91Ó°ÊÓ

A Cartesian equation is an expression that relates the x and y coordinates directly without involving a third variable or parameter like \( t \). Instead of relying on a parameter to define a curve, Cartesian equations allow us to understand the relationship between x and y directly.

In the given exercise, the parametric equations are \( x = e^t \) and \( y = 1 - e^{3t} \). To find the Cartesian equation, we eliminate the parameter \( t \). By expressing \( t \) in terms of \( x \) and substituting it into the \( y \)-function, we found that the Cartesian equation is \( y = 1 - x^3 \).

• This equation describes a simple relationship between x and y without any parameters.
• It directly shows how x determines y.

Understanding the Cartesian equation is crucial as it gives a clearer geometric perspective of how x and y interact to form the curve.

Eliminating the parameter means to remove the parameter (in this case, \( t \)) from the given parametric equations in order to derive a single equation involving only x and y.

Here, we start with the equations \( x = e^t \) and \( y = 1 - e^{3t} \). The task is to express \( t \) in terms of one of the variables, typically \( x \), and replace it in the other equation.

• We found \( t = \ln(x) \) from the equation \( x = e^t \).
• This was substituted into the \( y \) equation as \( y = 1 - e^{3\ln(x)} \).
• Simplifying this, we arrive at \( y = 1 - x^3 \).

By eliminating the parameter, we simplify the parametric equations into a more traditional form, helping us to visualize the entire curve as a function of two variables, x and y.

The natural logarithm, denoted as \( \ln \), is the inverse operation of taking an exponential with base \( e \). Essentially, it allows us to "undo" the exponential effect. When you see \( \ln(x) \), you are finding what power you must raise \( e \) to get \( x \).

In our exercise, this concept is key when we need to express \( t \) in terms of \( x \). Since \( x = e^t \), applying the natural logarithm on both sides gives us \( t = \ln(x) \).

• It's a handy tool when dealing with equations involving exponential terms.
• By transforming exponential expressions into linear ones, logarithms simplify calculations.

The natural logarithm plays a crucial role in the solution, transforming the parametric form into a Cartesian one.

The range of x and y refers to the possible values these variables can take, based on the given parameter's interval. In parametric equations, the parameter \( t \) often defines this range.

For our exercise, the interval \( 0 \leq t \leq 1 \) gives us specific ranges:

• For \( x = e^t \), since \( t \) varies from 0 to 1, we have the range \( 1 \leq x \leq e \).
• For \( y = 1 - x^3 \), we observe the bounds for x to determine y's range.
• As \( x \) increases from 1 to \( e \), \( y \) decreases, starting from a maximum value of 0, and reaching a minimum value when \( x \) is \( e \).

Knowing the range is essential for graphing the curve accurately. It ensures that we only consider the relevant portion of the curve as determined by the parameter's limits.