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Ch.12 - Parametric and Polar Curves
Briggs - Calculus: Early Transcendentals 3rd Edition
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Tutti i libri di testoBriggs 3rd EditionCh.12 - Parametric and Polar CurvesProblema 12.4.51d
Capitolo 12, Problema 12.4.51d

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
d. The point on a parabola closest to the focus is the vertex.

Guida verificata passo dopo passo
1
Recall the definition of a parabola: it is the set of all points equidistant from a fixed point called the focus and a fixed line called the directrix.
Understand that the vertex of the parabola is the point on the parabola that lies exactly halfway between the focus and the directrix.
To determine if the vertex is the closest point on the parabola to the focus, consider the distance from any point on the parabola to the focus and compare it to the distance from the vertex to the focus.
Use the geometric property that the distance from the vertex to the focus is the shortest distance from the parabola to the focus because the vertex lies on the axis of symmetry and minimizes this distance.
Conclude that the vertex is indeed the point on the parabola closest to the focus, since any other point on the parabola will be farther away due to the parabola's shape and definition.

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Definition of a Parabola and Its Focus

A parabola is the set of all points equidistant from a fixed point called the focus and a fixed line called the directrix. The vertex is the point on the parabola closest to the directrix and lies midway between the focus and directrix.
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Percorso guidato
05:43
Definition of the Definite Integral

Distance from a Point to the Focus

The distance from any point on the parabola to the focus varies along the curve. Understanding how this distance changes is essential to determine which point on the parabola is closest to the focus.
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6:58
Convert Points from Rectangular to Polar

Optimization and Minimizing Distance

To find the point on the parabola closest to the focus, one must analyze the distance function and use calculus techniques like differentiation to find its minimum value, confirming whether the vertex is indeed the closest point.
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09:07
Example 1: Minimizing Surface Area
Pratica correlata
Domanda del libro di testo

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.  


d. The point (3,π/2) lies on the graph of r=3 cos 2θ.  

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Tangents and normals: Let a polar curve be described by r = f(θ), and let ℓ be the line tangent to the curve at the point P(x,y) = P(r,θ) (see figure).

e. Prove that the values of θ for which ℓ is parallel to the y-axis satisfy tan θ = f(θ)/f'(θ).


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Domanda del libro di testo

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


e. There are two points on the curve x=−4 cos t, y=sin t, for 0≤t≤2π, at which there is a vertical tangent line.

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Domanda del libro di testo

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


d. The parametric equations x=cos t, y=sin t, for −π/2≤t≤π/2, describe a semicircle.

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Domanda del libro di testo

63–74. Arc length of polar curves Find the length of the following polar curves.


The curve r = sin³(θ/3), for 0 ≤ θ ≤ π/2

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Domanda del libro di testo

(Use of Tech) Finger curves: r = f(θ) = cos(aᶿ) - 1.5, where a = (1 + 12π)^(1/(2π)) ≈ 1.78933

d. Plot the curve with various values of k. How many fingers can you produce?

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