(a) Eliminate the parameter to find a Cartesian equation of...

(a) Eliminate the parameter to find a Cartesian equation of the curve.
(b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases.
$x=e^{t}-1, \quad y=e^{2 t}$

Key Concepts

Parametric Equations

These are representations of curves where both coordinates, x and y, are expressed as functions of an independent parameter. This approach offers flexibility in describing complex curves that might be difficult to represent with a single equation.

Elimination of Parameters

This process involves removing the independent parameter from the system of equations to obtain a direct relationship between x and y. It simplifies the analysis of the curve by converting the parametric form into a Cartesian equation that is often easier to graph and interpret.

Cartesian Equation

This is an equation that directly relates x and y in the coordinate plane without involving a parameter. It provides a clear and concise representation of the curve's shape and properties, making it easier to study its geometric features.

Sketching Parametric Curves with Direction

This concept involves plotting the curve defined by the Cartesian equation and marking the trajectory with an arrow to indicate the direction of increasing parameter value. Including the direction helps in understanding the motion along the curve, which is particularly useful in dynamic contexts such as physics and engineering.

Transcript

00:01 So in this problem, we're given a set of parametric equations, and the first thing we need to do is eliminate the parameter in order to come up with the cartesian equation for the curve.

00:12 So our parametric equations are x equals e to the t minus 1 and y equals e to the 2t.

00:22 Now in order to get rid of this t variable, the first thing we need to do is solve for t in terms of either x or y.

00:29 So let's solve our t in terms of x.

00:32 We have e to the t equals x plus 1.

00:37 We can take the natural log of both sides to get rid of this e.

00:41 So we have natural log of e to the t equals natural log of x plus 1.

00:48 So in order to reduce the left side of our equation, we can just bring this t over to the front so that it's t times the natural log of e.

00:58 Natural log of e is 1.

01:00 So we have t equals natural log of x plus one now we can plug this expression in to our equation for y so we have y equals e to the two times natural log of x plus one now we can do the inverse of what we did over here we can bring this two over so that it's the exponent so we have y equals e to the natural log of x plus one quantity squared so now we're going to have to use another property of logarithms.

01:39 Whenever you have e to the natural log of f of x, this is the same thing as just f of x.

01:48 So now we have our cartesian equation for the curve.

01:51 Y equals x plus 1 squared.

01:57 So now in the second part of our question, we need to sketch the curve and also find what direction the curve moves in as t increases.

02:06 Let's draw our cartesian plane here...

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