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5

The term Hergert Numbers is sometimes used in my specific region of the US for the values of $x$ where $f''(x) = 0$ or $f''(x)$ is undefined. This is in reference to Rodger Hergert, an Illinois community college professor who sometime in the 1990s became very frustrated with the fact that these numbers had no good name. But it doesn't really matter what you ...


2

The calculus textbook I'm currently using (Burzynski, Applied Calculus for Business, Life and Social Sciences, XYZ Textbooks) uses the term "hypercritical point". A quick web search indicates that a few other sources use it as well.


2

In my opinion whether this proof is correct or not depends on the actual wording of the question. This recursive argument shows that the statement is true for 100 numbers (since it is true for 2, 3, 4, so we eventually reach 100) It also shows that the statement is true for 1000 numbers (we eventually reach 1000). For any fixed natural number this is a ...


4

Logarithmic differentiation is used in mathematics when dealing with infinite products in complex analysis. Note the construction of $f’/f$ from $f$ does not require logarithms of $f$ as a middle step (except intuitively), so it is perfectly fine to use even when $f$ takes non-positive values. The prime number theorem and the link between prime numbers and ...


2

Building off the circle example, you can actually work out the centripetal acceleration formula by implicitly differentiating twice. If your students aren't familiar with vectors you can just plug in x = 0 and y = 1: $$x^2+y^2=r^2$$ Differentiate with respect to t: $$2x\cdot x' + 2y \cdot y' = 0$$ Plugging in x = 0, y = r, you can solve $y' = 0$. ...


6

For the expression $x^x$ we could focus on finding occurrences of $x\ln(x)$. One direction is Stirling's approximation $\ln(N!)\sim N\ln(N)$ so $N!$ is like $N^N$. Another direction is that the prime number theorem gives an estimate for the $n$-th prime $p_n$ as $p_n=n\ln(n)$. Yet another direction involves entropy. Entropy might be given by an expression ...


0

Linear approximation is taught as standard method for interpolating tabular data in many engineering areas. For instance steam tables. Sort of a crutch to give you more value than the resolution of the table.


1

For high-dimensional multivariable functions it's hard to imagine a geometric tangent, but the notion of linear approximation makes perfect sense. The derivative (usually called the Jacobian matrix https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant) still satisfies (for $\vec{x} \approx \vec{a}$): $$\vec{f}(\vec{x}) \approx J (\vec{x}-\vec{a}) + \...


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$n^n$ shows up in combinatorics (number of lists of numbers from 1-n of length n), but I doubt it will have applications for the physical world: $x$ cannot have units, since it doesn't make sense to raise something to a power with units.


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