7) Use your calculator to determine the point on the graph of y=Vx that is closest to the pointP (2, 3).

y=x3 and point (2,3) Find closest point on the graph y=x3 Every point on the curve has the form…

Answered: 7) Use your calculator to determine the…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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7) Use your calculator to determine the point on the graph of y=Vx that is closest to the point
P (2, 3).

Expert Solution

Step 1 given

$y = \sqrt[3]{x}$ and point (2,3)

Find closest point on the graph $y = \sqrt[3]{x}$

Every point on the curve has the form $(x, \sqrt[3]{x})$

The closest point is the one whose distance is minimum.

The distance between $(x, \sqrt[3]{x}) a n d (2, 3) i s d = \sqrt{{(x - 2)}^{2} + {(\sqrt[3]{x} - 3)}^{2}}$

Now we need to find derivative of d with respect to x

$\begin{array}{rcl} d' (x) & = & \frac{d}{d x} \sqrt{({(x - 2)}^{2} + {(\sqrt[3]{x} - 3)}^{2})} \\ = & \frac{- 3 + \sqrt[3]{x} + 3 x^{\frac{5}{3}} - 6 x^{\frac{2}{3}}}{3 x^{\frac{2}{3}} \sqrt{x^{2} - 4 x + x^{\frac{2}{3}} - 6 \sqrt[3]{x}} + 13} \end{array}$

now set d'(x) equal to zero and then find value of x

So we get

$\begin{array}{rcl} \frac{- 3 + \sqrt[3]{x} + 3 x^{\frac{5}{3}} - 6 x^{\frac{2}{3}}}{3 x^{\frac{2}{3}} \sqrt{x^{2} - 4 x + x^{\frac{2}{3}} - 6 \sqrt[3]{x} + 13}} & = & 0 \\ - 3 + \sqrt[3]{x} + 3 x^{\frac{5}{3}} - 6 x^{\frac{2}{3}} & = & 0 \\ x & = & 2.31895 \end{array}$

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