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
Ch.12 - Parametric and Polar Curves
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
Not the one you use?Change textbookSelect a different textbook
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 12.1.49
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
Verified step by step guidance1
Identify the two given points: P(1, 1) and Q(3, 5). These points lie on the line we want to parametrize.
Calculate the direction vector \( \vec{d} \) of the line by subtracting the coordinates of P from Q: \( \vec{d} = (3 - 1, 5 - 1) = (2, 4) \). This vector points in the direction of increasing x.
Set up the parametric equations using point P as the initial point and \( t \) as the parameter: \( x(t) = 1 + 2t \) and \( y(t) = 1 + 4t \). Here, \( t \) represents how far along the line you move from point P.
Determine the interval for \( t \) based on the problem context. Since the line extends infinitely in both directions, \( t \) can be any real number, but to keep the orientation in the direction of increasing x, \( t \) should be chosen such that \( x(t) \) increases as \( t \) increases, so \( t \in \mathbb{R} \).
Summarize the parametric form: \( x(t) = 1 + 2t \), \( y(t) = 1 + 4t \), with \( t \in (-\infty, \infty) \). This fully describes the line passing through P and Q oriented in the direction of increasing x.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
3mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Parametric Equations of a Line
Parametric equations express the coordinates of points on a curve as functions of a parameter, often denoted t. For a line through two points, the equations are derived by setting x and y as linear functions of t, representing movement from one point to another.
Recommended video:
Guided course
08:02
Parameterizing Equations
Vector Form and Direction of a Line
The direction vector of a line is found by subtracting the coordinates of the initial point from the terminal point. This vector determines the line's orientation, and the parameter t scales this vector to generate all points along the line.
Recommended video:
05:14Equations of Tangent Lines
Parameter Interval and Orientation
The parameter interval defines the range of t values for which the parametric equations are valid. Choosing an interval consistent with the direction of increasing x ensures the line is oriented correctly, reflecting the problem's requirement.
Recommended video:
Guided course
05:59Eliminating the Parameter
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.
4x² - y² = 16
1
views
Textbook Question
69–72. Tangent lines Find an equation of the line tangent to the following curves at the given point.
x² = -6y; (-6, -6)
Textbook Question
15–22. Sets in polar coordinates Sketch the following sets of points.
1 < r < 2 and π/6 ≤ θ ≤ π/3
1
views
Textbook Question
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
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