Mathematics Question Review (2025-12-16)

Question:

Show that the line x+y=1 is the axis of symmetry of both y=e^x and y=ln(x).

Solution:

The shortest distance between y=e^x and y=ln(x) is the distance between A(1,0) and B(0,1). Since the shortest distance must be straight line cutting y=x in equal halves, forming perpendicular bisector of line AB, the slope of the required equation must be -1.

QED.

What 7

Mutant form of the question:

Proofread version:

Another quesion:

Show that e^x and ln(x) diverge for x>1.

Solution:

Since e^x is the inverse function of ln(x),

if ln(x) diverges, e^x must diverge as well.

Now we are to show that ln(x) diverges:

For a increasing function to converge,

it must have a upper bound,

and it must be monotonic increasing.

Now we show ln(x) has no upper bound:

Assuming there is a upper bound y=l,

where l is a constant.

Since ln'(x) = 1/x >0,

and ln(x) is an analytic function for x>1,

the slope of ln(x) can never be zero,

i.e. for sufficiently large value of l,

ln(l) = l, i.e. they cut each other only once,

then given the domain x>1，

ln(x) is bijective and thus diverges.

i.e. for sufficiently large value of l,

ln(x) = l,

if x=l, then they must be infninty.

infinity*

Last question:

What is the use of partial differential equations (PDE) in Feng Shui?

Solution:

Differential Equations form direction fields. In Ordinary ones, the fields are in 2D. For PDE, the fields can be exhibited in 3D or beyond.

Ref: