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)?
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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).
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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).
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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.