Front
Derivative of $\arctan(x)$
Back
$\frac{1}{1+x^2}$
One of the most common inverse trig derivatives in integration.
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Calculus I Fundamentals
Master Calculus I with 50 cards covering limits, derivatives, integrals, and applications with formulas and worked examples.
AnyFlashcards50 cards
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Front
Limit definition of a function $f(x)$ at $c$
Back
$\lim_{x \to c} f(x) = L$
The value $f(x)$ approaches $L$ as $x$ gets arbitrarily close to $c$.
Front
Limit of $\frac{\sin(x)}{x}$ as $x \to 0$
Back
1
Essential for deriving the derivatives of trigonometric functions.
Front
Limit of $\frac{1 - \cos(x)}{x}$ as $x \to 0$
Back
0
Used in the proof of the derivative of $\cos(x)$.
Front
Squeeze Theorem (Sandwich Theorem)
Back
If $g(x) \le f(x) \le h(x)$ and $\lim g(x) = \lim h(x) = L$, then $\lim f(x) = L$
Used to find limits of functions trapped between two known limits.
Front
Definition of Continuity at a point $a$
Back
$\lim_{x \to a} f(x) = f(a)$
Requires the limit to exist, the function to be defined, and both to be equal.
Front
Intermediate Value Theorem (IVT)
Back
If $f$ is continuous on $[a, b]$, it takes every value between $f(a)$ and $f(b)$
Often used to prove the existence of roots.
Front
Infinite Limit: $\lim_{x \to 0^+} \frac{1}{x}$
Back
$\infty$
Indicates a vertical asymptote at $x=0$.
Front
Limit at Infinity: $\lim_{x \to \infty} \frac{1}{x}$
Back
0
Indicates a horizontal asymptote at $y=0$.
Front
Limit Law: Constant Multiple Rule
Back
$\lim_{x \to c} [k \cdot f(x)] = k \cdot \lim_{x \to c} f(x)$
Constants can be pulled out of the limit operation.
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