Find Order Notation of x√(1+x^2)

• coverband
In summary, the teacher discussed the use of big-O notation in finding the order of an expression, specifically x\sqrt{1+x^2}=x+\frac{1}{2}x^3...(= O(x)). They explained that the first equation is an expansion of the function x\sqrt{1 + x^2} in a Taylor series at zero, while the second equation represents the bound of the series by a constant multiple of x. This notation is only applicable near zero and is written as x\sqrt{1 + x^2} = O(x) as x approaches 0.
coverband
Teacher has:

$$x\sqrt{1+x^2}=x+\frac{1}{2}x^3...(= O(x))$$

in finding order notation of expression.

How is L.H.S. equal to R.H.S.?

The first equation is an expansion of the function $$x\sqrt{1 + x^2}$$ in a Taylor series at zero, using the binomial series.

The second equation is a statement that the series expansion is bounded by a constant multiple of $$x$$. This is true only near zero, not near infinity (near infinity, $$\sqrt{1 + x^2}$$ looks like $$|x|$$, so $$x\sqrt{1 + x^2}$$ grows quadratically). Near zero, the written-out version of the big-O notation is: there exist constants $$C > 0$$ and $$\delta > 0$$ so that, whenever $$|x| < \delta$$, $$|x\sqrt{1 + x^2}| < C|x|$$. This statement is abbreviated $$x\sqrt{1 + x^2} = O(x) \textrm{ as } x \to 0$$.

1. What is the purpose of finding the order notation of x√(1+x^2)?

Finding the order notation of a function helps us understand its growth rate and how it behaves as its input approaches infinity. This is useful in analyzing and comparing the efficiency of algorithms or solving problems in mathematics and computer science.

2. How do you determine the order notation of x√(1+x^2)?

The order notation of a function is determined by its dominant term, which is the term with the highest power. In this case, the dominant term is x√(1+x^2), so the order notation is √x or O(√x).

3. Can x√(1+x^2) be simplified further?

No, x√(1+x^2) is already in its simplest form. However, it can be rewritten as x^(3/2) + x^(1/2), which may be easier to understand in some cases.

4. What is the significance of the x√(1+x^2) function in mathematics?

The x√(1+x^2) function is commonly used in calculus and differential equations to model various physical phenomena, such as the velocity of a falling object under the influence of gravity.

5. Are there any real-world applications of x√(1+x^2)?

Yes, x√(1+x^2) can be used to calculate the distance between two points in a three-dimensional space. It is also used in calculating the arc length of a curve in calculus and in the calculation of electric potential in physics.

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