2 Find the curvature of the parabola given by Y = X = = x ² a+ x=2 Sketch the circle of curvature at (2,1) and find its center and radius

A highway has an exit ramp that begins at the origin of a coordinate system and follows the curve y = (1/ 32)x5/2 to the point (4, 1) (see figure). Then it follows a circular path whose curvature is that given by the curve at (4, 1). What is the radius of the circular arc? Explain why the curve and the circular arc should have the same curvature at (4, 1).

Find the curvature and the radius of curvature of the curve y = x + 5x2 – 2x3 at the origin (0,0).

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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:

Find the locus of the point of intersection of the tangents to the circle *2 + 12 = a2 at points whose parametric angles digger by pi/3.

Find the distance between the pointx1 =−1, y1 =1, z1 =3 and the point x2 =1, y2 =1, z2 =1.

Consider the equation 2.61x² + y² = 1 + 0.71x²y². In this question you will calculate the curvature, p. Fill out the table below (evaluate the derivative at the points described-you should get a decimal number and put that value into the input box). Point 1.62 y x = x = -0.50 y x = 0.16 Y x = 0.50 x = 1.55 Y Y 2.60 p= -0.65 p= p= p= p= 0.97 0.65 2.73 Curvature

Sketch one loop of the curve 9y² = x(3-x)² Find the area of the surface of revolution generated by one loop of the curve 9y2 = x(3x)2 about the y-axis and leave your answer in the exact form.

The equation of a curved road is given as y = 100+6x–0.015x² +0.00001111x? Determine the places where the bend is the sharpest and calculate the radius of curvature at these points. Determine the minimum Coulomb factor between the wheel and the road to prevent the vehicles traveling at 54 km / h from being skidded.