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dessyj1

  • one year ago

Calculus 1 Question is attached.

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  1. dessyj1
    • one year ago
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    Sorry, my internet connection suddenly slows down when i try to upload a picture.

  2. dessyj1
    • one year ago
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  3. freckles
    • one year ago
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    4d?

  4. dessyj1
    • one year ago
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    sorry number 5

  5. freckles
    • one year ago
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    lol oh that is a choice \[\lim_{h \rightarrow 0}\frac{1}{k}\ln(\frac{2+h}{2})\]

  6. freckles
    • one year ago
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    I didn't realize it was a multiple choice thingy

  7. freckles
    • one year ago
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    \[F'(x)=f(x) \\ \int\limits_a^b f(x) dx=?\] What does F'=f mean? I will give you a hint that F is the ____-derivative of f.

  8. freckles
    • one year ago
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    I will also give you another hint: fundamental theorem of calculus

  9. freckles
    • one year ago
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    for example: how do you evaluate this: \[\int\limits_{1}^{2}x^2 dx\]

  10. dessyj1
    • one year ago
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    Since i know that the derivative of F(x) is itself i can just switch them around in the integral equation right?

  11. freckles
    • one year ago
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    recall: \[\frac{d}{dx}(\frac{x^3}{3})=x^2 \text{ for all } x \\ \ \text{ so } \int\limits_1^2 x^2 dx=[\frac{x^3}{3}]_1^2 =\frac{2^3}{3}-\frac{1^3}{3}\] you are given \[\frac{d}{dx}(F)=f \text{ for all } x \\ \int\limits_a^b f dx=[ ? ]_a^b\] and that f is continuous which is another important thing

  12. dessyj1
    • one year ago
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    but wouldnt, f(x) in your example be the same function as its derivative?

  13. freckles
    • one year ago
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    are you saying f=f'?

  14. freckles
    • one year ago
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    math is case sensitive so when they say F they don't mean f

  15. freckles
    • one year ago
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    so no we aren't given f'=f

  16. freckles
    • one year ago
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    do you know usually to integrate you need to find the antiderivative of the expression that is the integrand ?

  17. freckles
    • one year ago
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    so if we are given F'=f that means the antiderivative of f is F since F'=f

  18. dessyj1
    • one year ago
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    They do not give is the functions. but lets assume the function is e^x

  19. freckles
    • one year ago
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    did you not understand the example I gave above?

  20. dessyj1
    • one year ago
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    i did not understand it

  21. freckles
    • one year ago
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    \[\frac{d}{dx}(\frac{x^3}{3})=x^2 \text{ for all } x \\ \ \text{ so } \int\limits_1^2 x^2 dx=[\frac{x^3}{3}]_1^2 =\frac{2^3}{3}-\frac{1^3}{3}\] I started off exactly as your question did

  22. freckles
    • one year ago
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    replace the x^2 with f and replace the x^3/3 with F you can do this since (x^3/3)'=x^2 and (F)'=f

  23. dessyj1
    • one year ago
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    the problem with that is the fact that F and f are not the same for all values of x like the question stated.

  24. freckles
    • one year ago
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    No it is saying F'(x)=f(x) for all x

  25. freckles
    • one year ago
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    Also why do F and f have to be the same? You are definitely not given that.

  26. freckles
    • one year ago
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    F' and f have to be the same for all x

  27. freckles
    • one year ago
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    which they are because when you differentiate (x^3/3) you do get x^2

  28. freckles
    • one year ago
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    x^2=x^2 for all x

  29. freckles
    • one year ago
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    http://mathworld.wolfram.com/FundamentalTheoremsofCalculus.html This is just the fundamental theorem of calculus

  30. dessyj1
    • one year ago
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    Okay, I think I have a hard time understanding this because we never learned the fundamental principle of calculus and I am currently studying for the final, so that means my teacher never intended to teach that concept.

  31. freckles
    • one year ago
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    so you guys never cover definite integrals?

  32. freckles
    • one year ago
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    covered*

  33. dessyj1
    • one year ago
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    We did, but we were given the rules.

  34. freckles
    • one year ago
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    so have you ever done the one or know how to do the one I mentioned before: \[\int\limits_1^2 x^2 dx\]

  35. dessyj1
    • one year ago
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    We were not taught how to evaluate an integral using the definition.

  36. freckles
    • one year ago
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    like how would you tackle that one then?

  37. dessyj1
    • one year ago
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    I can do definite integrals. Why did you choose x^2? as one of the functions?

  38. freckles
    • one year ago
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    I can choose 1 or x is doesn't matter it is just an example

  39. dessyj1
    • one year ago
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    would e^x work then?

  40. freckles
    • one year ago
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    \[\int\limits_1^2 1 dx=?\] sure we can use whatever function that is continuous and has a continuous derivative

  41. freckles
    • one year ago
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    I just want to see what you do to evaluate something like that if you never been taught the fundamental theorem of calculus

  42. freckles
    • one year ago
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    Like do you not normally find the antiderivative of the integrand ?

  43. dessyj1
    • one year ago
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    answer is 1

  44. freckles
    • one year ago
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    I know but I want to know how you get there

  45. freckles
    • one year ago
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    like what steps do you take

  46. dessyj1
    • one year ago
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    ill draw what i did

  47. dessyj1
    • one year ago
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    |dw:1434604206920:dw|

  48. freckles
    • one year ago
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    ok good but isn't the derivative of x, 1?

  49. dessyj1
    • one year ago
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    it is

  50. freckles
    • one year ago
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    by the way you use the fundamental of theorem of calculus above know it or not

  51. freckles
    • one year ago
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    (x)'=1 so you are given this

  52. freckles
    • one year ago
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    and then you said this \[\int\limits_1^2 1 dx=x|_1^2 \]

  53. dessyj1
    • one year ago
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    we were that a set of rules to deal with different types of equations such as exponents, and natural logs

  54. freckles
    • one year ago
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    in our question we are given (F)'=f so \[\int\limits_1^2 f dx=F|_1^2\]

  55. dessyj1
    • one year ago
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    Okay

  56. freckles
    • one year ago
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    I replaced the lower and upper with 1 and 2

  57. freckles
    • one year ago
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    do you not see this yet \[\int\limits_a^b f(x) dx=F(x)|_a^b=F(b)-F(a) \\ \]? you know the fundamental theorem of calculus even though you are saying you don't know because you just applied it just a sec ago

  58. freckles
    • one year ago
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    you know given the other stuff such as f is continuous and F'=f

  59. dessyj1
    • one year ago
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    this works if you are telling me the integral of f(x) is equal to F(prime)(x)

  60. freckles
    • one year ago
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    no I'm telling you that at all

  61. freckles
    • one year ago
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    \[f(x)=1 \\ F(x)=x +C \text{ where } C \text{ is a constant } \\ \text{ do you not agree that } F'(x)=f(x) ?\]

  62. freckles
    • one year ago
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    Since F'=f then F is the antiderivative of f

  63. dessyj1
    • one year ago
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    I do agree, but i feel like you are making the functions up now.

  64. freckles
    • one year ago
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    \[\int\limits_{a}^{b}f(x) dx \\ \text{ \to integrate this I need the antiderivative of } f \\ \text{ which is } F \\ \text{ since } F'=f \\ \int\limits_a^b f(x) dx=F(x)|_a^b\] I'm giving you examples

  65. dessyj1
    • one year ago
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    If were were able to seamlessly communicate this would be easier for me to grasp. But i cannot understand, i will just have to ask my teacher tomorrow. Regardless, thank you for all your help and effort.

  66. freckles
    • one year ago
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    if f(x)=1 the antiderivative let's call it F is F(x)=x+c we will just use F(x)=x since we have a definite integral anyways that we will be working with \[\int\limits_a^b 1 dx=x|_a^b=(b-a) \\ \int\limits_a^b f(x) dx=F(x)|_a^b=F(b)-F(a)\]

  67. freckles
    • one year ago
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    like in the example don't you see that (x)'=1 and in the question you are given (F)'=f

  68. freckles
    • one year ago
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    maybe @ganeshie8 can explain it better if you think maybe it is just me

  69. dessyj1
    • one year ago
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    No, i do not think it is just you.

  70. dessyj1
    • one year ago
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    is the antiderivate of F(prime)(x)= F(x) ?

  71. freckles
    • one year ago
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    yes!!!

  72. freckles
    • one year ago
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    \[F'=f \\ \int\limits_a^b f(x) dx=\int\limits_a^b F'(x) dx=F(x)|_a^b\]

  73. dessyj1
    • one year ago
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    I did not know the mathematical notation for the derivative of a capital function like F(x) was f(x)

  74. freckles
    • one year ago
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    well it could have had a different name like for example maybe they said where g'=f g'=f still means that g is the antiderivative of f

  75. freckles
    • one year ago
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    \[\int\limits_a^b f(x) dx=g(x)|_a^b=g(b)-g(a) \text{ since } g'=f \\ \text{ aka since } g \text{ is the antiderivative of } f \]

  76. freckles
    • one year ago
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    or I think the way you understand it better was like this: \[\int\limits_a^b f(x)=\int\limits_a^b g'(x) dx \text{ since } f=g' \\ \text{ now } \int\limits_a^b f(x) dx=\int\limits_a^b g'(x)dx=g(x)|_a^b =g(b)-g(a)\]

  77. freckles
    • one year ago
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    you know assuming g is continuous on [a,b] of course

  78. freckles
    • one year ago
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    I mean f also

  79. dessyj1
    • one year ago
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    so the answer to this question is D?

  80. freckles
    • one year ago
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    yep.

  81. dessyj1
    • one year ago
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    alright, i get it.

  82. freckles
    • one year ago
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    don't just say that if you don't believe it I will not be upset and @ganeshie8 is here he is awesome at explaining things if you do not feel I did the trick

  83. ganeshie8
    • one year ago
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    somehow i feel @dessyj1 you're confusing "anti derivatives" with "definite integrals"

  84. dessyj1
    • one year ago
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    I do understand the question now. The derivative of F(x) was equal to f(x). That means that the integral(indefinite integral, antiderivative) of f(x) is just like asking for the integral of F(prime)(x) which was F(x)

  85. dessyj1
    • one year ago
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    i think anti-derivative and indefinite integrals are the same thing.

  86. ganeshie8
    • one year ago
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    looks perfect!

  87. dessyj1
    • one year ago
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    Thank you.

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