#37

please show all work

#37 please explain all work a-e

In Exercises 31-38 you will use a CAS to explore the osculating circle at a point P on a plane curve where =0. Use a CAS to perform the following steps: a. Plot the plane curve given in parametric or function form over the specified interval to see what it looks like. b. Calculate the curvature K of the curve at the given value t0 using the appropriate formula from Exereise 5 or 6 . Use the parametrization x=t and y=f(t) if the curve is given as a function y=f(x). c. Find the unit normal vector N at t0. Notice that the signs of the components of N depend on whether the unit tangent veetor T is turning clockwise or counterelockwise at t=4 (Sec Exercise 7.) d. If C=ai+bj is the vector from the origin to the center (a,b) of the osculating circle, find the center C from the vector equation C=r(t0)+k(t0)1N(t0). The point P(x0,y5) on the curve is given by the position vector r(b0). e. Plot implicitly the equation (xa)2+(yb)2=1/2 of the osculating circle. Then plot the curve and osculating circle together. You may need to experiment with the size of the viewing window, but be sure the axes are equally sealed. 37. y=x2x1,2x5,x0=1 38. y=x(1x)23,1x2,x0=1/2 COMPUTER EXPLORATIONS In Exercises 31-38 you will use a CAS to explore the osculating circle at a point P on a plane curve where =0. Use a CAS to perform the following steps: a. Plot the plane curve given in parametric or function form over the specified interval to see what it looks like. b. Calculate the curvature of the curve at the given value t0 using the appropriate formula from Exercise 5 or 6 . Use the parametrization x=t and y=f(t) if the curve is given as a function y=f(x). c. Find the unit normal vector N at t0. Notice that the signs of the components of N depend on whether the unit tangent vector T is turning clockwise or counterclockwise at t=t0. (See Exercise 7.) d. If C=ai+bj is the vector from the origin to the center (a,b) of the osculating circle, find the center C from the vector equation C=r(t0)+(t0)1N(t0). The point P(x0,y0) on the curve is given by the position vector r(t0). e. Plot implicitly the equation (xa)2+(yb)2=1/K2 of the osculating circle. Then plot the curve and osculating circle together. You may need to experiment with the size of the viewing window, but be sure the axes are equally scaled. 37. y=x2x,2x5,x0=1 38. y=x(1x)2/5,1x2,x0=1/2