Question

Use a graph to estimate the coordinates of the rightmost point on the curve \(x = t - {t^6},y = {e^t}\).Then use calculus to find the exact coordinates.

Short answer

The exact coordinates of the rightmost point is: \(\left( {\left( {\frac{1}{{\sqrt(5){6}}} - \frac{1}{{6\sqrt(5){6}}}} \right),{e^{\frac{1}{{\sqrt(5){6}}}}}} \right)\) .

Step-by-step solution

Sketch the graph of the given curve

The graph of the given curve \(x = t - {t^6},y = {e^t}\) can be given by as follows:

From the graph we can see the rightmost point is: \((0.6,2)\)

Find the point for vertical tangent line 

Substitute \(\frac{{dx}}{{dt}} = 0\) to find the vertical line

\(\begin{array}{l}x = t - {t^6}\\\frac{{dx}}{{dt}} = 1 - 6{t^5}\\\frac{{dx}}{{dt}} = 0\\1 - 6{t^5} = 0\\6{t^5} = 1\\{t^5} = \frac{1}{6}\,\\t = \frac{1}{{\sqrt(5){6}}}\end{array}\)

Substitute \(t = \frac{1}{{\sqrt(5){6}}}\) and obtained point

\(\begin{array}{l}(x,y) = (t - {t^6},{e^t})\\(x,y) = \left( {\left( {\frac{1}{{\sqrt(5){6}}}} \right) - {{\left( {\frac{1}{{\sqrt(5){6}}}} \right)}^6},{e^{\frac{1}{{\sqrt(5){6}}}}}} \right)\\(x,y) = \left( {\left( {\frac{1}{{\sqrt(5){6}}} - \frac{1}{{6\sqrt(5){6}}}} \right),{e^{\frac{1}{{\sqrt(5){6}}}}}} \right)\end{array}\)

Hence, The exact coordinates of the rightmost point of the given curve is: \(\left( {\left( {\frac{1}{{\sqrt(5){6}}} - \frac{1}{{6\sqrt(5){6}}}} \right),{e^{\frac{1}{{\sqrt(5){6}}}}}} \right)\) .