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thuyvy

  • 2 years ago

Find equations of all tangents to the curve f(x)=9/x that have slope -1

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  1. ivanmlerner
    • 2 years ago
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    At first you have to take the derivative of f. Do you understand why is that?

  2. thuyvy
    • 2 years ago
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    yes I do, I got -9/x2

  3. ivanmlerner
    • 2 years ago
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    Ok, now what is the meaning of the derivative? Knowing this meaning can you guess what the next step would be?

  4. thuyvy
    • 2 years ago
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    that is when a function change as the input change, but I'm stuck at the 2nd step what shall I do next? plug in -1 slope as x?

  5. ivanmlerner
    • 2 years ago
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    Yes, in other words it is the rate of change of a function. But the meaning I was thinking of is the same, but a little diferent. When the input increases in a really small amount, the value changes too. Since those two changes are very small, the rate of change is approximately the change in the value divided by the increase in the input. Do you understand what I just said?

  6. thuyvy
    • 2 years ago
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    not really, can you lead me into solving this? what is the second step I should take?

  7. ivanmlerner
    • 2 years ago
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    You need to understand it before gettingg to the next step. But if you want to know, the derivative of a function at x is the slope of its tangent line at this point, that was what I was trying to show you. Then, you should put -9/x2=-1, that is the slope, to find in which points the line passes.

  8. thuyvy
    • 2 years ago
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    Yes I got to this step, when I was trying to solve for x I got +- 3

  9. thuyvy
    • 2 years ago
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    but the answer that it should be is y=-x-6 and y = -x+6

  10. thuyvy
    • 2 years ago
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    so I must have done something wrong right?

  11. ivanmlerner
    • 2 years ago
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    Wait, you just found out one point in wich the line passes, it is not the answer yet

  12. ivanmlerner
    • 2 years ago
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    What you did so far is correct

  13. thuyvy
    • 2 years ago
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    oh ic okay, so the other point is 3 since 9/x = y and x is 3 therefore y = 3 right?

  14. ivanmlerner
    • 2 years ago
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    or -3, but yes, thats correct

  15. thuyvy
    • 2 years ago
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    so now what's next?

  16. ivanmlerner
    • 2 years ago
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    Now, you know a point that belongs to the line, and its slope. The general formula for a line is y=ax+b, and you alrady have a, now put the point you know that belongs to it, and you will find b.

  17. FaisalAyaz
    • 2 years ago
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    use slope intercept form now

  18. thuyvy
    • 2 years ago
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    y = mx+b --> -3=-1x+b

  19. thuyvy
    • 2 years ago
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    sorry, I'm kinda slow at this point, how do I set this = to 9?

  20. ivanmlerner
    • 2 years ago
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    You dont need to, you know x and y, from the points (-3, -3) and (3, 3) that this tangent passes through, and you know a. Now: -3=-1*(-3)+b1 3=-1*3+b2

  21. thuyvy
    • 2 years ago
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    oh okay

  22. thuyvy
    • 2 years ago
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    I got it from here

  23. ivanmlerner
    • 2 years ago
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    Before, you found out in which points the tangent have this slope, then, you found out the value of the function at this point, and now you use those information to get the formulas for the tangent lines. You seem a little bit confused with what the numbers you are getting mean.

  24. thuyvy
    • 2 years ago
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    I got it thanks so much for being so patience :)

  25. ivanmlerner
    • 2 years ago
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    Your welcome

  26. thuyvy
    • 2 years ago
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    can I ask you something else?

  27. ivanmlerner
    • 2 years ago
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    Of course

  28. thuyvy
    • 2 years ago
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    same type of problem but it's f(x) = square root of (x+9)

  29. thuyvy
    • 2 years ago
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    slope 1/2 I did the derivatives and solve for x I got x= -1

  30. thuyvy
    • 2 years ago
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    I plugged x= -1 back to original equation I got y = +/- 2 sqrt 2 then I use slope intercept to find equations, I got y= 1/4+9/4 sqrt 2

  31. thuyvy
    • 2 years ago
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    but the answer should be y=1/4+13/4 where did I do wrong?

  32. ivanmlerner
    • 2 years ago
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    You got the x wrong.

  33. ivanmlerner
    • 2 years ago
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    Or the derivative

  34. thuyvy
    • 2 years ago
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    original fx is sqrt(x=9) derivative is 1/2(x+9)^1/2

  35. thuyvy
    • 2 years ago
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    Where did I do wrong.

  36. ivanmlerner
    • 2 years ago
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    its (1/2)(x+9)^(-1/2)

  37. thuyvy
    • 2 years ago
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    Yes i got that.. But solving for x=-1

  38. ivanmlerner
    • 2 years ago
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    Well, thats wrong, did you see the minus, in the exponent?

  39. thuyvy
    • 2 years ago
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    No i said that mine is 1/{2(x+9)^1/2 same thing

  40. ivanmlerner
    • 2 years ago
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    \[\frac{ 1 }{ 2 }=\frac{ 1 }{ 2\sqrt{x+9} }\rightarrow \sqrt{x+9}=1\rightarrow x+9=\pm1\rightarrow x=-10 or -8\]

  41. thuyvy
    • 2 years ago
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    the slope is 1/4 not 1/2

  42. ivanmlerner
    • 2 years ago
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    sorry, didnt ssee you posting. Then the left side of the equation will be 1/4, and sqrt(x+9)=2 and x+9=+-4, wich gives us: x=-5, and x=-13

  43. ivanmlerner
    • 2 years ago
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    Can you go from there?

  44. thuyvy
    • 2 years ago
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    let me try to do calculation, I did the calculation but got +/- 2 sqrt 2.. give me 1 minute please

  45. ivanmlerner
    • 2 years ago
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    Sure

  46. thuyvy
    • 2 years ago
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    I got it, I think I got messed up when I was trying to square to get rid of sq rt. Thanks Do you have time to help me on other problems?

  47. ivanmlerner
    • 2 years ago
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    I'm almost leaving actually, but I'm here almost every day.

  48. thuyvy
    • 2 years ago
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    quick question on solving this problem \[z(4z+7) - x(4x+7) / (z-x)\]

  49. thuyvy
    • 2 years ago
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    I can combine (z-x) then cancel top and bottom right?

  50. thuyvy
    • 2 years ago
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    so what is have left is (4z+7) * (4x+7) right?

  51. ivanmlerner
    • 2 years ago
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    No, be carefull, think of what is being divided, when you have something that is not being divided you cannot sum them withou a common denominator

  52. thuyvy
    • 2 years ago
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    oh maybe that's why. thanks

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