Textbook Question
31–36. Converting coordinates Express the following Cartesian coordinates in polar coordinates in at least two different ways.
(-4, 4√3)
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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 12.1.84
81–88. Arc length Find the arc length of the following curves on the given interval.
x = eᵗ sin t, y = eᵗ cos t; 0 ≤ t ≤ 2π
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 \(a \leq t \leq b\):
\[L = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt\]
Find the derivatives \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\) for the given functions:
\[x = e^t \sin t, \quad y = e^t \cos t\]
Use the product rule for differentiation since both \(x\) and \(y\) are products of \(e^t\) and trigonometric functions.
Compute \(\frac{dx}{dt}\):
\[\frac{dx}{dt} = \frac{d}{dt}(e^t \sin t) = e^t \sin t + e^t \cos t\]
Compute \(\frac{dy}{dt}\):
\[\frac{dy}{dt} = \frac{d}{dt}(e^t \cos t) = e^t \cos t - e^t \sin t\]
Substitute \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\) into the arc length formula under the square root:
\[\sqrt{(e^t \sin t + e^t \cos t)^2 + (e^t \cos t - e^t \sin t)^2}\]
Simplify the expression inside the square root by expanding and combining like terms.
Set up the integral for the arc length over the interval \(0 \leq t \leq 2\pi\):
\[L = \int_0^{2\pi} \sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2} \, dt\]
After simplification, evaluate or express the integral in a form ready for calculation.
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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 a curve as functions of a parameter, often denoted t. Instead of y as a function of x, both x and y depend on t, allowing the description of more complex curves. Understanding how to work with parametric forms is essential for calculating properties like arc length.
Recommended video:
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 given by the integral of the square root of (dx/dt)² + (dy/dt)² dt. This formula sums the infinitesimal distances along the curve, providing the total length over the interval.
Recommended video:
06:29Arc Length of Parametric Curves
Differentiation of Exponential and Trigonometric Functions
Calculating dx/dt and dy/dt requires differentiating functions involving products of exponentials and trigonometric terms. Mastery of the product rule and derivatives of e^t, sin t, and cos t is necessary to correctly find these derivatives and proceed with the arc length calculation.
Recommended video:
6:04Introduction to Trigonometric Functions
Related Practice
Textbook Question
81–88. Arc length Find the arc length of the following curves on the given interval.
x = 3 cos t, y = 3 sin t + 1; 0 ≤ t ≤ 2π
Textbook Question
13–30. Graphing conic sections Determine whether the following equations describe a parabola, an ellipse, or a hyperbola, and then sketch a graph of the curve. For each parabola, specify the location of the focus and the equation of the directrix; for each ellipse, label the coordinates of the vertices and foci, and find the lengths of the major and minor axes; for each hyperbola, label the coordinates of the vertices and foci, and find the equations of the asymptotes.
4x² - y² = 16
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Textbook Question
15–22. Sets in polar coordinates Sketch the following sets of points.
1 < r < 2 and π/6 ≤ θ ≤ π/3
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Textbook Question
37–52. Curves to parametric equations Find parametric equations for the following curves. Include an interval for the parameter values. Answers are not unique.
The line that passes through the points P(1, 1) and Q(3, 5), oriented in the direction of increasing x
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Textbook Question
37–48. Polar-to-Cartesian coordinates Convert the following equations to Cartesian coordinates. Describe the resulting curve.
r = 6 cos θ + 8 sin θ