Notes on Work #2

Read an interesting thread on HN. The comment that stood out the most to me was this one, and the ensuing discussion around it: Successful companies also attract gold diggers. When the company is small, unless everybody is going above and beyond the call of duty, it’s likely going to fail. As the company gets bigger, there is more and more latitude for failure. At some point it is successful enough that it can survive having people whose only goal is to direct a large amount of money into their pockets. …

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First Derivatives to Find Maximum and Minimum Values of Functions

If the graph of a function rises from the left to the right over an interval \(I\), the function is increasing on, or over, \(I\). If the graph drops from left to right, the function is decreasing on, or over, \(I\). Mathematically speaking, a function is increasing over an interval if, for every input \(a\) and \(b\) in the interval, the input \(a\) is less than the input \(b\), and the output \(f(a) < f(b)\). …

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Business Applications of Derivatives

A company is selling laptop computers. It determines that its total profit, in dollars, is given by \[P(x) = 0.08x^2 + 80x\] where \(x\) is the number of units produced and sold. Suppose that \(x\) is a function of time, in months, where \(x = 5t + 1\). (a) Find the total profit as a function of time \(t\). (b) Find the rate of change of total profit when \(t = 48\) months. …

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Higher-Order Derivatives

Let’s consider the function \[y = f(x) = x^5 - 3x^4 + x\] Its derivative \(f^\prime\) is \[y^\prime = f^\prime(x) = 5x^4 - 12x^3 + 1\] The derivative function \(f^\prime\) can also be differientiated. We can think of the derivative of \(f^\prime\) as rhe rate of change of the slope of the tange lines of \(f\). We use the notation \(f^{\prime\prime}\). \[f^{\prime\prime}(x) = \frac{d}{dx}f^\prime(x)\] We call this function the second derivative of \(f\). …

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Differentiation Techniques Pt.3

The extended power rule states that, supposing that \(g(x)\) is a differentiable function of \(x\). Then, for any real number \(k\), \[\frac{d}{dx}[g(x)]^k = k[g(x)]^{k-1} \cdot \frac{d}{dx}g(x)\] Taking a detour into function compositions, a composed function \(f \circ g\), the composition of \(f\) and \(g\), is defined as \[(f \circ g)(x) = f(g(x))\] Suppose we want to calculate how much it costs to heat a house on a particular day of the year. …

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