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Math Help - First Principles Calculus

  1. #1
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    First Principles Calculus

    Ok i got this homework sheet out.
    There are a LOT of questions to do.

    I have had one lesson on calculus 1st principles and I'm a bit confused.

    I've attached a file of the sheet. For the second question would I use first principles? It doesn't say and that is a LOT of working for the first question 2a) I was nearly over two pages.

    (Yeh I'm new to Advanced Higher course in scotland!)

    Attached Thumbnails Attached Thumbnails First Principles Calculus-1.12.jpg  
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  2. #2
    Bar0n janvdl's Avatar
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    Quote Originally Posted by shadowbeam View Post
    Ok i got this homework sheet out.
    There are a LOT of questions to do.

    I have had one lesson on calculus 1st principles and I'm a bit confused.

    I've attached a file of the sheet. For the second question would I use first principles? It doesn't say and that is a LOT of working for the first question 2a) I was nearly over two pages.

    (Yeh I'm new to Advanced Higher course in scotland!)

    With all due respect, you must be dreaming if you think we are going to do your homework for you.

    It would appear you do not have to use first principles for question 2.
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  3. #3
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    Post

    Also for question 1d) and the like I struggle to understand them.
    Here's my working so far:
    lim h->0 = 1/(x+h) - 1/x
    = x/x(x+h) - (x+h)/x(x+h)
    = x - (x+h)/x(x+h)
    =h/x(x+h)

    ???

    Am I right? Hard to type the answer. I could scan my working to show you.
    Oh and please i don't want people to do my homework for me! I just need a little guidance with question wording and what they're asking for.
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  4. #4
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by shadowbeam View Post
    Ok i got this homework sheet out.
    There are a LOT of questions to do.

    I have had one lesson on calculus 1st principles and I'm a bit confused.

    I've attached a file of the sheet. For the second question would I use first principles? It doesn't say and that is a LOT of working for the first question 2a) I was nearly over two pages.

    (Yeh I'm new to Advanced Higher course in scotland!)

    I assume that you mean f'(x)=\lim_{\Delta{x}\to{0}}\frac{f(x+\Delta{x})-f(x)}{\Delta{x}}

    And secondly I would suggest emailing your instructor, for there is no way we could no


    But if you have only been taught it this way, then I would assume so
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  5. #5
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    aye that's right!
    Except we use an h instead of the whole triangled x
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  6. #6
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by shadowbeam View Post
    Also for question 1d) and the like I struggle to understand them.
    Here's my working so far:
    lim h->0 = 1/(x+h) - 1/x
    = x/x(x+h) - (x+h)/x(x+h)
    = x - (x+h)/x(x+h)
    =h/x(x+h)

    ???

    Am I right? Hard to type the answer. I could scan my working to show you.
    Oh and please i don't want people to do my homework for me! I just need a little guidance with question wording and what they're asking for.
    \lim_{\Delta{x}\to{0}}\frac{\frac{1}{x+\Delta{x}}-\frac{1}{x}}{\Delta{x}}=\lim_{\Delta{x}\to{0}}\fra  c{\frac{-\Delta{x}}{x^2+x(\Delta{x})}}{\Delta{x}}=\lim_{\De  lta{x}\to{0}}\frac{-1}{x^2+x(\Delta{x})}=\frac{-1}{x^2}
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  7. #7
    Eater of Worlds
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    For first principles, plug in x+h for x for the f(x+h) portion. Like so:

    #1: \lim_{h\rightarrow{0}}\frac{\overbrace{3(x+h)^2+2(  x+h)}^{\text{f(x+h)}}-(\overbrace{3x^{2}+2x}^{\text{f(x)}})}{h}

    Simplify and cancel. Then you should see the derivative. Which is obviously

    6x+2
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  8. #8
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    Quote Originally Posted by galactus View Post
    For first principles, plug in x+h for x for the f(x+h) portion. Like so:

    #1: \lim_{h\rightarrow{0}}\frac{\overbrace{3(x+h)^2+2(  x+h)}^{\text{f(x+h)}}-(\overbrace{3x^{2}+2x}^{\text{f(x)}})}{h}

    Simplify and cancel. Then you should see the derivative. Which is obviously

    6x+2
    hmmm yes but that's a lot of working and I've got lots more questions over the page...
    I'm unsure if I'm being expected to 1st princ. it all
    Coz its a lot of simplifying
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  9. #9
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    It looks more ominous than it is. Give it a go. You'll see it as h-->0.
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  10. #10
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    Ah you're right once you've expanded and collected together h's
    it is more simple!

    Would I be expected to do the same with fractional powers??
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  11. #11
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    Oh wait I'm seeing now tooo!!
    in Q2c)
    if you add the powers 4/3 and 1/3 you'll get 5/3

    and you can take away 4/3
    to give 1/3

    ... change into a cube rooot?
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  12. #12
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by shadowbeam View Post
    Ah you're right once you've expanded and collected together h's
    it is more simple!

    Would I be expected to do the same with fractional powers??
    Fractional powers cannot be expanded at the level of beginning calculus, so no. You would not be expected to do it that way
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  13. #13
    Eater of Worlds
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    Let's do a fractional power. It's a little trickier in that you can use the conjugate.

    Take x^{\frac{3}{2}}. Let's find the derivative through first principles.

    \frac{(x+h)^{\frac{3}{2}}-x^{\frac{3}{2}}}{h}\cdot\frac{(x+h)^{\frac{3}{2}}+  x^{\frac{3}{2}}}{(x+h)^{\frac{3}{2}}+x^{\frac{3}{2  }}}

    Multiply through and simplifying:

    \lim_{h\to\rightarrow{0}}\frac{3hx^{2}+3xh^{2}+h^{  3}}{h((x+h)^{\frac{3}{2}}+x^{\frac{3}{2}})}

    Now, as we can see by h-->0, we have :

    \lim_{h\to\rightarrow{0}}\frac{3x^{2}}{(x+h)^{\fra  c{3}{2}}+x^{\frac{3}{2}}}+\lim_{h\to\rightarrow{0}  }\frac{3xh}{(x+h)^{\frac{3}{2}}+x^{\frac{3}{2}}}+\  lim_{h\to\rightarrow{0}}\frac{h^{2}}{(x+h)^{\frac{  3}{2}}+x^{\frac{3}{2}}}

    Which gives us:

    \frac{3x^{2}}{2x^{\frac{3}{2}}}=\boxed{\frac{3\sqr  t{x}}{2}}

    Which is as it should be.

    Does that help?. A wee bit anyway?.
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  14. #14
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    xD Cheers yeh it will help.
    Shall work through that tonight I can't take it in right now *fried brain*

    Thanks for all the help
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  15. #15
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    Urgent Help - Differentiation

    Hi

    Am totally lost, leaning from a book as I need to start my semester in finance next year Jan, am preparing for it, Please guide?

    c=0.004q(to the power3)+20q+5000

    and the demand function is p=450-4q

    Thanks

    Find the profit-maximizing output?
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