Exercise 1: Tangent Vector Points in the Direction of Increasing t

In this video I show that the derivative or tangent vector of a curve defined by a vector function always points in the direction in which the parameter t is increasing. I show this by analyzing how the curve behaves for positive or negative values of the term h in the definition of derivative for a vector function. For the positive h case, the result is obvious since the secant vector divided by h yields a positive value, hence the tangent vector is in the same direction as the secant vector, which itself is in the direction of increasing t. For the negative h case, the secant vector points in the direction of decreasing t, but dividing by the negative h yields the tangent vector in reverse direction, i.e. in the direction of increasing t!

#math #vectors #calculus #stem #education

Timestamps:

- Exercise 1: Tangent vector points in direction of increasing t: 0:00
- Solution: Reviewing the definition of the tangent vector or derivative: 1:10
- Case where h is greater than zero: 2:48
- Positive he means secant vector and tangent vector point in the same direction, of increasing t: 4:53
- Case where h is less than zero: 7:14
- Secant vector points in the direction of decreasing t, but negative h means tangent vector is in reverse direction and towards increasing t: 8:42
- Tangent vector is always pointing in the direction of increasing t: 10:23

Notes and playlists:

- Summary: https://inleo.io/threads/view/mes/re-leothreads-vqtxivr7
- Notes: https://peakd.com/hive-128780/@mes/hfsetrue
- Sections playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FgQ3jBd9IFaWOakOWbHous
- Vector Functions playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FgQ3jBd9IFaWOakOWbHous .

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