Results 1 to 4 of 4

Thread: Calculus ProofS Help

  1. #1
    Newbie
    Joined
    Nov 2006
    Posts
    22

    Calculus ProofS Help

    Hey, I need help on a couple of problems in my proof class. If anyone can help, it would be highly appreciated, Thank you!

    1. Prove that S = { n−1 |n ∈ N} is bounded above and that its supremumn
    is equal to 1.
    2.Use the Intermediate Value Theorem to show that the polynomial x4 +
    x3 − 9 has at least two real roots.
    3.Use the Mean Value Theorem to prove that |sin(b) − sin(a)| ≤ |b − a|
    for all a, b ∈ R.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Forum Admin topsquark's Avatar
    Joined
    Jan 2006
    From
    Wellsville, NY
    Posts
    11,134
    Thanks
    723
    Awards
    1
    Quote Originally Posted by Swamifez View Post
    2.Use the Intermediate Value Theorem to show that the polynomial x4 +
    x3 − 9 has at least two real roots.
    Let $\displaystyle f(x) = x^4+x^3-9$.

    Note that for large, negative x f(x) takes a positive value.
    f(0)= -9 is negative. Thus there is at least one root between a large negative x and x = 0.
    For large, positive x f(x) takes a positive value again. Thus there is at least one root between x = 0 and large a large positive x.

    Thus there are at least 2 real roots.

    -Dan
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Forum Admin topsquark's Avatar
    Joined
    Jan 2006
    From
    Wellsville, NY
    Posts
    11,134
    Thanks
    723
    Awards
    1
    Quote Originally Posted by Swamifez View Post
    1. Prove that S = { n−1 |n ∈ N} is bounded above and that its supremumn
    is equal to 1.
    N = {1, 2, 3,...} yes? Then S = {0, 1, 2, ...}. This set is NOT bounded above!

    -Dan

    Ah wait! Are you defining $\displaystyle S = \{ n^{-1} | n \in N \}$?

    Then show that for $\displaystyle p < q$ $\displaystyle p^{-1} > q^{-1}$ ($\displaystyle p, q \in N$). So the largest value in S is going to be for the smallest value in N: n = 1. Thus max(S) = 1. Since S has a maximum element sup(S) = max(S) (I believe. Someone please correct me if I'm wrong here!)

    -Dan
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    Quote Originally Posted by Swamifez View Post
    3.Use the Mean Value Theorem to prove that |sin(b) − sin(a)| ≤ |b − a|
    for all a, b ∈ R.
    If $\displaystyle b=a$ then the result is trivial (I love saying that).

    If $\displaystyle b\not = a$ then without loss of generality (I love saying that too) you can assume $\displaystyle b>a$
    Then, this $\displaystyle [a,b]$ is a closed interval.
    Consider the function, $\displaystyle f(x)=\sin x$ defined on this interval. The function is continous on $\displaystyle [a,b]$ and differenciable on $\displaystyle (a,b)$ this satisfies the conditions of Lagrange's Mean Value Theorem. Thus, there exists a $\displaystyle c\in (a,b)$ such as,
    $\displaystyle \frac{\sin b-\sin a}{b-a}=f'(c)=cos c$
    Thus,
    $\displaystyle \left| \frac{\sin b-\sin a}{b-a} \right|=|\cos c|\leq 1$
    Thus,
    $\displaystyle |\sin b-\sin a|\leq |b-a|$
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 5
    Last Post: Mar 10th 2013, 05:13 PM
  2. Replies: 1
    Last Post: Feb 11th 2010, 07:09 AM
  3. Replies: 1
    Last Post: Jun 23rd 2008, 09:17 AM
  4. More calculus ideas and proofs.
    Posted in the Calculus Forum
    Replies: 6
    Last Post: Nov 19th 2007, 10:16 AM
  5. Replies: 3
    Last Post: Oct 6th 2007, 02:01 PM

Search Tags


/mathhelpforum @mathhelpforum