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Thread: 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 $\displaystyle \frac {dy}{dx}$ if $\displaystyle x^4 +2x^2y^3 + y^2 = 21$

    2. Differentiate $\displaystyle 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 $\displaystyle 10 m^3/min$, 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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    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 $\displaystyle \frac {dy}{dx}$ if $\displaystyle x^4 +2x^2y^3 + y^2 = 21$
    this is an implicit differentiation problem

    $\displaystyle x^4 + 2x^2y^3 + y^2 = 21$

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

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

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

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

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

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

    $\displaystyle V = \frac 43 \pi r^3$

    differentiating implicitly with respect to time, we have

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

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

    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 $\displaystyle z$, the base is the distance of the boat from the dock, call this distance $\displaystyle x$. The height is 1. Now we are told that $\displaystyle \frac {dz}{dt} = -0.8$ (note that it is negative because the distance is decreasing) and we want $\displaystyle \frac {dx}{dt}$ when $\displaystyle x = 10$

    By Pythagoras' Theorem, we have that

    $\displaystyle z^2 = 1 + x^2$ ............................(1)

    $\displaystyle \Rightarrow x^2 = z^2 - 1$

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

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

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