Calculus: Rates of Change, Related Rates, Linear Approximations, Hyperbolic Functions, and the Mean Value Theorem

Comprehensive vocabulary flashcards covering rates of change, related rates, linear approximations, differentials, hyperbolic functions and identities, inverse hyperbolic functions, Rolle's Theorem, and the Mean Value Theorem.

Difference Quotient

The ratio $\frac{\Delta y}{\Delta x} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}$, which represents the average rate of change of $y$ with respect to $x$ over the interval $[x_1, x_2]$ and corresponds geometrically to the slope of the secant line $PQ$.

Instantaneous Rate of Change

The limit of the average rate of change as $\Delta x \to 0$, given by $f'(x) = \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x} = \frac{dy}{dx}$, which represents the slope of the tangent line at $P(x_1, f(x_1))$.

Instantaneous Velocity

For a particle moving in a straight line with position function $s = f(t)$, the derivative of position with respect to time, defined as $v(t) = \frac{ds}{dt}$, which represents the rate of change of displacement with respect to time.

Acceleration

The derivative of velocity with respect to time, given by $a(t) = v'(t) = s''(t)$.

Particle at Rest

The condition of a moving particle when its instantaneous velocity is zero, that is, $v(t) = 0$.

Related Rates Problem

A problem where the idea is to compute the rate of change of one quantity in terms of the rate of change of another quantity.

Linear Approximation (Tangent Line Approximation)

The approximation of a curve $y = f(x)$ when $x$ is near $a$ using the tangent line at $(a, f(a))$, written as $f(x) \approx f(a) + f'(a)(x - a)$.

Linearization

The linear function whose graph is the tangent line to $f$ at $a$, defined by $L(x) = f(a) + f'(a)(x - a)$.

Differential $dx$

For a differentiable function $y = f(x)$, an independent variable that can be assigned any given real number value.

Differential $dy$

The dependent differential defined in terms of $dx$ and the derivative of $f$ by the equation $dy = f'(x)\,dx$.

Hyperbolic Sine Function ($\sinh(x)$)

The hyperbolic function defined by $\sinh(x) = \frac{e^x - e^{-x}}{2}$, with domain $(-\infty, \infty)$ and range $(-\infty, \infty)$.

Hyperbolic Cosine Function ($\cosh(x)$)

The hyperbolic function defined by $\cosh(x) = \frac{e^x + e^{-x}}{2}$, with domain $(-\infty, \infty)$ and range $[1, \infty)$.

Hyperbolic Tangent Function ($\tanh(x)$)

The hyperbolic function defined by $\tanh(x) = \frac{\sinh(x)}{\cosh(x)} = \frac{e^x - e^{-x}}{e^x + e^{-x}}$, with domain $(-\infty, \infty)$ and range $(-1, 1)$.

Reciprocal Hyperbolic Functions

The functions defined as $\text{csch}(x) = \frac{1}{\sinh(x)}$, $\text{sech}(x) = \frac{1}{\cosh(x)}$, and $\coth(x) = \frac{\cosh(x)}{\sinh(x)}$.

Fundamental Hyperbolic Identity

The identity $\cosh^2(x) - \sinh^2(x) = 1$.

Hyperbolic Tangent-Secant Identity

The identity obtained by dividing $\cosh^2(x) - \sinh^2(x) = 1$ by $\cosh^2(x)$, given by $1 - \tanh^2(x) = \text{sech}^2(x)$.

Derivatives of $\sinh(x)$ and $\cosh(x)$

The differentiation rules $\frac{d}{dx}(\sinh(x)) = \cosh(x)$ and $\frac{d}{dx}(\cosh(x)) = \sinh(x)$.

Inverse Hyperbolic Sine Function ($\sinh^{-1}(x)$)

The inverse function defined by $y = \sinh^{-1}(x) \iff \sinh(y) = x$, having domain $(-\infty, \infty)$ and range $(-\infty, \infty)$, with logarithmic formula $\sinh^{-1}(x) = \ln(x + \sqrt{x^2 + 1})$ for all $x \in \mathbb{R}$.

Inverse Hyperbolic Cosine Function ($\cosh^{-1}(x)$)

The inverse function defined by $y = \cosh^{-1}(x) \iff \cosh(y) = x$, having domain $[1, \infty)$ and range $[0, \infty)$, with logarithmic formula $\cosh^{-1}(x) = \ln(x + \sqrt{x^2 - 1})$ for $x \ge 1$.

Inverse Hyperbolic Tangent Function ($\tanh^{-1}(x)$)

The inverse function defined by $y = \tanh^{-1}(x) \iff \tanh(y) = x$, having domain $(-1, 1)$ and range $(-\infty, \infty)$, with logarithmic formula $\tanh^{-1}(x) = \frac{1}{2}\ln\left(\frac{1 + x}{1 - x}\right)$ for $-1 < x < 1$.

Derivative of $\sinh^{-1}(x)$

The formula $\frac{d}{dx}(\sinh^{-1}(x)) = \frac{1}{\sqrt{1 + x^2}}$.

Derivative of $\cosh^{-1}(x)$

The formula $\frac{d}{dx}(\cosh^{-1}(x)) = \frac{1}{\sqrt{x^2 - 1}}$.

Derivative of $\tanh^{-1}(x)$

The formula $\frac{d}{dx}(\tanh^{-1}(x)) = \frac{1}{1 - x^2}$.

Rolle's Theorem

Let $f$ be a function that satisfies three hypotheses: (1) $f$ is continuous on the closed interval $[a, b]$, (2) $f$ is differentiable on the open interval $(a, b)$, and (3) $f(a) = f(b)$. Then there exists a number $c$ in $(a, b)$ such that $f'(c) = 0$.

The Mean Value Theorem

Let $f$ be a function that satisfies: (1) $f$ is continuous on the closed interval $[a, b]$, and (2) $f$ is differentiable on the open interval $(a, b)$. Then there exists a number $c$ in $(a, b)$ such that $f'(c) = \frac{f(b) - f(a)}{b - a}$, or equivalently $f(b) - f(a) = f'(c)(b - a)$.

Geometric Interpretation of the Mean Value Theorem

There is at least one point $(c, f(c))$ on the graph of $f$ where the slope of the tangent line is the same as the slope of the secant line passing through points $A(a, f(a))$ and $B(b, f(b))$.

Main Significance of the Mean Value Theorem

It enables us to obtain information about a function from information about its derivative.

Zero-Derivative Theorem (Section 4.2)

If $f'(x) = 0$ for all $x$ in an interval $(a, b)$, then $f$ is constant on $(a, b)$.

Equal Derivatives Corollary (Section 4.2)

If $f'(x) = g'(x)$ for all $x$ in an interval $(a, b)$, then $f - g$ is constant on $(a, b)$; that is, $f(x) = g(x) + c$ where $c$ is a constant.