Use Green's Theorem to evaluate the line integral of F = (x5,2x) around the boundary of the parallelogram in the following figure (note the orientation). (xo.Yo) (2x-Yo) With o= 5 and yo= 5. (X0.0) Sex dx + 2x dy = x

4) a) Show that the equation y = x3 + 3x2 + 16 has exactly one real root. What is it?

Determine the area of the triangle formed by the tangent to the graph of the function y=3−x2 drawn at the point (1,2) and the coordinate axes (Figure 11).

Evaluate This Integral if curve C consists of curve C₁ which is a parabola y=x² from point (0,0) to point (2,4) and curve C₂ which is a vertical line segment from point (2,4) to point (2,6) if a and b are each constant.

find the tangent line to the parabola (x + 2y)^2+ 2x – y – 3 = 0, parallel to the linex – 4y = 2.

Find the tangent plane to the equation z = z = 3x² - y² - 3y at the point (4, −3, 48)

Find the equation of the tangent line to the curve of intersection of the surface z = x² - y² with the plane x = 9 at the point (9, 1,80). (Express numbers in exact form. Use symbolic notation and fractions where needed.) equation: