Question

Indicate whether the first function of each of the following pairs has a smaller, same or larger order of growth (to within a constant multiple) than the second function. Justify your answer
1. n(n+1) and 2000 n
2 + 34 n 2. In n and Ign
3. 2n-1 and 2n
4. 2 n² and 0.001 n3 - 2 n

Answer 1

Answer:

1. The first function $n^2 + n$ has the same order of growth as the second function $2000n^2 + 34n$ within a constant multiple.

2. The first $ln(n)$  and the second $log(n)$ logarithmic functions have the same order of growth within a constant multiple.

3. The first function $\frac{1}{2}2^{n}$ has the same order of growth as the second function $2^n$ within a constant multiple.

4. The first function $2n^2$ has a smaller order of growth as the second function $0.001n^3 - 2n$ within a constant multiple.

Explanation:

The given functions are

1. $n(n +1 )$ and $2000n^2 + 34n$

2. $ln(n)$ and $log(n)$

3. $2^{n-1}$ and $2^n$

4. $2n^2$ and $0.001n^3 - 2n$

The First pair:

$n(n +1 )$ and $2000n^2 + 34n$

The first function can be simplified to

$n(n +1 ) \\\\(n \times n) + (n\times1)\\\\n^2 + n$

Therefore, the first function $n^2 + n$ has the same order of growth as the second function $2000n^2 + 34n$ within a constant multiple.

The Second pair:

$ln(n)$ and $log(n)$

As you can notice the difference between these two functions is of logarithm base which is given by

$log_a \: n = log_a \: b\: log_b \: n$

Therefore, the first $ln(n)$  and the second $log(n)$ logarithmic functions have the same order of growth within a constant multiple.

The Third pair:

$2^{n-1}$ and $2^n$

The first function can be simplified to

$2^{n-1} \\\\\frac{2^{n}}{2} \\\\\frac{1}{2}2^{n}$

Therefore, the first function $\frac{1}{2}2^{n}$ has the same order of growth as the second function $2^n$ within a constant multiple.

The Fourth pair:

$2n^2$ and $0.001n^3 - 2n$

As you can notice the first function is quadratic and the second function is cubic.

Therefore, the first function $2n^2$ has a smaller order of growth as the second function $0.001n^3 - 2n$ within a constant multiple.