Le^x-(1+x/1+x^2/2)l <= e/6 for all x E [0,1].

squenshl · May 5, 2009

I have a problem.
How do I show that le^x-(1+x/1!+x^2/2!)l <= e/6 for all x E [0,1].

Phrak · May 5, 2009

Show that the equation is true at the boundries. Show that the term on the left is monotonic in x (if, indeed, it is).

adriank · May 6, 2009

What a strange coincidence. I saw this thread only after posting my example in this thread.

If you don't want to see the full solution, here's a hint: Taylor's theorem.

ice109 · May 8, 2009

how to show that a function is monotonic increasing? show that f(x+1) >= f(x) for all x?

dx · May 8, 2009

ice109 said:

how to show that a function is monotonic increasing? show that f(x+1) >= f(x) for all x?

No, that won't work. A function is monotonic increasing if f(x) ≥ f(y) whenever x ≥ y.

Le^x-(1+x/1+x^2/2)l <= e/6 for all x E [0,1].

Discussion

Similar threads

Solve Image of f(x)=-2(x+1)/(x-1)^2 in x from [0,1)U(1,3]

Solving ln((x-1)/(x-3))=2

Undergrad How to think about x/x, or (x-1)/(x-1) etc.

Undergrad Why is [x^x^x]' NOT (xln(x)^2+ln(x)+xln(x)+1)*x^(x+x^x)?

Natural logs solve ln⁡((x-1)/(x-3))=2

Undergrad Why is ∫x/(1+x^2)dx = 1/2ln(1+x^2)?

Graduate Integrating (1 + x²)/((1 - x²)√(1 + x⁴))

Finding the rate of change of f(x) = 1/(x² + 1) at x = 2

Proving convexity of f(x)=(1+x)^n and estimating with tangent line

Range of $\frac{1}{x} + x = 1/x^2 + x$ is all R