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1. Find the Taylor polynomial of degree 3 about x = 0 for the function f (x) = x + e−x . If 0 ≤ x ≤ 1 what is an error bound in using T3 (x) to approximate f (x)? − 2. Find the curvature of → r (t) = t2 x̂ + ln tŷ + t ln tẑ at the point (1, 0, 0). p 3. Let f (x, y) = xy + 4 − y 2 . Find and sketch the domain of f ? 4. Find the line through the point (0, 14, −10) and parallel to the line x = 2t − 1, y = 6 − 3t, z = 3 + 9t. 5. Find the equation for the plane that passes through the origin and the points (2, −4, 6) and (5, 1, 3). − 6. Find the unit tangent vector to the curve → r (t) = sin tx̂ + 2 cos tŷ at the point where t = π/4. √ 7. Change (−1, − 3, 2) from Cartesian to cylindrical coordinates. 8. Determine the equations for a cylinder oriented in the y-direction with radius 3 whose axis passes through the point (4, 5, 2). − 9. Find the length of the curve → r (t) = 2tx̂ + t2 ŷ + t3 3 ẑ where 0 ≤ t ≤ 1. 10. Find a vector function that represents the curve defined by the intersection of the cylinder x2 + y 2 = 1 and the plane y + z = 2.