2 questions, 1 proof of e and 1 find dimension of a box?

Limits, differentiation, related rates, integration, trig integrals, etc.
trey5498
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2 questions, 1 proof of e and 1 find dimension of a box?

Here are the 2 questions I am having issues with. I have no idea on number 1

Question1:
A rectangular box without top is made from a metal sheet of 20x20 by removing a square from each corner and by folding the four side as in the following figure. Find the dimension of the square being removed that maximizes the volume of the box. Each corner is x by x.

Question 2:
Use lim n->inf (1+1/n) = e or its variation and logarithm to prove lim n->inf (1+x/n) = e^x for all x.

Any help would be great appreciated.

nona.m.nona
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Re: 2 questions, 1 proof of e and 1 find dimension of a box?

Question1: A rectangular box without top is made from a metal sheet of 20x20 by removing a square from each corner and by folding the four side as in the following figure. Find the dimension of the square being removed that maximizes the volume of the box. Each corner is x by x.
This exercise requires nothing past algebra. The set-up would be similar to that which is displayed in the last example on this page. Once you have established your expression for the volume in terms of the variable, you would find the vertex to find the maximizing value.
Question 2: Use lim n->inf (1+1/n) = e or its variation and logarithm to prove lim n->inf (1+x/n) = e^x for all x.
One method might be to substitute 1/m for x/n, so that n = mx, where we assume:

$\displaystyle{\lim_{m\, \to\, \infty}\, \left(1\, +\, \frac{1}{m}\right)^m\, =\, e}$

Then:

$\displaystyle{\lim_{n\, \to\, \infty}\, \left(1\, +\, \frac{x}{n}\right)^n\, =\, \lim_{m\, \to\, \infty}\, \left(1\, +\, \frac{1}{m}\right)^{mx}\, =\, \left(\lim_{m\, \to\, \infty}\, \left(1\, +\, \frac{1}{m}\right)^m\right)^x}$

However, since you are required to use logarithms, you may want instead to proceed along these lines:

$\displaystyle{\left(1\, +\, \frac{x}{n}\right)^n\, =\, L}$

for some value $L$. Take logs of either side:

$\displaystyle{\ln\left( \left(1\, +\, \frac{x}{n}\right)^n\right)\, =\, \ln(L)}$

$\displaystyle{n\, \times\, \left(\ln\left(1\, +\, \frac{x}{n}\right)\right)\, =\, \ln(L)}$

$\displaystyle{\frac{ \ln\left( \left(1\, +\, \frac{x}{n}\right)\right)}{\frac{1}{n}}\, =\, \ln(L)}$

Then try apply l'Hospital's Rule and take the limit of the resulting expression.

trey5498
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Re: 2 questions, 1 proof of e and 1 find dimension of a box?

$\displaystyle{\lim_{m\, \to\, \infty}\, \left(1\, +\, \frac{1}{m}\right)^m\, =\, e}$
The (1+1/m) would have an exponent even if the original did not have an exponent?

nona.m.nona
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Re: 2 questions, 1 proof of e and 1 find dimension of a box?

The (1+1/m) would have an exponent even if the original did not have an exponent?
I'm sorry, but I don't know what you mean by the above. Kindly please reply with clarification. Thank you.

trey5498
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Re: 2 questions, 1 proof of e and 1 find dimension of a box?

the original limit function was (1+x/n) = e and in your work it shows (1+1/n)^x = e. Is there a difference in the two?

nona.m.nona
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Re: 2 questions, 1 proof of e and 1 find dimension of a box?

the original limit function was (1+x/n) = e
No; the original expression, being the limit expression for "e", had no "x" but did have an exponent.

trey5498
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Re: 2 questions, 1 proof of e and 1 find dimension of a box?

Sorry I meant (1+x/n) = e^x is the same as your work? if so what about the (1+1/n) = e?

trey5498
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Re: 2 questions, 1 proof of e and 1 find dimension of a box?

Checking my math on the rectangle problem:

V(x)=x(20-2x)^2
=4x^3-80x^2+400x
V'(x)=12^2-160x+400
4(x-10)(3x-10)=0
3x-10=0
3x=10
x=10/3
x=3.33

I also have another question as well: Find an equation of the tangent line of f(x) = sin^2(x) at 1/2

Here is what I got so far:
sin'x(sinx)+(sinx)sin'x
=cosxsinx+sinxcosx
=2sinxcosx = sin2x = sin2(1/2) = sin(1)

y = sin^2(sin)

there is where I get stuck. I know I have to solve to find y1 and then plug it in y-y1 = sin(x+1/2). See anywhere I messed up or where I can change?

nona.m.nona
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Re: 2 questions, 1 proof of e and 1 find dimension of a box?

Sorry I meant (1+x/n) = e^x is the same as your work?
I'm sorry, but I don't understand this run-on of incomplete sentences.
if so what about the (1+1/n) = e?
I'm sorry, but I don't understand what you are saying. Yes, the two different expressions stand for the two different values. How do you feel this to be incorrect?
Checking my math on the rectangle problem:

V(x)=x(20-2x)^2
=4x^3-80x^2+400x
V'(x)=12^2-160x+400
This should read "V'(x) = 12x^2 - 160x + 400".
4(x-10)(3x-10)=0
3x-10=0
3x=10
x=10/3
x=3.33
Leave this answer in exact form, rather than introducing round-off errors. Your instructor may also require that you check the other solution, confirming it to be invalid within context.
I also have another question as well: Find an equation of the tangent line of f(x) = sin^2(x) at 1/2

Here is what I got so far:
sin'x(sinx)+(sinx)sin'x
What is this? How does it relate to f(x) = (sin(x))^2?
=cosxsinx+sinxcosx
=2sinxcosx = sin2x = sin2(1/2) = sin(1)
Do not differentiate and evaluate at the same time. Do one, and only then do the other.
y = sin^2(sin)
What is this?
there is where I get stuck. I know I have to solve to find y1 and then plug it in y-y1 = sin(x+1/2).
What is "y1"? How does it relate to either of "f(x)" or "y"? What is the source of "sin(x + 1/2)"?

My supposition is that you are attempting to do two or three different things at once, and the resulting confusion is causing you to lose track of your progress and/or goal. Tis better to proceed methodically.

Before beginning, however, one must first determine the meaning of "at 1/2". Is this the value of x or of y? If not specified, kindly please consult with your instructor. Thank you.

trey5498
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Re: 2 questions, 1 proof of e and 1 find dimension of a box?

Good to know that my math, minus the the typo of leaving the x out, is good on the box question. I guess I will will restate the other two questions as we have gotten a bit far from the original.

Question 1:
Use lim n->inf (1+1/n) = e or its variation and logarithm to prove lim n->inf (1+x/n) = e^x for all x.

Again I got confused when you put an exponent on the (1+X/n) side. I have no idea where it can from so I apologize for my confusion. I am trying to understand so I can solve a question like this for any problem. My professor is not the greatest at explaining and this is not in my calculus book.

Question 2:
Find an equation of the tangent line of f(x) = sin^2(x) at 1/2

I know the steps to solve this. Find the derivative, place derivative into the the original equation to find x (which I believe I can skip that part since I know x is 1/2), place X into derivative to find Y. Finally put into Y intercept form ( y-y1=m(x+x1) ). Since I have never done this question with sin I am just not sure about it.

Thank you again for your patience and assistance. I swear I am smarter that I may seem, I just ask questions to make sure I know the problem from all aspects.