Find the Equation for the Plane Tangent to the Parametrized

question 66

Short Answer

Find the equation for the plane tangent to the parametrized surface S at the point P.
- $$S \text { is the cylinder } r ( \theta , z ) = 6 \cos ^ { 2 } \theta \mathbf { i } + 3 \sin 2 \theta \mathbf { j } + \mathrm { zk } ; \mathrm { P } \text { is the point corresponding to } ( \theta , z ) = \left( \frac { \pi } { 4 } , - 1 \right)$$

---

Definitions:

Truth Table

A mathematical table used in logic to determine the veracity of various propositions under all possible scenarios.

Hypothetical Syllogism (HS)

A form of deductive reasoning that connects two conditional statements to arrive at a conclusion, if P then Q; if Q then R, therefore if P then R.

Modus Ponens (MP)

A valid form of argument where if a conditional statement ("if p then q") is accepted, and the antecedent (p) is true, then the consequent (q) must also be true.

Modus Tollens (MT)

A form of logical argument where, from "If P then Q" and "Not Q," it follows that "Not P."