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ASSIGNMENT 1·10 There are two parts to this assignment. The first part is on WeBWorK — the link is available on the course webpage. The second part consists of the questions on this page. You are expected to provide full solutions with complete arguments and justifications. You will be graded on the correctness, clarity and elegance of your solutions. Your answers must be typeset or very neatly written. They must be stapled, with your name and student number at the top of each page. 1. Find the equations of all lines that are tangent to the curve y = 1−x and pass through the point (−1, 0). 1+x 2. A common mistake is to assume that if f and g are differentiable, then d d d (f (x)g(x)) = f (x) g(x). dx dx dx Write down two functions f and g that do satisfy the equality above. 3. Pascal’s triangle is a triangular array of numbers with many deep mathematical connections. The first eleven rows are shown below. It is constructed as follows: every row begins and ends with the entry “1”; every other entry is equal to the sum of the two entries immediately above it. The top row is called “row 0”, the next row is called “row 1”, and so on. Let f and g be infinitely differentiable. (“Infinitely differentiable” simply means that you can differentiate f and g as many times as you wish.) Let Fk and Gk denote the k th derivatives of f (x) and g(x), respectively. For example, F0 = f (x), F1 = f 0 (x), F2 = f 00 (x), and F3 = f 000 (x). Prove, using mathematical induction, that the nth derivative of f (x)g(x) is equal to an Fn G0 + an−1 Fn−1 G1 + an−2 Fn−2 G2 + · · · + a2 F2 Gn−2 + a1 F1 Gn−1 + a0 F0 Gn where the coefficients an , an−1 , an−2 , . . . , a2 , a1 , a0 are the numbers in row n of Pascal’s triangle.