A curve is given by the parametric equations
x=e^t, y=5-e^(3t)
so what?
y = 5-x^3
The given curve is represented by the parametric equations:
x = e^t
y = 5 - e^(3t)
To understand the nature of this curve, let's first analyze the individual equations.
1. Equation x = e^t: This equation represents the x-coordinate of a point on the curve as a function of the parameter t. For each value of t, we raise the constant e (approximately equal to 2.71828) to the power of t, resulting in a corresponding value of x.
2. Equation y = 5 - e^(3t): This equation represents the y-coordinate of a point on the curve as a function of the parameter t. Similar to the x equation, we raise e to the power of 3t and subtract it from 5 to obtain the value of y.
To plot the curve, we need to find the relationship between x and y. By eliminating the parameter t, we can express y in terms of x.
Starting with the equation x = e^t, we can take the natural logarithm of both sides to get:
ln(x) = t
Now substitute this value of t in the second equation:
y = 5 - e^(3t)
= 5 - e^(3ln(x))
= 5 - e^(ln(x^3))
= 5 - x^3
Hence, the relationship between x and y is y = 5 - x^3.
This means that every point (x, y) on the curve satisfies this equation. To obtain the plot of the curve, you can choose various values for x and use the equation y = 5 - x^3 to calculate the corresponding y-coordinate. By connecting these points, you will get the desired curve.