   Chapter 10.2, Problem 21E

Chapter
Section
Textbook Problem

# Use a graph to estimate the coordinates of the rightmost point on the curve x = t − t6, y = et. Then use calculus to find the exact coordinates.

To determine

To plot: The graph for the given parametric equations x=tt6 and y=et to estimate the rightmost point on the curve and exact coordinates.

Explanation

Given:

The parametric equation for the variable x is as follows.

x=tt6 (1)

The parametric equation for the variable y is as follows.

y=et (2)

Calculation:

Differentiate the parametric equation x with respect to t .

x=tt6dxdt=16t5

Differentiate the parametric equation y with respect to t .

y=etdydt=d(et)dt=et

Write the chain rule for dydx .

dydx=dydtdxdt

Substitute (et) for dydt and (16t5) for dxdt in the above equation.

dydx=(et)(16t5)

Substitute 1 for t in equation (1).

x=tt6=(1)(1)6=0

Substitute 1 for t in equation (2).

y=et=e(1)=2.718

The values of x and y for each step value of t is tabulated in the below table.

 t 0.5 1 1.5 2 x 0

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