🚀 go-pugleaf

Unless something got distorted during transmission, I saw that an

                inequality (namely exp(x)>x  --  see below)
was called an
                equation.

For your students' sake, get the terminology straight. (Some of my
students do that, too; even calling expressions such as 3*x-5 equations. 
I wonder why...)

 And: some inequalities are satisfied everywhere (such as exp(x)>x for 
real x), some on proper subsets of the declared or assumed domain.
 How does one prove that exp(x)>x everywhere? Not merely by plotting many
points of the graph (that is good as a hint), but in this case, by 
finding (or estimating from below) the minimum of exp(x)-x. This is a 
good place for using Calculus. (BTW, the minimum is 1. Why?)

 And yes, solving inequalities such as exp(x)>5*x is helped by finding the 
solutions of exp(x)=5*x first, but the students should have heard 
at least something about continuity, differentiability and their 
consequences (Rolle's Theorem to show that there are no more than 2 
solutions, etc.)

 How much to put into the exam depends, of course, on how much Calculus
information and skills the students have acquired. Is it really Calculus,
or are the students still in the (often regrettably underpracticed)
pre-Calculus part? 

Good luck, ZVK (Slavek).

In article <5k5u04$kv2$1@qualcuno.nettuno.it>,
Stefano Baraldi <baro@pt.tizeta.it> wrote:
:Hi all!
:
:I'm going to give my Calculus exam next month, i still have real
:doubts about solving equations of this type (i don't knoe their exact
:name)
:
:exp(x)>x,    ln(x+a)<tan(x), ....
:
:and so on...
:I know the solution should be found graphically, please could you
:point me to books (available online) or tell me the method to do it?
:
:Up to now i made a simple comparing algorithm on my sci-calculator,
:but that's not so ortodox.. :))
:
:thank you and bye!
[...]