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Math Help - Calculus: Derivatives + Limits

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    Calculus: Derivatives + Limits

    Studying for a test, and have a few questions I'm confused about... Some of them are probably really easy, I just can't seem to get them right.

    1. Find \frac {dy}{dx} if x^4 +2x^2y^3 + y^2 = 21

    2. Differentiate  y = \sqrt \frac {x}{(1 + x^2)}

    3. A spherical balloon is being inflated. If the volume of the balloon is increasing at a rate of 10 m^3/min, how fast is the radius increasing when the radius is 3m?

    4. A boat is pulled into a dock by a rope attached to the bow of the boat and passing through a pulley on the dock that is 1m higher than the bow of the boat. If the boat is pulled in at a rate of 0.8 m/s, how fast does the boat approach the dock when it is 10m from the dock?

    So this would make a right-angle triangle, one side is 1m, the other 10m. I assume this would have something to do with Pythagorean theorem, but not sure how to apply it.


    Thank you very much.
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Cougar22 View Post
    Studying for a test, and have a few questions I'm confused about... Some of them are probably really easy, I just can't seem to get them right.

    1. Find \frac {dy}{dx} if x^4 +2x^2y^3 + y^2 = 21
    this is an implicit differentiation problem

    x^4 + 2x^2y^3 + y^2 = 21

    \Rightarrow 4x^3 + 4xy^3 + 6x^2y^2 ~\frac {dy}{dx} + 2y ~\frac {dy}{dx} = 0 (note that we used the product rule for the middle term on the left hand side)

    now the rest is algebra. solve for \frac {dy}{dx}

    2. Differentiate  y = \sqrt \frac {x}{(1 + x^2)}
    we have y = \left( \frac x{1 + x^2} \right)^{\frac 12}, we proceed by the chain rule

    \Rightarrow y' = \frac 12 \left( \frac x{1 + x^2} \right)^{- \frac 12} \cdot \frac {1 - x^2}{(1 + x^2)^2}

    note that the last factor is the derivative of \frac x{1 + x^2} by the quotient rule

    3. A spherical balloon is being inflated. If the volume of the balloon is increasing at a rate of 10 m^3/min, how fast is the radius increasing when the radius is 3m?
    the volume, V, of a sphere with radius r is given by

    V = \frac 43 \pi r^3

    differentiating implicitly with respect to time, we have

    \frac {dV}{dt} = 4 \pi r^2~\frac {dr}{dt}

    we are told \frac {dV}{dt} = 10 and we want r = 3, so plug these in and solve for \frac {dr}{dt}

    4. A boat is pulled into a dock by a rope attached to the bow of the boat and passing through a pulley on the dock that is 1m higher than the bow of the boat. If the boat is pulled in at a rate of 0.8 m/s, how fast does the boat approach the dock when it is 10m from the dock?
    yes, draw a diagram. note that it makes a right triangle, the hypotenuse is the length of the rope from the pulley to the boat, call this length z, the base is the distance of the boat from the dock, call this distance x. The height is 1. Now we are told that \frac {dz}{dt} = -0.8 (note that it is negative because the distance is decreasing) and we want \frac {dx}{dt} when x = 10

    By Pythagoras' Theorem, we have that

    z^2 = 1 + x^2 ............................(1)

    \Rightarrow x^2 = z^2 - 1

    \Rightarrow 2x~\frac {dx}{dt} = 2z \frac {dz}{dt}

    \Rightarrow \frac {dx}{dt} = \frac zx~\frac {dz}{dt}

    now plug in everything on the right to find \frac {dx}{dt}. note that you can find the value of z from equation (1)
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