MATH 1320 : Spring 2014
Lab 6
Lab Instructor : Kurt VanNess
Name:
Score:
Write all your solutions on a separate sheet of paper.
1. If f (x) = sin(x3 ), find f (15) (0). [Hint: Consider the Maclaurin series for sin x.]
sin x − x
.
x−→0
x3
2. Use the series to evaluate the following limit: lim
3. Write inequalities to describe the following regions.
(a) The region consisting of all points between, but not on, the spheres of radius r and R centered at
the origin, where r < R.
(b) The solid upper hemisphere of the sphere of radius 2 centered at the origin.
4. Find a linear equation (only involving linear terms) of the set of all points equidistant from the points
A(−1, 5, 3) and B(6, 2, −2). Write the equation in the form ax + by + cz + d = 0.
5. (a) If v lies in the third quadrant and makes an angle π/3 with the negative x-axis and |v| = 4, find
v in component form.
(b) Suppose a is a three-dimensional unit vector in the first octant that starts at the origin and makes
angles of 60◦ and 45◦ with the positive x-axes and y-axes, respectively. Express a in terms of its
components.
6. Find the angle between a diagonal of a cube and one of its edges.
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