Math 162

Class 3

We will study this function:

\[f(x) = \int_1^x \frac{1}{t}\;dt\]

What are its properties?

Properties

  • Domain:

  • Derivative:

x y (1 , 1) 1 x, 1 x x f ( x ) 1 x ,x 1 x − f 1 x x, 1 x x11 x, 1 x x f ( x ) + 1 1 x ,x 1 xx1 − f 1 x + 1 x, 1 x x f ( x ) + 1 1 x ,x x − f 1 x + 1 − f 1 x + 1 = f ( x ) + 1

Limit at \(\infty\)

x (1 , 1) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 2 , 1 2 4 , 1 4 8 , 1 8 16 , 1 16

\(f(2) > 1\cdot \frac{1}{2} = \frac{1}{2}\)

\(f(4) > \frac{1}{2} + 2\cdot \frac{1}{4} = \frac{2}{2}\)

\(f(8) > \frac{2}{2} + 4\cdot \frac{1}{8} = \frac{3}{2}\)

\(f(16) > \frac{3}{2} + 8\cdot \frac{1}{16} = \frac{4}{2}\)

\(f\left(2^n\right) > \frac{1}{2} + \frac{1}{2} + \cdots + \frac{1}{2} = \frac{n}{2}\)

Also, \(f\) is increasing, so if \(x > 2^n\), \(f(x) > \frac{n}{2}\)

Therefore \[ \lim_{x\to\infty} f(x) = \infty\]

The Range

Another interesting property

x (1 , 1) 1 xyx

\(\displaystyle f(xy) = \int_1^{xy} \frac{1}{t}\;dt\) \(\displaystyle {} = \int_1^x \frac{1}{t}\;dt + \int_x^{xy} \frac{1}{t}\;dt\)

A similar property

If \(r\) is a rational number, what is \(f\left(x^r\right)\)?

Properties

  • Domain: \((0,\infty)\)

  • \(f(1) = 0\)

  • \(f'(x) = \frac{1}{x}\)

  • \(f\) is increasing.

  • \(\displaystyle \lim_{x\to\infty} f(x) = \infty\)

  • \(\displaystyle \lim_{x\to 0} f(x) = -\infty\)

  • \(f(xy) = f(x) + f(y)\)

  • If \(r\) is rational, then \(f\left(x^r\right) = rf(x)\).

This function already has a name: the natural logarithm!