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Chapter 11: Q. 37 (page 860)
Evaluate and simplify the indicated quantities in Exercises 35鈥�41.
The simplification of is .
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Given a twice-differentiable vector-valued function , what is the definition of the principal unit normal vector ?
The DNA molecule takes the shape of a double helix鈥攖wo 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.).
In Exercises 19鈥�21 sketch the graph of a vector-valued function with the specified properties. Be sure to indicate the direction of increasing values oft.
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Let be a differentiable scalar function and be a differentiable vector function. Prove that . (This is Theorem 11.11 (b).)
Evaluate the limits in Exercises 42鈥�45.
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