Higher Derivatives.

[pandaLOVE]

Hello, I need help with a problem.
The problem first asks me to calculate the first five derivatives of f(x)=sqrt(x), which I was able to do.
f(x)'=1/2x[sup:2235jim7]-1/2[/sup:2235jim7]
f(x)''=-1/4x[sup:2235jim7]-3/2[/sup:2235jim7]
f(x)'''=3/8x[sup:2235jim7]-5/2[/sup:2235jim7]
f(x)[sup:2235jim7]4[/sup:2235jim7]=-15/16x[sup:2235jim7]-7/2[/sup:2235jim7]
and so on..

Now I need help with the question that follows it.
a) show that f[sup:2235jim7](n)[/sup:2235jim7](x) is a multiple of x[sup:2235jim7]-n+1/2[/sup:2235jim7]
b) show that f[sup:2235jim7](n)[/sup:2235jim7](x) alternates in sign as (-1)n-1 for n > 1

I just don't understand what the questions are asking me to do.
Thank you in advanced!

 

[Deleted member 4993]

Hello, I need help with a problem.
The problem first asks me to calculate the first five derivatives of f(x)=sqrt(x), which I was able to do.
f(x)'=1/2x[sup:1232pxai]-1/2[/sup:1232pxai]

f[sup:1232pxai](1)[/sup:1232pxai](x) = ½ * x[sup:1232pxai](-1 + ½)[/sup:1232pxai]

f(x)''=-1/4x[sup:1232pxai]-3/2[/sup:1232pxai]

f[sup:1232pxai](2)[/sup:1232pxai](x) = (-1)[sup:1232pxai]2-1[/sup:1232pxai] * ¼ * x[sup:1232pxai](-2 + ½)[/sup:1232pxai]

f(x)'''=3/8x[sup:1232pxai]-5/2[/sup:1232pxai]

f[sup:1232pxai](3)[/sup:1232pxai](x) = (-1)[sup:1232pxai]3-1[/sup:1232pxai] * 3/8 * x[sup:1232pxai](-3 + ½)[/sup:1232pxai]

f(x)[sup:1232pxai]4[/sup:1232pxai]=-15/16x[sup:1232pxai]-7/2[/sup:1232pxai]
and so on..

Now I need help with the question that follows it.
a) show that f[sup:1232pxai](n)[/sup:1232pxai](x) is a multiple of x[sup:1232pxai]-n+1/2[/sup:1232pxai]
b) show that f[sup:1232pxai](n)[/sup:1232pxai](x) alternates in sign as (-1)n-1 for n > 1

I just don't understand what the questions are asking me to do.
Thank you in advanced!

Have you done proof by induction - that would be another method....

 

[pandaLOVE]

Sorry for the late reply, but thank you for your help!
Based on what you've given me, I think I understand what the question is trying to ask me now

As for the proof by induction method.. I don't think we've gone over that in class yet..
so I wouldn't know how to use it here.

 

[BigGlenntheHeavy]

Is this what you wanted?

\(\displaystyle \sum_{n=0}^{\infty}\frac{(-1)^{n}(2n)!}{2^{2n+1}(n!)x^{(2n+1)/2}} = \frac{1}{2x^{1/2}}-\frac{1}{4x^{3/2}}+\frac{3}{8x^{5/2}}-\frac{15}{16x^{7/2}}+\frac{105}{32x^{9/2}}-\frac{945}{64x^{11/2}}+\frac{10395}{128x^{13/2}}-\frac{135135}{256x^{15/2}}+...\)

\(\displaystyle The \ above \ should \ answer \ all \ your \ questions.\)