Calculus/Multivariable and differential calculus:Exercises
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Parametric Equations
edit1. Find parametric equations describing the line segment from P(0,0) to Q(7,17).
2. Find parametric equations describing the line segment from to .
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.
Polar Coordinates
edit20. 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.
22.
23.
24.
Sketch the following sets of points.
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26.
Calculus in Polar Coordinates
editFind points where the following curves have vertical or horizontal tangents.
40.
41.
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
edit60. 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 if and
63. Find all unit vectors parallel to
64. Prove one of the distributive properties for vectors in :
65. Find all unit vectors orthogonal to in
66. Find all unit vectors orthogonal to in
67. Find all unit vectors that make an angle of with the vector
Cross Product
editFind and
80. and
81. and
Find the area of the parallelogram with sides and .
82. and
83. and
84. Find all vectors that satisfy the equation
85. Find the volume of the parallelepiped with edges given by position vectors , , and
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.
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89.
Calculus of Vector-Valued Functions
edit100. Differentiate .
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 .
104. Find if and .
105. Evaluate
Motion in Space
edit120. 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
editFind the length of the following curves.
140.
141.
Parametrization and Normal Vectors
edit142. 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⋅N=0.
Equations of Lines And Planes
edit160. Find an equation of a plane passing through points