2. (Based on problem 32. Section 2.6 from Stewart's 2nd Edition of Essential Calculus} The following curve is sometimes called a bouncing wagon. 2y3+y2y5=:r42;r3+:c2 {a} Make a plot of this equation for a: between -1 and 2 and y between -1.5 and 1.75. (b) Does the name of the curve seem appropriate? Is this curve a function? Explain. (c) Use your plot to estimate the points where the vertical and horizontal tangents are and then use the implicit derivative to nd these points analytically with the help of Mathematics. If strange results are found remember all techniques of solving, try the command NSolve or after the output for Solve on the next line type NW0] {see examples above). Also, it may help to plot the graph of the piece you are setting equal to 0 and then zoom in to nd intervals of I-values or y-values that the root lies between and use the command FindRoot. Make sure to state all of the point(s) for the vertical and horizontal tangents. CALCULUS 1 FALL 2020 (d) Is there only one tangent line at x = 1.5? Are there more than one? Find the slopes and the equations of the tangent line(s] when a: = 1.5. (Hint, substitute a: = 1.5 into the implicit function and solve for y as done in the discussion above). (e) Plot the implicit function along with its tangent line(s) at :1: = 1.5 all on the same plot. CALCULUS I FALL 2020 EXERCISES 1. For the following lemniscates: 2(:::2 + y2}2 = 3x2 3y2 2(3:2 + y2}2 332 3y2 (a) Make 3 labeled plots; one plot for each individual implicit function and a joint plot including both implicit functions. (1)} Use your plot to estimate the points where the vertical and horizontal tangents are and then use the implicit derivative to nd these points analytically with the help of Mathematica. If strange results are found remember all techniques of solving, try the command NSolve or after the output for Solve on the next line type N[%] (see examples above). Also, it may help to plot the graph of the piece you are setting equal to 0 and then zoom in to nd intervals of x-values or y-values that the root lies between and use the command FindRoot. Make sure to state all of the point(s) for the vertical and horizontal tangents. (c) How are the lemniscates related to one another? How are the points where vertical and horizontal tangents are located of the two lemniscates related to one another