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ASSIGNMENT 9 for SECTION 001 This assignment is to be handed in. There are two parts: Part A and Part B. Part A will be graded for completeness. You will receive full marks only if every question has been completed. Part B will be graded for correctness. You will receive full marks on a question only if your answer is correct and your reasoning is clear. In both parts, you must show your work. Please submit Part A and Part B separately, with your name on each part. Part A From Calculus: Early Transcendentals: From section 3.5, complete questions: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32(a) and 32(b), 34, 36, 40, 60, 62 Part B 1. Consider the curve x2 + y 2 3 = 8x2 y 2 . Find the equation of the tangent line at the point (−1, 1). 2. Let P be a point on a curve, and l be the tangent line at P . The normal line at P is the line perpendicular to l and passing through P . Consider the curve x2 + (y − x)3 = 9. Find the equation of the normal line at x = 1. y 3. The curve x2 − xy + y 2 = 3 describes a tilted ellipse, pictured to the right. Find the area of the smallest box (with sides parallel to the axes) containing this ellipse. x 4. A (non-tilted) ellipse centred on the origin is described by the equation x2 y2 + 2 = 1. 2 a b (2a and 2b are the “width” and “height” of the ellipse.) Prove that if the normal line at every point on the ellipse passes through the origin, then the ellipse is a circle. 5. Describe, in the form of a haiku, how calculus has changed your life.