I'm in an analysis class currently discussing differentiation. I'm pretty sure I need to use Taylor's Theorem on these but I'm not sure. Could someone work me through them?
First, if x is greater than zero, show that (1+x)^(1/3)-(1+(1/3)x-(1/9)x^2) is less than or equal to (5/81)x^3.
Second, if abs(x) is less than or equal to one, show that abs([sin(x)-(x-(x^3)/6+(x^5)/120]) is less than (1/5040).
Sorry. I know it's quite complicated, but I have no idea. Anyways, thank you for any help you can give.
I really appreciate the help.
For the second one, what would the derivative of abs(sin x) be?
Yeah I end up with at the end that
abs([sin(x)-(x-(x^3)/6+(x^5)/120]) is less than or equal to abs((x^7)/5040).
How would I show that this is just greater than the expression on the left (and not equal)?
As x^7<=1 for all x such that |x|<=1, so you have abs([sin(x)-(x-(x^3)/6+(x^5)/120])<=1/5040.
Originally Posted by 12MED34
Equality can only occur when |x|=1, as if |x|<1, then the bound can be made tighter. So you need only check that equality does not apply at x=+/-1.