QUESTION 1

. . . . . . . . d . . . . . a. When usmg lmpllClt differentiation, in the formula for i it is always necessary to eliminate y to get a formula purely in terms x. v b. Use implicit differentiation to find the xcoordinate of the point(s) on the curve: 3 x2+yz=lxy with horizontal tangent lines. If there is more than one answer, separate your answers by commas. x = c. When performing implicit differentiation, we don't need to worry about the possibility of having a denominator equal to 0. V Question 1 should be completed in Web Work by 11:59PM, Monday 3 October. Web Work is ac- cessed via Assignment 7 Web Work in the Canvas Assignments section. You should upload a scan of your solutions to Questions 2 and 3 to Gradescope, accessed via Assignment 7 Written Part in the Canvas Assignments section. There is no need to include your answers to Question 1 in the solutions for the Web Work part. 1. [4 marks] See Web Work. 2. In Questions 2 and 3 we consider the the path P of the parametric curve r : R - R2 defined by r(t) = 2 cos(t) sin?(t)i + sin(t)j so P = range(r) and the set C = {(x, y) ER? | x2 + 496 = 4y4}. In Assignment 1, you proved that P C C. Here we use the approach of Question 2 of Practice class 9 (which you should review) to prove that C C P, completing the proof that P = C. (a) Find the x intercept of C (there is only one) and explain why it is in P.(b) Assuming y y 0, rearrange the equation for C into the form 1L2 | y2 = 1 where u is expressed in terms of a: and y. You should give a formula for u. (c) In (b), you showed that (u, y) is in the unit circle. Use this to complete the proof that C Q P. 3. (a) Plot the vectors r'(0),r'(), r'(1r) and t?) in the diagram below. In each case, base the vector at the point t). Example 3.42 in lectures illustrates this. (b) The curve C has vertical tangent lines at i \\/E i 1/? _i 2 _i _ 2 (3V6! 3)?(3V11 3 1 3V6? 3 1 Iii/El 3 ' Using this information and the vectors you plotted in (a) to sketch P = C in the diagram below. Indicate the vertical tangent points clearly