## 18 Mar 2013 Notes and Examples. These notes contain subsections on. • Parametric equations. • Sketching a parametric curve. • Finding the cartesian

t the parameter, and the trajectory traced out is a parametric curve. Any graph y For example, here is a parametric equation for the ellipse centered at (0,0),. In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations EXAMPLE A Investigate the family of curves with parametric equations. What do these curves have in common? How does the shape change as increases? Let's consider an example. Suppose we have the following parametric equation: x = cos(t) y = sin(t). We know from the pythagorean theorem that this parametric CALCULUS WITH PARAMETRIC CURVES. EXAMPLE A Find an equation of the tangent line to the parametric curve at the point . Where does this curve have parametric curve γ is given, the point γ(a) is the initial point of γ, and γ(b) is the terminal point(or end point) of γ. Example 11.1.2. For a parametric equation of the where the variable t is called a parameter. For example, two functions. {x=Rcosty=Rsint. describe in parametric form the equation of a circle centered at the origin

Lecture 1 – Explicit, Implicit and Parametric Equations. 2 2) Extend the example by adding: A good example of an implicit equation is the equation of the. 5 Feb 2017 10.1 - Curves Defined By Parametric Equations. Definition 1: Parametric Equations In this example (x(4),y(4)) is called the Terminal Point. Example: A parametric equation for a circle of radius 1 and center (0,0) is: x = cost, y = sint. The equations x = f(t), y = g(t) are called parametric equations. Bridge to Calculus 1 Parametric Practice. 1. Sketch the graph determined by the parametric equations. In what direction is the graph traced out as the value of t EXAMPLE 3 What curve is represented by the given parametric equations? Examples 2 and 3 show that different sets of parametric equations can represent Example. Find the slope and equation of the tangent line for the following parametric equations at t = 1. x = t3 − t y = t4 − 5t2 + 4. 1. 2. 3. 4. 5. 6 x. -2. -1. 1. 2. 3. 4.

242 Chapter 10 Polar Coordinates, Parametric Equations. Just as we EXAMPLE 10.1.3 Find the equation of the line y = 3x + 2 in polar coordinates. We. t in /a, b0 and then eliminate the parameter t in hopes of obtaining a familiar equation in x and y. EXAMPLE 1 Sketch the curve parameterized by. r t! - ($ t2, %t2. Example 1 Draw and identify the parametric curve given by the parametric Example 4 Covert the following parametric equation to an equation relating x and y:. Mechanics gives and . Time is a parameter. □. Given parameter . Then. , are parametric equations for a curve in the -plane. Example. ,. Draw the curve in the Lecture 1 stopped here. Example 1.5. Eliminate the parameter to find a Cartesian equation of the curve for r(t)=(x(t),y(t)) = (t2. 3,t + 2), t 2 [3, 3]. Solution: From. parametric equations, we usually call it a parametrized curve. We have already worked with some interesting examples of parametric equations. bug starts We start with a simple example. Example 6. Plot the curve x = 2 cos t y = 3 sint. ︸. ︷︷. ︸. 0 ≤ t ≤ π. 2. ︸ ︷︷ ︸ parametric equations of the curve parameter

## Use parametric equations to represent projectile motion, as applied in Example 4. Τ To solve real-life problems, such as modeling the path of a leaping dolphin.

t in /a, b0 and then eliminate the parameter t in hopes of obtaining a familiar equation in x and y. EXAMPLE 1 Sketch the curve parameterized by. r t! - ($ t2, %t2. Example 1 Draw and identify the parametric curve given by the parametric Example 4 Covert the following parametric equation to an equation relating x and y:. Mechanics gives and . Time is a parameter. □. Given parameter . Then. , are parametric equations for a curve in the -plane. Example. ,. Draw the curve in the Lecture 1 stopped here. Example 1.5. Eliminate the parameter to find a Cartesian equation of the curve for r(t)=(x(t),y(t)) = (t2. 3,t + 2), t 2 [3, 3]. Solution: From. parametric equations, we usually call it a parametrized curve. We have already worked with some interesting examples of parametric equations. bug starts We start with a simple example. Example 6. Plot the curve x = 2 cos t y = 3 sint. ︸. ︷︷. ︸. 0 ≤ t ≤ π. 2. ︸ ︷︷ ︸ parametric equations of the curve parameter We have seen parametric equations for lines. Now we will look at parametric equations of Next we will give a series of examples of parametrized curves.