### 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 $f\left(c\right)\ge f\left(x\right)$ for all $x$ in the domain of $f,$ we say $f$ has an absolute maximum at $c$

- absolute minimum
- if $f\left(c\right)\le f\left(x\right)$ for all $x$ in the domain of $f,$ we say $f$ has an absolute minimum at $c$

- antiderivative
- a function $F$ such that ${F}^{\prime}\left(x\right)=f\left(x\right)$ for all $x$ in the domain of $f$ is an antiderivative of $f$

- concave down
- if $f$ is differentiable over an interval $I$ and ${f}^{\prime}$ is decreasing over $I,$ then $f$ is concave down over $I$

- concave up
- if $f$ is differentiable over an interval $I$ and ${f}^{\prime}$ is increasing over $I,$ then $f$ is concave up over $I$

- concavity
- the upward or downward curve of the graph of a function

- concavity test
- suppose $f$ is twice differentiable over an interval $I;$ if ${f}^{\u2033}>0$ over $I,$ then $f$ is concave up over $I;$ if ${f}^{\u2033}<0$ over $I,$ then $f$ is concave down over $I$

- critical number
- if $f\prime \left(c\right)=0$ or $f\prime \left(c\right)$ is undefined, we say that $c$ is a critical number of $f$

- critical point
- the point $\left(c\text{,}f\right(c\left)\right)$ a critical point of $f$

- differential
- the differential $dx$ is an independent variable that can be assigned any nonzero real number; the differential $dy$ is defined to be $dy=f\prime (x)dx$

- differential form
- given a differentiable function $y=f\prime (x),$ the equation $dy=f\prime (x)dx$ 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 \text{\u2212}\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 ${f}^{\prime}$ changes sign from positive to negative as $x$ increases through $c,$ then $f$ has a local maximum at $c;$ if ${f}^{\prime}$ changes sign from negative to positive as $x$ increases through $c,$ then $f$ has a local minimum at $c;$ if ${f}^{\prime}$ does not change sign as $x$ increases through $c,$ then $f$ does not have a local extremum at $c$

- horizontal asymptote
- if $\underset{x\to \infty}{\text{lim}}f\left(x\right)=L$ or $\underset{x\to \text{\u2212}\infty}{\text{lim}}f\left(x\right)=L,$ then $y=L$ is a horizontal asymptote of $f$

- indefinite integral
- the most general antiderivative of $f\left(x\right)$ is the indefinite integral of $f;$ we use the notation $\int f\left(x\right)dx$ to denote the indefinite integral of $f$

- indeterminate forms
- when evaluating a limit, the forms $\frac{0}{0},$ $\infty \text{/}\infty ,$ $0\xb7\infty ,$ $\infty -\infty ,$ ${0}^{0},$ ${\infty}^{0},$ 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 $\left(c,f\left(c\right)\right)$ is an inflection point of $f$

- initial value problem
- a problem that requires finding a function $y$ that satisfies the differential equation $\frac{dy}{dx}=f\left(x\right)$ together with the initial condition $y\left({x}_{0}\right)={y}_{0}$

- iterative process
- process in which a list of numbers ${x}_{0},{x}_{1},{x}_{2},{x}_{3}\text{\u2026}$ is generated by starting with a number ${x}_{0}$ and defining ${x}_{n}=F\left({x}_{n-1}\right)$ for $n\ge 1$

- L’Hôpital’s rule
- if $f$ and $g$ are differentiable functions over an interval $a,$ except possibly at $a,$ and $\underset{x\to a}{\text{lim}}f\left(x\right)=0=\underset{x\to a}{\text{lim}}g\left(x\right)$ or $\underset{x\to a}{\text{lim}}f\left(x\right)$ and $\underset{x\to a}{\text{lim}}g\left(x\right)$ are infinite, then $\underset{x\to a}{\text{lim}}\frac{f\left(x\right)}{g\left(x\right)}=\underset{x\to a}{\text{lim}}\frac{{f}^{\prime}\left(x\right)}{{g}^{\prime}\left(x\right)},$ assuming the limit on the right exists or is $\infty $ or $\text{\u2212}\infty $

- limit at infinity
- the limiting value, if it exists, of a function as $x\to \infty $ or $x\to \text{\u2212}\infty $

- linear approximation
- the linear function $L(x)=f(a)+f\prime (a)(x-a)$ 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 $f\left(c\right)\ge f\left(x\right)$ for all $x\in I,$ we say $f$ has a local maximum at $c$

- local minimum
- if there exists an interval $I$ such that $f\left(c\right)\le f\left(x\right)$ 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 $\left(a,b\right),$ then there exists $c\in \left(a,b\right)$ such that

$${f}^{\prime}\left(c\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}$$

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

- oblique asymptote
- the line $y=mx+b$ if $f\left(x\right)$ approaches it as $x\to \infty $ or $x\to \text{\u2212}\infty $

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

- percentage error
- the relative error expressed as a percentage

- 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 $\text{\Delta}q$ for a particular quantity, $\frac{\text{\Delta}q}{q}$ is the relative error.

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

- second derivative test
- suppose ${f}^{\prime}\left(c\right)=0$ and ${f}^{\u2033}$ is continuous over an interval containing $c;$ if ${f}^{\u2033}\left(c\right)>0,$ then $f$ has a local minimum at $c;$ if ${f}^{\u2033}\left(c\right)<0,$ then $f$ has a local maximum at $c;$ if ${f}^{\u2033}\left(c\right)=0,$ 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 $x=a$