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anonymous

  • one year ago

Let 'f' be differentiable at c. Let y = ax + b be the equation of the tangent line to the graph of 'f' at point [c,f(c)]. Prove that Lim F(x) - (ax + b) = 0 x->c x - c

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  1. dan815
    • one year ago
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    whats with the x-c on the bottom there

  2. anonymous
    • one year ago
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    Treat that limit as a fraction i.e x -c is the denominator

  3. dan815
    • one year ago
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    is f and F the same function in the question

  4. anonymous
    • one year ago
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    Ah sorry, i must have typed it up wrong >< it's f(x) - (ax + b)

  5. dan815
    • one year ago
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    so u know this tangent line passes through the point c,f(c)

  6. dan815
    • one year ago
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    we have an indeterminate form 0/0 u can use lhopitals

  7. anonymous
    • one year ago
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    okay

  8. dan815
    • one year ago
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    or split into 2 cases, when f(x) degree <1 and f(x) degree>1

  9. dan815
    • one year ago
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    |dw:1433324646253:dw|

  10. dan815
    • one year ago
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    |dw:1433324690567:dw|

  11. anonymous
    • one year ago
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    Ah okay

  12. dan815
    • one year ago
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    |dw:1433324730086:dw|

  13. dan815
    • one year ago
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    are you allowed to use L'hopitals though?

  14. anonymous
    • one year ago
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    yeah, I was about to substitute the points on the tangent into the differentiated limit to solve for: lim x->c

  15. anonymous
    • one year ago
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    yeah absolutely

  16. dan815
    • one year ago
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    ah i see

  17. dan815
    • one year ago
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    well here is the intuition for it

  18. dan815
    • one year ago
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    when you are very close to the c

  19. dan815
    • one year ago
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    you are travelling on your graph close to the tangent line, where as in the denomiator u are travelling horizotally towards C

  20. dan815
    • one year ago
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    so technically if u are going towards C on the tangent line itself the numerator shud be approaching 0 much faster than the denominator

  21. dan815
    • one year ago
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    eh that doesnt sound very clear lol xD forget it, if its confusing u

  22. anonymous
    • one year ago
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    yeah I know what you mean, I graphically picture this Thanks for the help by the way ^^

  23. dan815
    • one year ago
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    |dw:1433324933036:dw|

  24. anonymous
    • one year ago
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    no no, I know what's going on. It's just this particular question, I haven't gotten much practice, since I like doing the easy L'hopital questions haha

  25. dan815
    • one year ago
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    like u see the values on the graph are much closed to the tangent line as small distances away

  26. dan815
    • one year ago
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    closer*

  27. dan815
    • one year ago
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    at small distances

  28. dan815
    • one year ago
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    because the smaller distances it gets the better a linear approximation is to this graph f(x)

  29. dan815
    • one year ago
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    infact it gets infintessimally better than the horizotal way of approaching on the number line

  30. dan815
    • one year ago
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    |dw:1433325051734:dw|

  31. dan815
    • one year ago
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    there are excepts to this way of looking at it though, for example of the graph f(x) being a horizontal line too

  32. dan815
    • one year ago
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    k well cheers

  33. amilapsn
    • one year ago
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    use these: \[F^/(c)=a=\lim_{x\rightarrow c }\frac{f(x)-f(c)}{x-c}\\\frac{y-F(c)}{x-c}=a\text{ which will be the equation of the tangent line..... }\]

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