Calculus Essentials

85 flashcards by MentisLibrary

Limits, derivatives, integrals, and series, with every rule stated cleanly and applied in one line. Covers the theorems, the techniques, and the applications that show up on every exam.

What does $\lim_{x\to a} f(x) = L$ mean?

As $x$ gets arbitrarily close to $a$ from either side, $f(x)$ gets arbitrarily close to $L$

When does a limit fail to exist?

When the one-sided limits differ, the function oscillates without settling, or it grows without bound

What are the main limit laws?

Limits distribute over sums, differences, products, quotients with nonzero denominator, and powers

What are the common indeterminate forms?

$\frac{0}{0}$, $\frac{\infty}{\infty}$, $0 \cdot \infty$, $\infty - \infty$, $1^\infty$, $0^0$, $\infty^0$

How do you evaluate a $\frac{0}{0}$ limit algebraically?

Factor and cancel, rationalize, or simplify a complex fraction, then substitute

What is the squeeze theorem?

If $g(x) \le f(x) \le h(x)$ near $a$ and $g$ and $h$ share the limit $L$, then $f$ has limit $L$ too

What is $\lim_{x\to 0}\frac{\sin x}{x}$?

$1$, with $x$ in radians

What is $\lim_{x\to 0}\frac{1-\cos x}{x}$?

$0$

How do you find $\lim_{x\to\infty}$ of a rational function?

Compare degrees: lower numerator gives $0$, equal degrees give the leading coefficient ratio, higher numerator diverges

What are the three conditions for $f$ to be continuous at $x=c$?

$f(c)$ exists, $\lim_{x\to c}f(x)$ exists, and they are equal

What is the Intermediate Value Theorem used for in calculus?

Guaranteeing a root exists between two points where the continuous function changes sign

What is the limit definition of the derivative?

$f'(x) = \lim_{h\to0}\dfrac{f(x+h)-f(x)}{h}$

What does the derivative mean geometrically and physically?

The slope of the tangent line, and the instantaneous rate of change

What is the power rule?

$\dfrac{d}{dx}x^n = nx^{n-1}$ for any real $n$

What is the product rule?

$(fg)' = f'g + fg'$

What is the quotient rule?

$\left(\dfrac{f}{g}\right)' = \dfrac{f'g - fg'}{g^2}$

What is the chain rule?

$\dfrac{d}{dx}f(g(x)) = f'(g(x)) \cdot g'(x)$

What are the derivatives of $\sin x$, $\cos x$, and $\tan x$?

$\cos x$, $-\sin x$, and $\sec^2 x$

What are the derivatives of $\sec x$, $\csc x$, and $\cot x$?

$\sec x\tan x$, $-\csc x\cot x$, and $-\csc^2 x$

What are the derivatives of $e^x$ and $a^x$?

$\dfrac{d}{dx}e^x = e^x$ and $\dfrac{d}{dx}a^x = a^x\ln a$

What are the derivatives of $\ln x$ and $\log_a x$?

$\dfrac{1}{x}$ and $\dfrac{1}{x\ln a}$

What are the derivatives of $\arcsin x$ and $\arctan x$?

$\dfrac{1}{\sqrt{1-x^2}}$ and $\dfrac{1}{1+x^2}$

What is implicit differentiation?

Differentiating both sides with respect to $x$, treating $y$ as a function of $x$ and applying the chain rule

What is logarithmic differentiation?

Take $\ln$ of both sides, differentiate implicitly, then solve for $y'$

What does the second derivative tell you?

The rate of change of the slope, giving concavity and, in motion, acceleration

What is a critical point?

A point in the domain where $f'(x) = 0$ or $f'(x)$ does not exist

What is the first derivative test?

If $f'$ changes from positive to negative at a critical point there is a local maximum, and the reverse gives a local minimum

What is the second derivative test?

At a critical point, $f''<0$ gives a local max, $f''>0$ gives a local min, and $f''=0$ is inconclusive

What is the Extreme Value Theorem?

A continuous function on a closed interval attains an absolute maximum and minimum

What is the Mean Value Theorem?

If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then $f'(c) = \dfrac{f(b)-f(a)}{b-a}$ for some $c$