Chapter 11: Q. 37 (page 860)
Evaluate and simplify the indicated quantities in Exercises 35–41.
The simplification of is .
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Find the unit tangent vector and the principal unit normal vector at the specified value of t.
Imagine that you are driving on a twisting mountain road. Describe the unit tangent vector, principal unit normal vector, and binomial vector as you ascend, descend, twist right, and twist left.
Given a vector-valued function r(t) with domain what is the relationship between the graph of r(t) and the graph of kr(t), where k > 1 is a scalar?
Evaluate and simplify the indicated quantities in Exercises 35–41.
The DNA molecule takes the shape of a double helix—two helices that stay a roughly uniform distance apart.
(a) Neglecting actual dimensions, we can model one strand of DNA using the vector function .
Sketch the graph of . What is the effect of the parameter ?
(b) The second strand of DNA can be constructed by shifting the first. Does the graph of ever intersect that of ?
(c) The distance between two curves is the minimum distance between any two points on the curves. What is the distance between and if ? (Hint: Write two points on the curves using parameters and , expand the formula for the distance between them, and then use a trigonometric identity for addition. Then let
and minimize.).
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