![]() Textbook content produced by OpenStax is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License. We recommend using aĪuthors: Gilbert Strang, Edwin “Jed” Herman Use the information below to generate a citation. Then you must include on every digital page view the following attribution: If you are redistributing all or part of this book in a digital format, For example, if we know a parameterization of a given curve, is it possible. Then you must include on every physical page the following attribution: 7.2.1 Determine derivatives and equations of tangents for parametric curves. If you are redistributing all or part of this book in a print format, EXAMPLE 1.4 More Circles and Ellipses Defined by Parametric Equations Identify the plane curves (a) x 2cost,y 3sint,(b) x 2 +4cost, y 3 +4sint and (c) x 3cos2t,y 3sin2t. ![]() We explore this in example 1.4 and the exercises. Want to cite, share, or modify this book? This book uses theĬreative Commons Attribution-NonCommercial-ShareAlike License Simple modications to the parametric equations in example 1.3 will produce a variety of circles and ellipses. Consider the plane curve defined by the parametric equations We start by asking how to calculate the slope of a line tangent to a parametric curve at a point. Furthermore, we should be able to calculate just how far that ball has traveled as a function of time. Find an equation for the line tangent to the graph of the parametric equations from Example 1 at the point where t. If the position of the baseball is represented by the plane curve ( x ( t ), y ( t ) ), ( x ( t ), y ( t ) ), then we should be able to use calculus to find the speed of the ball at any given time. ![]() For example, if we know a parameterization of a given curve, is it possible to calculate the slope of a tangent line to the curve? How about the arc length of the curve? Or the area under the curve?Īnother scenario: Suppose we would like to represent the location of a baseball after the ball leaves a pitcher’s hand. Now that we have introduced the concept of a parameterized curve, our next step is to learn how to work with this concept in the context of calculus. 7.2.4 Apply the formula for surface area to a volume generated by a parametric curve.7.2.3 Use the equation for arc length of a parametric curve.7.2.2 Find the area under a parametric curve.7.2.1 Determine derivatives and equations of tangents for parametric curves.
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