Suppose that we are interested in the quantity $ g(x) = 2 + x - 3x^3 + 4 \sqrt{x} $ for large $ x $'s. Write the "best" big-oh notation. (Here, "best" means that it must be the simplest and best describe the quantity.)

For large x, the dominant term in . Therefore, the best big-oh notation for g(x) is O().To see this…

Answered: Suppose that we are interested in the…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Suppose that we are interested in the quantity $ g(x) = 2 + x - 3x^3 + 4 \sqrt{x} $ for large $ x $'s. Write the "best" big-oh notation. (Here, "best" means that it must be the simplest and best describe the quantity.)

Expert Solution

Step 1: problem solution

For large $x$ , the dominant term in $g left parenthesis x right parenthesis equals 2 plus x minus 3 x cubed plus 4 square root of x space text is end text space minus 3 x cubed$ . Therefore, the best big-oh notation for $g (x)$ is $x cubed$ .

To see this more formally, we can use the following inequality: $long dash 2 space plus space x space minus space 3 x cubed space plus space 4 square root of x space less or equal than space vertical line x vertical line space plus space vertical line 3 x cubed vertical line space plus space vertical line 4 square root of x vertical line$

For large $x$ , the terms |x| and |4x| are asymptotically negligible compared to $open vertical bar 3 x cubed close vertical bar$ . Therefore, we can say that:

$open vertical bar long dash 2 space plus space x space minus space 3 x cubed space plus space 4 square root of x close vertical bar space less or equal than space space vertical line 3 x cubed vertical line$

This means that $x cubed$ .

In other words, for any positive constant $c$ , there exists a positive constant $k$ such that for all $x> k$ , we have:

$open vertical bar g left parenthesis x right parenthesis close vertical bar less or equal than c open vertical bar x cubed close vertical bar$