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- The semi-hyperbola H(X)=(sqrt(x^2-1)) is continous at which x interval?Show that the path given by r(t) = (cos t,cos(2t), sint) intersects the xy-plane infinitely many times, but the underlying space curve intersects the xy-plane only twice.A normal is drawn to the hyperbola x^2/a^2+y^2/b^2=1 meets the transverse axis at G. If perpendicular from G onthe asymptote meets it at L, then show that LP is parallel toconjugate axis.
- The paraboloid z = x2 + y2 - 4x + 2y + 5 has a local minimum at (2, -1). Verify the conclusion of as shown for this function.Give a parametric representation for the cylinder x^2+z^2=121Find the appropriate parametrization for the given piecewise smooth-curvein R^2, with the implied orientation. The curve C, which goes along the circle of radius 3, from the point(3,0) above the x-axis to the point (-3,0), and then in a straight line along the x-axis back to (3,0).
- Consider the right triangle withvertices (0, 0), (0, b), and (a, 0), where a > 0 and b > 0. Showthat the average vertical distance from points on the x-axis to thehypotenuse is b/2, for all a > 0.Is z=0 is not singularity ?Show that the tangent line at a point P = (x0, y0) on the hyperbola(x/a)2−(y/b)2= 1 has equation Ax − By = 1 where A = x0 a2 and B = y0b2 .