1. Find parametric equations describing the line segment from P(0,0) to Q(7,17).
2. Find parametric equations describing the line segment from
3. Find parametric equations describing the ellipse centered at the origin with major axis of length 6 along the x-axis and the minor axis of length 3 along the y-axis, generated clockwise.
20. Convert the equation into Cartesian coordinates:
21. Find an equation of the line y=mx+b in polar coordinates.
Sketch the following polar curves without using a computer.
Sketch the following sets of points.
Calculus in Polar Coordinates
Find points where the following curves have vertical or horizontal tangents.
Sketch the region and find its area.
42. The region inside the limaçon
43. The region inside the petals of the rose
and outside the circle
Vectors and Dot Product
60. Find an equation of the sphere with center (1,2,0) passing through the point (3,4,5)
61. Sketch the plane passing through the points (2,0,0), (0,3,0), and (0,0,4)
62. Find the value of
63. Find all unit vectors parallel to
64. Prove one of the distributive properties for vectors in
65. Find all unit vectors orthogonal to
66. Find all unit vectors orthogonal to
67. Find all unit vectors that make an angle of
with the vector
Find the area of the parallelogram with sides and .
84. Find all vectors that satisfy the equation
85. Find the volume of the parallelepiped with edges given by position vectors
86. A wrench has a pivot at the origin and extends along the x
-axis. Find the magnitude and the direction of the torque at the pivot when the force
is applied to the wrench n
units away from the origin.
Prove the following identities or show them false by giving a counterexample.
Calculus of Vector-Valued Functions
101. Find a tangent vector for the curve
at the point
102. Find the unit tangent vector for the curve
103. Find the unit tangent vector for the curve
at the point
Motion in Space
120. Find velocity, speed, and acceleration of an object if the position is given by
121. Find the velocity and the position vectors for
if the acceleration is given by
Length of Curves
Find the length of the following curves.
Parametrization and Normal Vectors
142. Find a description of the curve that uses arc length as a parameter:
143. Find the unit tangent vector T
and the principal unit normal vector N
for the curve
Check that T
Equations of Lines And Planes
160. Find an equation of a plane passing through points