Key Terms

absolute extremum: if $f$ has an absolute maximum or absolute minimum at $c,$ we say $f$ has an absolute extremum at $c$

absolute maximum: if $f (c) \geq f (x)$ for all $x$ in the domain of $f,$ we say $f$ has an absolute maximum at $c$

absolute minimum: if $f (c) \leq f (x)$ for all $x$ in the domain of $f,$ we say $f$ has an absolute minimum at $c$

antiderivative: a function $F$ such that $F^{'} (x) = f (x)$ for all $x$ in the domain of $f$ is an antiderivative of $f$

concave down: if $f$ is differentiable over an interval $I$ and $f^{'}$ is decreasing over $I,$ then $f$ is concave down over $I$

concave up: if $f$ is differentiable over an interval $I$ and $f^{'}$ is increasing over $I,$ then $f$ is concave up over $I$

concavity test: suppose $f$ is twice differentiable over an interval $I;$ if $f^{″} > 0$ over $I,$ then $f$ is concave up over $I;$ if $f^{″} < 0$ over $I,$ then $f$ is concave down over $I$

critical number: if $f' (c) = 0$ or $f' (c)$ is undefined, we say that $c$ is a critical number of $f$

differential: the differential $d x$ is an independent variable that can be assigned any nonzero real number; the differential $d y$ is defined to be $d y = f' (x) d x$

differential form: given a differentiable function $y = f' (x),$ the equation $d y = f' (x) d x$ is the differential form of the derivative of $y$ with respect to $x$

end behavior: the behavior of a function as $x \to \infty$ and $x \to − \infty$

extreme value theorem: if $f$ is a continuous function over a finite, closed interval, then $f$ has an absolute maximum and an absolute minimum

Fermat’s theorem: if $f$ has a local extremum at $c,$ then $c$ is a critical point of $f$

first derivative test: let $f$ be a continuous function over an interval $I$ containing a critical point $c$ such that $f$ is differentiable over $I$ except possibly at $c;$ if $f^{'}$ changes sign from positive to negative as $x$ increases through $c,$ then $f$ has a local maximum at $c;$ if $f^{'}$ changes sign from negative to positive as $x$ increases through $c,$ then $f$ has a local minimum at $c;$ if $f^{'}$ does not change sign as $x$ increases through $c,$ then $f$ does not have a local extremum at $c$

horizontal asymptote: if $lim_{x \to \infty} f (x) = L$ or $lim_{x \to − \infty} f (x) = L,$ then $y = L$ is a horizontal asymptote of $f$

indefinite integral: the most general antiderivative of $f (x)$ is the indefinite integral of $f;$ we use the notation $\int f (x) d x$ to denote the indefinite integral of $f$

indeterminate forms: when evaluating a limit, the forms $\frac{0}{0},$ $\infty / \infty,$ $0 \cdot \infty,$ $\infty - \infty,$ $0^{0},$ $\infty^{0},$ and $1^{\infty}$ are considered indeterminate because further analysis is required to determine whether the limit exists and, if so, what its value is

infinite limit at infinity: a function that becomes arbitrarily large as x becomes large

inflection point: if $f$ is continuous at $c$ and $f$ changes concavity at $c,$ the point $(c, f (c))$ is an inflection point of $f$

initial value problem: a problem that requires finding a function $y$ that satisfies the differential equation $\frac{d y}{d x} = f (x)$ together with the initial condition $y (x_{0}) = y_{0}$

iterative process: process in which a list of numbers $x_{0}, x_{1}, x_{2}, x_{3} …$ is generated by starting with a number $x_{0}$ and defining $x_{n} = F (x_{n - 1})$ for $n \geq 1$

L’Hôpital’s rule: if $f$ and $g$ are differentiable functions over an interval $a,$ except possibly at $a,$ and $lim_{x \to a} f (x) = 0 = lim_{x \to a} g (x)$ or $lim_{x \to a} f (x)$ and $lim_{x \to a} g (x)$ are infinite, then $lim_{x \to a} \frac{f (x)}{g (x)} = lim_{x \to a} \frac{f^{'} (x)}{g^{'} (x)},$ assuming the limit on the right exists or is $\infty$ or $− \infty$

limit at infinity: the limiting value, if it exists, of a function as $x \to \infty$ or $x \to − \infty$

linear approximation: the linear function $L (x) = f (a) + f' (a) (x - a)$ is the linear approximation of $f$ at $x = a$

local extremum: if $f$ has a local maximum or local minimum at $c,$ we say $f$ has a local extremum at $c$

local maximum: if there exists an interval $I$ such that $f (c) \geq f (x)$ for all $x \in I,$ we say $f$ has a local maximum at $c$

local minimum: if there exists an interval $I$ such that $f (c) \leq f (x)$ for all $x \in I,$ we say $f$ has a local minimum at $c$

mean value theorem: if $f$ is continuous over $[a, b]$ and differentiable over $(a, b),$ then there exists $c \in (a, b)$ such that

$f^{'} (c) = \frac{f (b) - f (a)}{b - a}$

Newton’s method: method for approximating roots of $f (x) = 0;$ using an initial guess $x_{0};$ each subsequent approximation is defined by the equation $x_{n} = x_{n - 1} - \frac{f (x_{n - 1})}{f' (x_{n - 1})}$

oblique asymptote: the line $y = m x + b$ if $f (x)$ approaches it as $x \to \infty$ or $x \to − \infty$

optimization problems: problems that are solved by finding the maximum or minimum value of a function

propagated error: the error that results in a calculated quantity $f (x)$ resulting from a measurement error dx

related rates: are rates of change associated with two or more related quantities that are changing over time

relative error: given an absolute error $Δ q$ for a particular quantity, $\frac{Δ q}{q}$ is the relative error.

rolle’s theorem: if $f$ is continuous over $[a, b]$ and differentiable over $(a, b),$ and if $f (a) = f (b),$ then there exists $c \in (a, b)$ such that $f^{'} (c) = 0$

second derivative test: suppose $f^{'} (c) = 0$ and $f^{″}$ is continuous over an interval containing $c;$ if $f^{″} (c) > 0,$ then $f$ has a local minimum at $c;$ if $f^{″} (c) < 0,$ then $f$ has a local maximum at $c;$ if $f^{″} (c) = 0,$ then the test is inconclusive

tangent line approximation (linearization): since the linear approximation of $f$ at $x = a$ is defined using the equation of the tangent line, the linear approximation of $f$ at $x = a$ is also known as the tangent line approximation to $f$ at $x = a$