Calc 2: Curves Defined by parametric equations

In summary, when trying to find the Cartesian equation of a curve, you can solve for either x or y first and then substitute for t. As long as you follow the path of least effort, it doesn't matter which variable you solve for first.
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MillerGenuine
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these are fairly simple questions that only require you to plot points and then find a Cartesian equation of the curve. My question is when trying to solve for the Cartesian Equation, whether to solve for X first or Y? I will give an example.

X=3t-5 , y=2t + 1
Graphing this is simple, solving for Cartesian equation of the curve by eliminating the parameter is fairly simple,

x=3t-5 therefore t=(x+5)/3

Subsitute t into y=2t+1 and you get y=2/3(x+5) + 1

easy enough.

Now in the following problem

x=(t^2)-2 , y=5-2t

again, graphing is simple, but this time to find the Cartesian equation they first solve for t like so..

y=5-2t therefore t=(y-5)/2

then x= 1/4 (y-5)^2 - 2

So does it matter whether I have my Cartesian equation in the Form of X=... or Y=...
or do i just take the easiest path to solve for "t" and substitute?
 
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FAQ: Calc 2: Curves Defined by parametric equations

1. What are parametric equations in Calculus 2?

Parametric equations are a set of equations that express the coordinates of a point in terms of one or more independent variables, called parameters. In Calculus 2, these equations are commonly used to describe curves or surfaces in three-dimensional space.

2. How are curves defined by parametric equations?

Curves are defined by parametric equations by assigning a separate equation for each coordinate of a point on the curve. The independent variable, or parameter, is usually denoted by t. As t varies, the coordinates of the point will change, tracing out a curve in space.

3. What is the significance of the parameter in parametric equations?

The parameter in parametric equations serves as a way to track the movement of a point on a curve. It allows for a more flexible and intuitive way to describe curves, as opposed to traditional equations that only use x and y coordinates.

4. How do you graph a curve defined by parametric equations?

To graph a curve defined by parametric equations, you can plot points by choosing different values for the parameter, t. These points can then be connected to form the curve. Alternatively, you can use graphing software to plot the curve for a range of values of t.

5. What are some common applications of parametric equations in Calculus 2?

Parametric equations are commonly used in Calculus 2 to describe the motion of objects in space, such as projectiles or planets. They are also used in vector calculus to describe the path of a particle moving in three-dimensional space. Additionally, they are useful in computer graphics and animation to create smooth, curved shapes.

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