Chapter 11: Q. 3 (page 889)
Define what it means for a curve to be parametrized by arc length
The length of every segment of the curve is equal to the change in the value of the parameter.
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Let be a vector-valued function, where a < b are real numbers and the functions x(t), y(t), and z(t)are continuous. Explain why the graph of r is contained in some sphere centered at the origin.
Let be a vector-valued function, where a is a real number. Under what conditions would the graph of r have a vertical asymptote as t → ∞? Provide an example illustrating your answer.
Let , , , and be differentiable scalar functions. Prove that the dot product of the vector-valued functions role="math" localid="1649579098744" and role="math" localid="1649579122624" is a differentiable scalar function.
Carefully outline the steps you would use to find the equation of the osculating plane at a point on the graph of a vector function.
Find and graph the vector function determined by the differential equation
. ( HINT: What familiar pair of functions have the given properties ?)
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