Math Is Fun Forum

samuel12

Guest

calculus

Hi just have a quick question and need some guidance,

Use calculus to show that x<=e^x   (for any x element of R)

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: calculus

Hi samuel12;

For x < 0 it is obvious, the LHS is negative and the RHS is positive:

For x>=0, Taylorize e^x. The Taylor series for e^x converges for all x ∈ R.

Subtract x from both sides:

which is obviously true.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,583

Re: calculus

'Use calculus'

When x = 0, e^x = 1 which is greater.

d(x)/dx = 1 for all x

d(e^x)/dx = e^x > 0 for all x => y = e^x is an increasing function.(always goes up never down)

So y = e^x starts bigger and goes on getting bigger (at a faster rate) so y = x will never overtake it.

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

samuel12

Guest

Re: calculus

thanks for the help

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: calculus

Hi samuel12;

Welcome to the forum!

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.