Math 31A Calculus - Lessons 1-15

Vocabulary flashcards covering core concepts, limit laws, continuity, differentiation rules, higher derivatives, implicit differentiation, and inverse/transcendental functions from Math 31A Calculus.

Limit of a Function

The specific value $L$ that a function $f(x)$ approaches as the input $x$ approaches a real number $c$, denoted as $\lim_{x \to c} f(x) = L$, which does not depend on the actual value of $f(c)$.

Left-Hand Limit

The value that a function $f(x)$ approaches as $x$ approaches $c$ from values strictly less than $c$, denoted as $\lim_{x \to c^-} f(x)$.

Right-Hand Limit

The value that a function $f(x)$ approaches as $x$ approaches $c$ from values strictly greater than $c$, denoted as $\lim_{x \to c^+} f(x)$.

Limit Existence Condition

The two-sided limit $\lim_{x \to c} f(x)$ exists if and only if both the left-hand limit $\lim_{x \to c^-} f(x)$ and the right-hand limit $\lim_{x \to c^+} f(x)$ exist and are equal.

Limit Failure Scenarios

The three conditions under which a limit fails to exist (DNE): the left-hand and right-hand limits are unequal, the limit approaches $\pm \infty$, or the function oscillates infinitely near the point.

Continuity at a Point

A condition satisfied by a function $f(x)$ at $x = c$ if and only if $\lim_{x \to c} f(x)$ exists, $f(c)$ is defined, and $\lim_{x \to c} f(x) = f(c)$.

Removable Discontinuity

A discontinuity at $x = c$ where the limit $\lim_{x \to c} f(x)$ exists, but either $\lim_{x \to c} f(x) \neq f(c)$ or $f(c)$ is undefined (DNE).

Jump Discontinuity

A discontinuity at $x = c$ where both one-sided limits exist as finite numbers, but $\lim_{x \to c^-} f(x) \neq \lim_{x \to c^+} f(x)$.

Infinite Discontinuity

A discontinuity at $x = c$ where the function approaches $+\infty$ or $-\infty$ from one or both sides of $c$.

Intermediate Value Theorem (IVT)

A theorem stating that if a function $f(x)$ is continuous on a closed interval and $f(a) < f(b)$, then for every value between $f(a)$ and $f(b)$, there exists at least one $x$ in the interval where $f(x)$ attains that value.

Indeterminate Form

An undefined algebraic expression resulting from direct substitution, such as $\frac{0}{0}$, $\frac{\infty}{\infty}$, $\infty \cdot 0$, or $\infty - \infty$, which requires algebraic simplification before the limit can be determined.

Fundamental Trigonometric Limits

The standard evaluated trigonometric limits: $\lim_{\theta \to 0} \frac{\sin(\theta)}{\theta} = 1$ and $\lim_{\theta \to 0} \frac{1 - \cos(\theta)}{\theta} = 0$.

Secant Line

A straight line passing through two distinct points $P(a, f(a))$ and $Q(x, f(x))$ on the graph of $y = f(x)$, whose slope equals the average rate of change $\frac{f(x) - f(a)}{x - a}$.

Tangent Line

The limiting line that a secant line approaches as $x \to a$, touching the curve at $P(a, f(a))$ with a slope equal to the derivative $f'(a)$, written as $y = f'(a)(x - a) + f(a)$.

Derivative at a Point

The instantaneous rate of change of a function $f(x)$ at $x = a$, defined as $f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$ or equivalently $f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$.

Differentiability Implies Continuity

The foundational theorem stating that if a function $f(x)$ is differentiable at $x = a$, then $f(x)$ is guaranteed to be continuous at $x = a$.

Power Rule

The differentiation rule stating that for any real exponent $n$, $\frac{d}{dx}[x^n] = n x^{n-1}$.

Product Rule

The differentiation rule stating that the derivative of the product of two differentiable functions is $(fg)'(x) = f'(x)g(x) + f(x)g'(x)$.

Quotient Rule

The differentiation rule stating that the derivative of the quotient of two differentiable functions is $(\frac{f}{g})'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}$, provided $g(x) \neq 0$.

Chain Rule

The differentiation rule for composite functions stating that $(f \circ g)'(x) = f'(g(x)) g'(x)$, or in Leibniz notation, $\frac{dy}{dx} = \frac{dy}{du} \frac{du}{dx}$.

Implicit Differentiation

A technique used to find $\frac{dy}{dx}$ from an equation relating $x$ and $y$ by differentiating both sides with respect to $x$, viewing $y$ as a function of $x$, and applying the chain rule.

One-to-One Function (1:1)

A function $f: X \to Y$ for which every output corresponds to at most one input ($f(x_1) \neq f(x_2)$ whenever $x_1 \neq x_2$), geometrically verified by passing the horizontal line test.

Inverse Function

The unique function $f^{-1}$ associated with a one-to-one function $f$ satisfying $f^{-1}(f(x)) = x$ for all $x$ in the domain and $f(f^{-1}(y)) = y$ for all $y$ in the range.

Derivative of an Inverse Function

The theorem stating that if $f$ is differentiable and $f'(f^{-1}(b)) \neq 0$, then $(f^{-1})'(b) = \frac{1}{f'(f^{-1}(b))}$.

Euler's Number ($e$)

The unique mathematical constant approximately equal to $2.718$ defined such that $\lim_{h \to 0} \frac{e^h - 1}{h} = 1$, giving the exponential function the derivative property $\frac{d}{dx}[e^x] = e^x$.

Logarithmic Function

The inverse function of the exponential function with base $b$ ($b > 0, b \neq 1$), defined by $\log_b(y) = x$ if and only if $b^x = y$, having domain $(0, \infty)$ and range $(-\infty, \infty)$.

Derivatives of Sine and Cosine

The primary trigonometric derivatives: $\frac{d}{dx}[\sin(x)] = \cos(x)$ and $\frac{d}{dx}[\cos(x)] = -\sin(x)$.

Derivatives of Tangent and Secant

The derivative formulas: $\frac{d}{dx}[\tan(x)] = \sec^2(x)$ and $\frac{d}{dx}[\sec(x)] = \sec(x)\tan(x)$.

Derivatives of Cotangent and Cosecant

The derivative formulas: $\frac{d}{dx}[\cot(x)] = -\csc^2(x)$ and $\frac{d}{dx}[\csc(x)] = -\csc(x)\cot(x)$.

Derivative of Inverse Sine

The derivative of the inverse sine function: $\frac{d}{dx}[\sin^{-1}(x)] = \frac{1}{\sqrt{1 - x^2}}$, where $x \in (-1, 1)$.

Derivative of Inverse Tangent

The derivative of the inverse tangent function: $\frac{d}{dx}[\tan^{-1}(x)] = \frac{1}{1 + x^2}$, defined for all real numbers $x$.

Derivative of Inverse Cotangent

The derivative of the inverse cotangent function: $\frac{d}{dx}[\cot^{-1}(x)] = -\frac{1}{1 + x^2}$, defined for all real numbers $x$.

Linearity of Differentiation

The properties allowing term-by-term differentiation: the sum/difference rule $(f \pm g)' = f' \pm g'$ and the constant multiple rule $(kf)' = k f'$.

Inverse Sine Function Domain Restriction

The restriction of the domain of $f(\theta) = \sin(\theta)$ to $[-\frac{\pi}{2}, \frac{\pi}{2}]$ to make it one-to-one, yielding the inverse function $\theta = \sin^{-1}(x)$ with domain $[-1, 1]$ and range $[-\frac{\pi}{2}, \frac{\pi}{2}]$.

Inverse Cosine Function Domain Restriction

The restriction of the domain of $f(\theta) = \cos(\theta)$ to $[0, \pi]$ to make it one-to-one, yielding the inverse function $\theta = \cos^{-1}(x)$ with domain $[-1, 1]$ and range $[0, \pi]$.

Exponential Function Growth and Decay

The behavior of $y = a^x$: for base $a > 1$ the function is strictly increasing with $\lim_{x \to \infty} a^x = \infty$ and $\lim_{x \to -\infty} a^x = 0$, whereas for $0 < a < 1$ the function is strictly decreasing with $\lim_{x \to \infty} a^x = 0$ and $\lim_{x \to -\infty} a^x = \infty$.