- #1
asif zaidi
- 56
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I have two problem relating to integrals and derivatives.
Integrals: I can do but I am confused about the limit
Derivative: I am having problem. Need some help with that. I have a solution but I think it is wrong.
For each problem I have posted step by step what I am doing. Hopefully it will be easier in answering.
Problem 1: Integral
What is [tex]\int e ^{-t}[/tex] (limit is from 0 (on bottom) to 1 (on top))
Problem 1: My solution
My issue is step 6
1. Take u = -t
2. Use Chain Rule: dy/dx = (dy/du)(du/dx)
3. du/dt = -1. Therefor -du = dt
4. Find limits:
For t=1: u= -1
For t=0: u= 0
5. New integral becomes (-1) [tex]\int e ^{u} du [/tex].
6. My question/issue is about the new limits:
Do I have to evaluate the integral with u=-1 on bottom of integral and u=0 on top of integral or the other way.
Problem 2 Statement:
y= [tex]x^{x^x}[/tex]. Find the minimum of y.
Problem 2 Solution
I am not really sure that I am doing this right. So please advise
I know how to find the derivative of [tex] x^x [/tex].
So here are my steps
1. make a = [tex] x^x [/tex] (I am taking the first two x's)
2. Therefore y = [tex] a^x [/tex]
3. dy/dx = (dy/da)(da/dx)
4. Find dy/da:
6. Multiple from step 4,5 and equate to 0.
My question is: is step 4 right? Actually now that I am looking at it, I am wondering if step 5 is right also. Plz advise on this.
Thanks
Asif
Integrals: I can do but I am confused about the limit
Derivative: I am having problem. Need some help with that. I have a solution but I think it is wrong.
For each problem I have posted step by step what I am doing. Hopefully it will be easier in answering.
Problem 1: Integral
What is [tex]\int e ^{-t}[/tex] (limit is from 0 (on bottom) to 1 (on top))
Problem 1: My solution
My issue is step 6
1. Take u = -t
2. Use Chain Rule: dy/dx = (dy/du)(du/dx)
3. du/dt = -1. Therefor -du = dt
4. Find limits:
For t=1: u= -1
For t=0: u= 0
5. New integral becomes (-1) [tex]\int e ^{u} du [/tex].
6. My question/issue is about the new limits:
Do I have to evaluate the integral with u=-1 on bottom of integral and u=0 on top of integral or the other way.
Problem 2 Statement:
y= [tex]x^{x^x}[/tex]. Find the minimum of y.
Problem 2 Solution
I am not really sure that I am doing this right. So please advise
I know how to find the derivative of [tex] x^x [/tex].
So here are my steps
1. make a = [tex] x^x [/tex] (I am taking the first two x's)
2. Therefore y = [tex] a^x [/tex]
3. dy/dx = (dy/da)(da/dx)
4. Find dy/da:
y = [tex] a^x [/tex]
= [tex] e^{xlna} [/tex]
dy/da = [tex] a^x [/tex] d(xlna)/da = [tex] (a^x) [/tex] [tex] x^{1-x} [/tex]
5. Find da/dxa = [tex] x^x [/tex] = [tex] e^{xlnx} [/tex]
By productrule da/dx = [tex] x^x[/tex] (ln x + 1)
6. Multiple from step 4,5 and equate to 0.
My question is: is step 4 right? Actually now that I am looking at it, I am wondering if step 5 is right also. Plz advise on this.
Thanks
Asif