Textbook Question
57–64. Graphing polar curves Graph the following equations. Use a graphing utility to check your work and produce a final graph.
r = sin 3θ
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 12.1.80
77–80. Slopes of tangent lines Find all points at which the following curves have the given slope.
x = 2 + √t, y = 2 - 4t; slope = -8
Verified step by step guidance1
Identify the given parametric equations: \(x = 2 + \sqrt{t}\) and \(y = 2 - 4t\).
Recall that the slope of the tangent line to a parametric curve is given by \(\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}\).
Compute the derivatives with respect to \(t\): \(\frac{dx}{dt} = \frac{d}{dt}(2 + \sqrt{t})\) and \(\frac{dy}{dt} = \frac{d}{dt}(2 - 4t)\).
Set the slope equal to the given value: \(\frac{dy}{dx} = -8\), which means \(\frac{\frac{dy}{dt}}{\frac{dx}{dt}} = -8\).
Solve the resulting equation for \(t\), then substitute back into the original parametric equations to find the corresponding points \((x, y)\).
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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 as t. Here, x and y are given in terms of t, allowing us to analyze the curve's behavior by studying these functions.
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Guided course
08:02
Parameterizing Equations
Derivative of Parametric Curves
The slope of the tangent line to a parametric curve is found by computing dy/dx = (dy/dt) / (dx/dt). This requires differentiating both x(t) and y(t) with respect to t and then dividing the results to find the instantaneous rate of change.
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06:49Differentiation of Parametric Curves
Finding Points with a Given Slope
To find points where the curve has a specific slope, set the derivative dy/dx equal to the given slope and solve for the parameter t. Substituting t back into x(t) and y(t) gives the coordinates of the points with that slope.
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05:45Understanding Slope Fields
Related Practice
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.
25y² - 4x² = 100
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Textbook Question
33–40. Areas of regions Make a sketch of the region and its bounding curves. Find the area of the region.
The region inside one leaf of r = cos 3θ
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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 = √t + 4, y = 3√t; 0 ≤ t ≤ 16
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.
10x² - 7y² = 140
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Textbook Question
53–56. Circular motion Find parametric equations that describe the circular path of the following objects. For Exercises 53–55, assume (x, y) denotes the position of the object relative to the origin at the center of the circle. Use the units of time specified in the problem. There are many ways to describe any circle.
The tip of the 15-inch second hand of a clock completes one revolution in 60 seconds.
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