Find both parametric and rectangular representations for the plane tangent to r(u,v)=u2i+ucos(v)j+usin(v)kr(u,v)=u2i+ucos⁡(v)j+usin⁡(v)k at the point P(4,−2,0)P(4,−2,0).One possible parametric representation has the form⟨4−4u⟨4−4u , , 4v⟩4v⟩(Note that parametric representations are not unique. If your first and third components look different than the ones presented here, you will need to adjust your parameters so that they do match, and then the other components should match the ones expected here as well.)The equation for this plane in rectangular coordinates has the form x+x+ y+y+ z+z+ =0 **Topic: Tangent Plane Parametric and Rectangular Representations****Problem Statement:**Find both parametric and rectangular representations for the plane tangent to $ \mathbf{r}(u,v) = u^2 \mathbf{i} + u \cos(v) \mathbf{j} + u \sin(v) \mathbf{k} $ at the point $ P(4, -2, 0) $.---**Parametric Representation:**One possible parametric representation has the form:\[ (4 - 4u, \quad \underline{\hspace{50px}}, \quad 4v) \](Note that parametric representations are not unique. If your first and third components look different than the ones presented here, you will need to adjust your parameters so that they do match, and then the other components should match the ones expected here as well.)---**Rectangular Representation:**The equation for this plane in rectangular coordinates has the form:\[ \underline{\hspace{50px}}x + \underline{\hspace{50px}}y + \underline{\hspace{50px}}z + \underline{\hspace{50px}} = 0 \](Be sure your coefficients have been set so that (1) the coefficient of $ x $ is positive, (2) all coefficients are integers, and (3) there are no more common factors that can still be divided out.)---In this educational exercise, the focus is on finding both parametric and rectangular descriptions of the tangent plane. By following the guidelines for filling in the blanks, you will learn how to explore different forms of plane equations and ensure they meet specific standard criteria.

Answered: Find both parametric and rectangular…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Find both parametric and rectangular representations for the plane tangent to r(u,v)=u2i+ucos(v)j+usin(v)kr(u,v)=u2i+ucos⁡(v)j+usin⁡(v)k at the point P(4,−2,0)P(4,−2,0).

One possible parametric representation has the form

⟨4−4u⟨4−4u , , 4v⟩4v⟩

(Note that parametric representations are not unique. If your first and third components look different than the ones presented here, you will need to adjust your parameters so that they do match, and then the other components should match the ones expected here as well.)

The equation for this plane in rectangular coordinates has the form

x+x+ y+y+ z+z+ =0

**Topic: Tangent Plane Parametric and Rectangular Representations**

**Problem Statement:**
Find both parametric and rectangular representations for the plane tangent to $ \mathbf{r}(u,v) = u^2 \mathbf{i} + u \cos(v) \mathbf{j} + u \sin(v) \mathbf{k} $ at the point $ P(4, -2, 0) $.

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**Parametric Representation:**
One possible parametric representation has the form:

\[ (4 - 4u, \quad \underline{\hspace{50px}}, \quad 4v) \]

(Note that parametric representations are not unique. If your first and third components look different than the ones presented here, you will need to adjust your parameters so that they do match, and then the other components should match the ones expected here as well.)

---

**Rectangular Representation:**
The equation for this plane in rectangular coordinates has the form:

\[ \underline{\hspace{50px}}x + \underline{\hspace{50px}}y + \underline{\hspace{50px}}z + \underline{\hspace{50px}} = 0 \]

(Be sure your coefficients have been set so that (1) the coefficient of $ x $ is positive, (2) all coefficients are integers, and (3) there are no more common factors that can still be divided out.)

---

In this educational exercise, the focus is on finding both parametric and rectangular descriptions of the tangent plane. By following the guidelines for filling in the blanks, you will learn how to explore different forms of plane equations and ensure they meet specific standard criteria.

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