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math1a - quiz 3 - solns

# math1a - quiz 3 - solns - Math 1A Quiz 3 February 15th 2008...

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Math 1A Quiz 3 February 15th, 2008 Name SID 1. (Version 1:) Find a degree-3 polynomial p ( x ) such that the graph of p ( x ) has a horizontal tangent line at the point (0 , 0) and a tangent line with slope 1 at the point (1 , 3). (Version 2:) Find a degree-3 polynomial p ( x ) such that the graph of p ( x ) has a horizontal tangent line at the point (0 , 0) and a tangent line with slope 1 at the point (2 , 3). Solution to Version 2: Let p ( x ) = Ax 3 + Bx 2 + Cx + D . We want to figure out what A, B, C, and D should be in order to have p satisfy the given conditions. Note that p ( x ) = 3 Ax 2 + 2 Bx + C . The given conditions tell us four things: the graph of p passes through the points (0 , 0) and (2 , 3) – so p (0) = 0 and p (2) = 3 – and the slope of the tangent to the graph of p is 0 for x = 0 and 1 for x = 2, i.e. p (0) = 0 and p (2) = 1. Each of these conditions gives us an equation involving A, B, C, and D: 0 = p (0) = 0 A + 0 B + 0 C + D = D 0 = p (0) = 0 · 3 A + 0 · 2 B + C = C 3 = p (2) = 8 A + 4 B + 2 C + D 1 = p (2) = 4 · 3 A + 2 · 2 B + C Using the first two equations (i.e. C = 0 = D ), we reduce the second two equations to 3 = 8

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math1a - quiz 3 - solns - Math 1A Quiz 3 February 15th 2008...

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