Math 162
Class 3
We will study this function:
f(x)=∫x11tdt
What are its properties?
Properties
Domain:
Derivative:
Limit at ∞
f(2)>1⋅12=12
f(4)>12+2⋅14=22
f(8)>22+4⋅18=32
f(16)>32+8⋅116=42
f(2n)>12+12+⋯+12=n2
Also, f is increasing, so if x>2n, f(x)>n2
Therefore limx→∞f(x)=∞
The Range
Another interesting property
f(xy)=∫xy11tdt =∫x11tdt+∫xyx1tdt
A similar property
If r is a rational number, what is f(xr)?
Properties
Domain: (0,∞)
f(1)=0
f′(x)=1x
f is increasing.
limx→∞f(x)=∞
limx→0f(x)=−∞
f(xy)=f(x)+f(y)
If r is rational, then f(xr)=rf(x).
This function already has a name: the natural logarithm!
Math 162 Class 3