Answer
Estimated coordinates: $(0.6, 2)$
Exact coordinates: $(0.58, 2.01)$
Work Step by Step
1. Graph the equation and estimate the coordinates of the rightmost point on that curve.
- Insert the parametric equation and then zoom in that rightmost point, and take note of the coordinates.
- Mine was $(0.6,2)$.
2. Calculate $\frac{dy}{dt}$ and $\frac{dx}{dt}$:
$\frac{dy}{dt} = \frac{d(e^t)}{dt} = e^t$
$\frac{dx}{dt} = \frac{d(t-t^6)}{dt} = 1 -6t^5$
Thus: $\frac{dy}{dx} = \frac{e^t}{1-6t^5}$
The rightmost point is on a vertical tangent, therefore, $1-6t^5= 0$ and $e^t \neq 0$
$1-6t^5 = 0$
$1 = 6t^5$
$\frac 1 6 =t^5$
$\sqrt [5] {(\frac 1 6)} = t$
$t \approx 0.69883$
- Check if $e^t \neq 0$:
- That expression will never equal 0.
3. Find the coordinates for: $t = 0.69883$
$x = ( 0.69883) - ( 0.69883)^6 = 0.58$
$y = e^{( 0.69883)} = 2.01$
