Multivariable Calculus

Math 251 Name:
Homework: Sections 13.3-14.3
1. Let r(t) = ht, 5 sin t, 5 costi.
(a) Find T(π)
(b) Find N(π)
(c) Find B(π)
(d) Find κ(π)
(e) Find κ(
π
2
).
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2. The path of a particle is given by the following function r(t) = h4 sin (3t), 3t, 4 cos (3t)i.
(a) Precisely sketch this curve and label important points. Use arrows to indicate the
direction of increasing t.
(b) Suppose the particle travels 10 units as it moves along the curve from position (4,
π
2
, 0)
to position (a, b, c). Determine the exact coordinates of position (a, b, c).
*Warning, your answer will be a bit messy. This does not mean you have made a
mistake.*
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3. Let f(x, y) = x
2 + y
2 and g(x, y) = p
x
2 + y
2.
(a) State the domain and range of both functions.
(b) Sketch the contour maps for each functions. Be as precise as possible.
(c) Discuss the similarities and differences between the two contour maps.
(d) Sketch f(x, y) and g(x, y). Describe how the contour maps relate to the shape of the
graph.
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