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4.5: Optimization

  • Page ID
    88661
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    Sometimes we have a function and we just really want to know what its high point or low point is in terms of \(y\)-value. The high point is called a maximum, and the low point is called a minimum. For most functions, these points occur when the derivative is zero or undefined (we talked about why this is briefly in a previous section).

    Optimization

    The height of a baseball follows the function \(h(t) = -5t^2 + 20t + 10\), where \(h\) is measured in meters and \(t\) is measured in seconds.. What value of \(t\) maximizes the height?

    We will follow the maxim “optimization happens when the derivative is zero”. First we find the derivative using the power rule \(h'(t) = -10t + 20\). Then we set this equal to zero, so we solve

    \[\begin{align*} -10t + 20 & = 0 \\ -10t & = - 20 \\ t & = \boxed{2} \end{align*}\]

    Hence, the height is maximized when the time is equal to \(2\) seconds. At this point, the height of the ball is \(h(2) = -5(2)^2 + 20(2) + 10 = -20 + 40 + 10 = 30\) meters. Here’s a rough sketch based on what we know about this function:

    notes-pic36.svgfixme

    Let’s do anther optimization example:

    Optimization 2

    The cost per item of producing Super Hero Action Figures, if \(x\) are produced, is given by

    \(C(x) = \frac{500}{x} + 0.001x.\)

    At what value \(x\) is the cost per item minimized? What is the cost at this value of \(x\)?

    To solve this problem, we find \(C'(x)\):

    \(C'(x) = \frac{d}{dx} 500 x^{-1} + 0.001 x = -500x^{-2} + 0.001\)

    We then set this equal to zero and solve:

    \[\begin{align*} -500x^{-2} + 0.001 & = 0 \\ -500x^{-2} & = -0.001 \\ x^{-2} & = 0.000002 \\ 1 & = 0.000002 x^2 \\ 500,000 & = x^2 \\ \sqrt{500,000} & = x \approx \boxed{707.10} \end{align*}\]

    So we see about \(707\) action figures is the best number to choose. The cost at this point (by plugging \(x = 707\) into the original equation) is just \(\boxed{\$1.41}\). Not too bad!


    This page titled 4.5: Optimization is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Tyler Seacrest via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.