1,3) to Q(8,6).

Find the tangent plane to the equation 2y cos(5z – 4y) at the point (4,5,10)

Find a parametric description for the curve x=y^2-9 from (0, 3) to (-5, -2) such that t=0 corresponds to (0, 3).

a) Plot the curve x^3 - 4xy + 2y^3 = 0 as x varies between -3 and 3 with the aid of theimplicitplot command.b) Find dy/dx at all points (x,y) on the curve for which x = 1.c) Find the equations of the lines tangent to the curve at the points you found in part b.d) Plot the curve and the tangent lines on the same set of axes.