Calculus & Analysis — Community Channel — RIDA MATH

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University
What does each quantifier in the ε–δ definition protect against?
Take the definition of $\lim_{x\to a} f(x) = L$ . Swap the order of “for every $\varepsilon$ ” and “there exists $\delta$ ”. What does the new statement say, and which functions satisfy it?
A strong contribution: States the swapped definition precisely, gives an example that satisfies it, and explains why it is not a limit.
- Read: How to Read a Mathematical Definition
University
Differentiating under the integral sign
Let $I(s) = \int_0^1 \cos(ts)\,dt$ . Compute $I(s)$ directly, then compute $I''(s)$ in two ways: by differentiating your closed form, and by differentiating under the integral sign. Which conditions justify the second method?
A strong contribution: Both computations agree; the answer cites continuity of the integrand and its partial derivatives on a compact interval.
- Lab: Derivative Lab
- Read: The Structure Behind Higher-Order Derivatives
University
One formula instead of four cases
Show that $\dfrac{d^n}{dx^n}\sin(ax) = a^n \sin\!\left(ax + \tfrac{n\pi}{2}\right)$ for every $n \ge 0$ . Why does the phase shift remove the need to treat $n \bmod 4$ separately?
A strong contribution: A short induction, plus an explanation that differentiation rotates the phase by a quarter turn.
- Lab: Derivative Lab

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