Calculus Volume 3 · free preview
Vector-Valued Functions and Space Curves
Vector-valued functions map a parameter to vectors, tracing space curves, with limits, continuity, and componentwise structure extending the function concept to three-dimensional trajectories.
1. Vector-valued functions
Vector-valued functions
2. Limits and continuity
Limits and continuity
3. Graphing space curves
Graphing space curves
4. Domains and standard parametrizations
Domains and standard parametrizations
5. Componentwise limit
The limit of r(t) = (f(t), g(t), h(t)) as t approaches a equals what?
6. Helix components
The helix (cos t, sin t, t) has which projection onto the xy plane?
7. Continuity definition
The vector function r is continuous at a when which condition holds?
8. Domain intersection
The domain of r(t) = (1/t, root t, ln t) is which interval?
9. Magnitude continuity insufficiency
Why is continuity of |r(t)| alone not enough for r's continuity?
10. Twisted cubic middle component
The twisted cubic (t, t^2, t^3) has which middle component? Answer like t^2.
11. Helix rise per turn
The helix (a cos t, a sin t, ct) rises by 2 pi c per turn. As a multiple of pi c, what is the rise? Answer with a number.
12. Domain intersection count
The domain of (1/t, root t, ln t) excludes zero and negatives. As a number, how many components restrict it? Answer with a number.
13. Curve versus path distinction
The helix's projection being a circle illustrates which distinction?
14. Position vector viewpoint
A space curve's points are the tips of which vectors as the parameter varies? Answer with two words.
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