Find the Equation for the Plane Tangent to the Parametrized

question 119

Essay

Find the equation for the plane tangent to the parametrized surface S at the point P.
- $$\mathrm { S } \text { is the sphere } \mathbf { r } ( \theta , \varphi ) = 7 \cos \theta \sin \varphi \mathbf { i } + 7 \sin \theta \sin \varphi \mathbf { j } + 7 \cos \varphi \mathbf { k } ; \mathrm { P } \text { is the point corresponding to } ( \theta , \varphi ) = \left( \frac { \pi } { 3 } , \frac { \pi } { 4 } \right) \text {. }$$

---

Definitions:

Carboxylic Acid

An organic compound containing a carboxyl group, characterized by the presence of at least one carbon atom double-bonded to an oxygen atom and also bonded to a hydroxyl group.

O-H Stretch

A specific stretching vibration in infrared spectroscopy that denotes the presence of an OH (hydroxyl) group in a molecule.

IR Spectrum

A graph demonstrating the infrared absorption and transmission of a substance, providing distinctive molecular fingerprints.

Carboxylic Acid

An organic acid characterized by the presence of a carboxyl group, with the formula R-COOH.