Updated picture. Please provide explanations for 29, 31, 41

152 Chapter 3 Derivatives Second Derivatives In Exercises 19-26, use implicit differentiation to find dy / dx and then day/ dx2. Write the solutions in terms of x and y only. 19. x2 + y = 1 20. x2/3 + 2/3 = 1 21. y2 = x2 + 2x 22. y2 - 2x = 1 - 2y 23. 2Vy = x -y 24. xy + yz = 1 25. 3 + sin y = y - x3 26. sin y = x cos y - 2 27. If x3 + y3 = 16, find the value of day / dx2 at the point (2, 2). 28. If xy + y? = 1, find the value of day / dx2 at the point (0, -1). In Exercises 29 and 30, find the slope of the curve at the given points. 29. y2 + x2 = y4 - 2x at (-2, 1) and (-2, -1) 30. (x2 + y2)2 = (x - y)2 at (1, 0) and (1, -1) Slopes, Tangents, and Normals In Exercises 31-40, verify that the given point is on the curve and find the lines that are (a) tangent and (b) normal to the curve at the given point. 31. x2 + xy - y = 1, '(2, 3) 32. x2 + y2 = 25, (3, -4) 33. x2y2 = 9, (-1, 3) 34. y2 - 2x - 4y - 1 = 0, (-2, 1) 35. 6x2 + 3xy + 2y2 + 17y - 6 = 0, (-1, 0) 36. x2 - V3xy + 2y2 = 5, (V3, 2) 37. 2xy + 7 sin y = 217, (1, TT /2) 38. x sin 2y = y cos 2x, (TT / 4, TT / 2) 39. y = 2 sin (7x - y), (1, 0) 40. x2 cos2 y - sin y = 0, (0, TT) 41. Parallel tangents Find the two points where the curve x2 + xy + y = 7 crosses the x-axis, and show that the tangents to the curve at these points are parallel. What is the common slope of these tangents? 42. Normals parallel to a line Find the normals to the curve xy + 2x - y = 0 that are parallel to the line 2x + y = 0. 43. The eight curve Find the slopes of the curve y* = y - x at the two points shown here