Textbook Question
31–38. Equations of parabolas Find an equation of the following parabolas. Unless otherwise specified, assume the vertex is at the origin.
A parabola that opens to the right with directrix x = -4
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Ch.12 - Parametric and Polar Curves
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 12, Problem 12.1.88
81–88. Arc length Find the arc length of the following curves on the given interval.
x = sin t, y = t - cos t; 0 ≤ t ≤ π/2
Verified step by step guidance1
Recall the formula for the arc length of a parametric curve given by \(x = x(t)\) and \(y = y(t)\) over the interval \(a \leq t \leq b\):
\[L = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt\]
Identify the given functions:
\(x(t) = \sin t\)
\(y(t) = t - \cos t\)
and the interval:
\(0 \leq t \leq \frac{\pi}{2}\)
Compute the derivatives of \(x(t)\) and \(y(t)\) with respect to \(t\):
\[\frac{dx}{dt} = \cos t\]
\[\frac{dy}{dt} = 1 + \sin t\]
Substitute the derivatives into the arc length formula under the square root:
\[\sqrt{(\cos t)^2 + (1 + \sin t)^2} = \sqrt{\cos^2 t + (1 + \sin t)^2}\]
Simplify the expression inside the square root as much as possible before integrating, then set up the integral for the arc length:
\[L = \int_0^{\frac{\pi}{2}} \sqrt{\cos^2 t + (1 + \sin t)^2} \, dt\]
This integral can then be evaluated to find the arc length.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Parametric Equations
Parametric equations express the coordinates of points on a curve as functions of a parameter, often denoted t. Here, x and y are given in terms of t, allowing the curve to be analyzed by studying these functions over the specified interval.
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Guided course
08:02
Parameterizing Equations
Arc Length Formula for Parametric Curves
The arc length of a curve defined parametrically by x(t) and y(t) from t = a to t = b is found by integrating the square root of the sum of the squares of the derivatives: ∫ from a to b √[(dx/dt)² + (dy/dt)²] dt. This formula measures the distance along the curve.
Recommended video:
06:29Arc Length of Parametric Curves
Differentiation of Parametric Functions
To apply the arc length formula, one must compute the derivatives dx/dt and dy/dt accurately. Differentiation rules for trigonometric and polynomial functions are used here to find these derivatives, which are essential for evaluating the integral.
Recommended video:
06:49Differentiation of Parametric Curves
Related Practice
Textbook Question
85–87. Grazing goat problems Consider the following sequence of problems related to grazing goats tied to a rope. (See the Guided Project Grazing goat problems.)
A circular corral of unit radius is enclosed by a fence. A goat inside the corral is tied to the fence with a rope of length 0≤a≤2 (see figure). What is the area of the region (inside the corral) that the goat can graze? Check your answer with the special cases a=0 and a=2.
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Textbook Question
15–30. Working with parametric equations Consider the following parametric equations.
a. Eliminate the parameter to obtain an equation in x and y.
b. Describe the curve and indicate the positive orientation.
x = 8 + 2t, y = 1; −∞ < t < ∞
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Textbook Question
90–94. Focal chords A focal chord of a conic section is a line through a focus joining two points of the curve. The latus rectum is the focal chord perpendicular to the major axis of the conic. Prove the following properties.
Let L be the latus rectum of the parabola y ² =4px for p>0. Let F be the focus of the parabola, P be any point on the parabola to the left of L, and D be the (shortest) distance between P and L. Show that for all P, D+|FP|+ is a constant. Find the constant.
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Textbook Question
Given three polar coordinate representations for the origin.
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Textbook Question
11–20. Slopes of tangent lines Find the slope of the line tangent to the following polar curves at the given points.
r = 1 - sin θ; (1/2, π/6)
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