Maximizing Limits and Integrals: Solving Three Challenging Problems

In summary: So in summary, for the first question, the limit is equal to half pi, for the second question, the limit is equal to 1/(a+1), and for the third question, the given equality can be shown by using either the product rule or a change of dummy variable.
  • #1
MathematicalPhysicist
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i have three questions:
1) find the limit of [tex]b_n=\frac{1}{\sqrt n^2}+\frac{1}{\sqrt(n^2-1)}+...+\frac{1}{\sqrt(n^2-(n-1)^2)}[/tex]
2) if a is any number greater than -1, evaluate [tex]\lim_{n\rightarrow\infty} \frac{1^a+2^a+...+n^a}{n^{a+1}}[/tex]
3) prove that [tex]\int_{0}^{x}[\int_{0}^{u}f(t)dt]du=\int_{0}^{x}f(u)(x-u)du[/tex]

for the first i got: half pi, and for the second question i got 1/(a+1) is this correct?

for the third question, here's what i did:
[tex]\int_{0}^{x}u'[\int_{0}^{u}f(t)dt]du=[\int_{0}^{u}f(t)dtu]_{0}^{x}-\int_{0}^{x}uf(u)du[/tex] now my question is can i use here a change of dummy variable here for the first integral, from f(t)dt to f(u)du and to get the equality?
 
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  • #2
no one has got anything to say?
 
  • #3
How did you get the first one?
 
  • #4
i used riemann sums here, we have the sum:
[tex]\sum_{k=1}^{n-1}\frac{1}{n}\frac{1}{\sqrt{1-(\frac{k}{n})^2}}+\frac{1}{n}[/tex]
is this correct?
 
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  • #5
loop quantum gravity said:
i have three questions:
1) find the limit of [tex]b_n=\frac{1}{\sqrt n^2}+\frac{1}{\sqrt(n^2-1)}+...+\frac{1}{\sqrt(n^2-(n-1)^2)}[/tex]
2) if a is any number greater than -1, evaluate [tex]\lim_{n\rightarrow\infty} \frac{1^a+2^a+...+n^a}{n^{a+1}}[/tex]
3) prove that [tex]\int_{0}^{x}[\int_{0}^{u}f(t)dt]du=\int_{0}^{x}f(u)(x-u)du[/tex]

for the first i got: half pi, and for the second question i got 1/(a+1) is this correct?

for the third question, here's what i did:
[tex]\int_{0}^{x}u'[\int_{0}^{u}f(t)dt]du=[\int_{0}^{u}f(t)dtu]_{0}^{x}-\int_{0}^{x}uf(u)du[/tex] now my question is can i use here a change of dummy variable here for the first integral, from f(t)dt to f(u)du and to get the equality?
Your first two answers look good.

For the third, I can't make sense of what you've done. What is u'?
Heres a hint:
Define the functions F and G as

[tex]F(x) = \int_{0}^{x} \left( \int_{0}^{u} f(t)dt \right) du[/tex]

[tex]G(x) = \int_{0}^{x}f(u)(x-u)du [/tex]

Find the derivatives of F and G with respect to x. Deduce from this that there is a constant C such that F = G + C.
 
  • #6
u' is the derivative of u wrt u.
i.e du/du=1.
 
  • #7
Oh I see, you used the product rule (integration by parts). That'll work too!

Sure, you can always substitute the dummy variable so long as its different from the one used for the limit of integration, so in this case you'd have to first evaluate the expression,

[tex] \left[ \int_{0}^{u}uf(t)dt \right]_{0}^{x} = \int_{0}^{x} x f(t)dt[/tex]

and then make the substitution.
 
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1. What are the three challenging problems in maximizing limits and integrals?

The three challenging problems in maximizing limits and integrals are the optimization problem, the related rates problem, and the area/volume calculation problem.

2. How do you solve the optimization problem?

The optimization problem involves finding the maximum or minimum value of a function. To solve it, you will need to take the derivative of the function and set it equal to zero. Then, solve for the value of the variable that makes the derivative equal to zero. This value will be the maximum or minimum value of the function.

3. What is the related rates problem and how is it solved?

The related rates problem involves finding the rate of change of one variable with respect to another variable. This can be solved by setting up a proportion between the two variables and using the chain rule to find the derivative. Then, substitute in the given values to solve for the unknown rate of change.

4. How do you solve the area/volume calculation problem?

The area/volume calculation problem involves finding the area or volume of a complex shape or object. This can be solved by breaking the shape or object into smaller, simpler shapes and using integration to find the total area or volume.

5. What are some tips for solving challenging problems in maximizing limits and integrals?

Some tips for solving challenging problems in maximizing limits and integrals include practicing with different types of problems, understanding the concepts and formulas, and breaking the problem into smaller, manageable steps. It is also helpful to double check your work and use a graphing calculator or software to visualize the problem.

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