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Bob Miller's Calc for the Clueless: Calc I & II, 1997
The first calc study guides that really give students a clue. Bob Miller's student-friendly Calc for the Clueless features quickly-absorbed, fun-to-use information and help. Students will snap up Calc for the Clueless as they discover: Bob Miller's painless and proven techniques to learning Calculus Bob Miller's way of anticipating problems.
This is called "A Gentle Introduction to the Hessian Matrix"
Hessians are somewhere between #linearalgebra #calculus and #rstats but still a core aspect of #datascience
All in all, building and deriving things like these are probably only useful when developing a unique solution. For the vast majority of cases, having a general understanding is enough.
... actually, I am pretty sure that there is a #python library for just such an occasion (I have never looked though so ymmv)
Integrals of inverse functions!
Proof without words (see image; credit: Jonathan Steinbuch, CC BY-SA 3.0, via Wikimedia Commons)...
For any montonic and invertible function \(f(x)\) in the interval \([a,b]\):
\[\displaystyle\int_a^bf(x)~ \mathrm dx+\int_{f(a)=c}^{f(b)=d}f^{-1}(x)~\mathrm dx=b\cdot f(b)-a\cdot f(a)=bd-ac\]
If \(F\) is an antiderivative of \(f\), then the antiderivatives of \(f^{-1}\) are:
\[\boxed{\displaystyle\int f^{-1}(y)~\mathrm dy=yf^{-1}(y)-F\circ f^{-1}(y)+C}\]
where \(C\) is an arbitrary constant (of integration), and \(\circ\) is the composition operator (function composition).
For example:
\[\begin{align*}\displaystyle\int \sin^{-1}(y) \, \mathrm dy &= y\sin^{-1}(y) - (-\cos(\sin^{-1}(y)))+C\\ &=y\sin^{-1}(y)+\sqrt{1-y^2}+C\end{align*}\]
\[\displaystyle\int \ln(y) \, dy = y\ln(y)-\exp(\ln(y)) + C= y\ln(y)-y + C.\]
#Function #InverseFunction #InverseFunctions #Functions #Integral #Integrals #Antiderivative #Integration #Calculus #FunctionComposition #CompositeFunction)
Lo, a problem in a business calculus textbook:
"Sarah is paid $500 for working up to 40 hours per week. For work beyond 40 hours per week she is paid $18/hour."
The neurotypical interpretation of the above specification results in Sarah *owing* the company money if she works less than 12.222... hours in a week.
I shit you not. The equation describing the specification above is:
Pay=18*(hrs-40)+500
If you're a business calculus teacher, and you want to be taken seriously by your colleagues, you gotta do better than this.
Deriving the Reactance Formulas - If you’ve dealt with reactance, you surely know the two equations for computing in... - https://hackaday.com/2025/04/28/deriving-the-reactance-formulas/ #radianfrequency #mischacks #reactance #calculus #math
If you’ve dealt with reactance, you surely know the…
HackadayA cycloidal pendulum - one suspended from the cusp of an inverted cycloid - is isochronous, meaning its period is constant regardless of the amplitude of the swing. Please find the proof using energy methods: Lagrange's equations (in the images attached to the reply).
Background:
The standard pendulum period of \(2\pi\sqrt{L/g}\) or frequency \(\sqrt{g/L}\) holds only for small oscillations. The frequency becomes smaller as the amplitude grows. If you want to build a pendulum whose frequency is independent of the amplitude, you should hang it from the cusp of a cycloid of a certain size, as shown in the gif. As the string wraps partially around the cycloid, the effect decreases the length of the string in the air, increasing the frequency back up to a constant value.
In more detail:
A cycloid is the path taken by a point on the rim of a rolling wheel. The upside-down cycloid in the gif can be parameterized by \((x, y)=R(\theta-\sin\theta, -1+\cos\theta)\), where \(\theta=0\) corresponds to the cusp. Consider a pendulum of length \(L=4R\) hanging from the cusp, and let \(\alpha\) be the angle the string makes with the vertical, as shown (in the proof).
#Pendulum #Cycloid #Period #Frequency #SHM #TimePeriod #CycloidalPendulum #Lagrange #Cusp #Energy #KineticEnergy #PotentialEnergy #Lagrangian #Length #Math #Maths #Physics #Mechanics #ClassicalMechanics #Amplitude #CircularFrequency #Motion #Vibration #HarmonicMotion #Parameter #ParemeterizedEquation #GoverningEquations #Equation #Equations #DifferentialEquations #Calculus
Some people use math just like they use people, to lie, cheat or steal. But math itself never lies, or cheats, or steals from me, thats why I love math.
My graph of the day. Inspired by the fact that
\[ \sum_{k=1}^{\infty}{\sin k} \]
is bounded but diverges, I've been exploring other similar sums. Here's the start of
\[ \sum_{k=1}^{\infty}{\tan k} \]
It's not shocking that it looks like \(y = \ln(\|\cos x\| ) \), but I definitely wonder if at some point it blows up down the line!
I've made my lecture notes for applied calculus 2 publicly available. The content of that course is mostly "multidimensional real functions", including all the practical formulas for computing surfaces and volumes, slopes, optimization etc. The lecture notes contain a lot of practice problems with solutions. Maybe they're helpful for someone either learning or teaching #calculus and #math
https://paulbalduf.com/teaching/
It's really fucking aggravating when you try to compute an integral, get to a solution, and then when you look at the instructor's solution, you find him using a formula that WAS NEVER INTRODUCED BEFORE JUST NOW!