 Calculus Volume 1

# Key Terms

### Key Terms

average velocity
the change in an object’s position divided by the length of a time period; the average velocity of an object over a time interval $[t,a][t,a]$ (if $t or $[a,t][a,t]$ if $t>a)t>a)$, with a position given by $s(t),s(t),$ that is $vave=s(t)−s(a)t−avave=s(t)−s(a)t−a$
constant multiple law for limits
the limit law $limx→acf(x)=c·limx→af(x)=cLlimx→acf(x)=c·limx→af(x)=cL$
continuity at a point
A function $f(x)f(x)$ is continuous at a point a if and only if the following three conditions are satisfied: (1) $f(a)f(a)$ is defined, (2) $limx→af(x)limx→af(x)$ exists, and (3) $limx→af(x)=f(a)limx→af(x)=f(a)$
continuity from the left
A function is continuous from the left at b if $limx→b−f(x)=f(b)limx→b−f(x)=f(b)$
continuity from the right
A function is continuous from the right at a if $limx→a+f(x)=f(a)limx→a+f(x)=f(a)$
continuity over an interval
a function that can be traced with a pencil without lifting the pencil; a function is continuous over an open interval if it is continuous at every point in the interval; a function $f(x)f(x)$ is continuous over a closed interval of the form $[a,b][a,b]$ if it is continuous at every point in $(a,b),(a,b),$ and it is continuous from the right at a and from the left at b
difference law for limits
the limit law $limx→a(f(x)−g(x))=limx→af(x)−limx→ag(x)=L−Mlimx→a(f(x)−g(x))=limx→af(x)−limx→ag(x)=L−M$
differential calculus
the field of calculus concerned with the study of derivatives and their applications
discontinuity at a point
A function is discontinuous at a point or has a discontinuity at a point if it is not continuous at the point
epsilon-delta definition of the limit
$limx→af(x)=Llimx→af(x)=L$ if for every $ε>0,ε>0,$ there exists a $δ>0δ>0$ such that if $0<|x−a|<δ,0<|x−a|<δ,$ then $|f(x)−L|<ε|f(x)−L|<ε$
infinite discontinuity
An infinite discontinuity occurs at a point a if $limx→a−f(x)=±∞limx→a−f(x)=±∞$ or $limx→a+f(x)=±∞limx→a+f(x)=±∞$
infinite limit
A function has an infinite limit at a point a if it either increases or decreases without bound as it approaches a
instantaneous velocity
The instantaneous velocity of an object with a position function that is given by $s(t)s(t)$ is the value that the average velocities on intervals of the form $[t,a][t,a]$ and $[a,t][a,t]$ approach as the values of t move closer to $a,a,$ provided such a value exists
integral calculus
the study of integrals and their applications
Intermediate Value Theorem
Let f be continuous over a closed bounded interval $[a,b];[a,b];$ if z is any real number between $f(a)f(a)$ and $f(b),f(b),$ then there is a number c in $[a,b][a,b]$ satisfying $f(c)=zf(c)=z$
intuitive definition of the limit
If all values of the function $f(x)f(x)$ approach the real number L as the values of $x(≠a)x(≠a)$ approach a, $f(x)f(x)$ approaches L
jump discontinuity
A jump discontinuity occurs at a point a if $limx→a−f(x)limx→a−f(x)$ and $limx→a+f(x)limx→a+f(x)$ both exist, but $limx→a−f(x)≠limx→a+f(x)limx→a−f(x)≠limx→a+f(x)$
limit
the process of letting x or t approach a in an expression; the limit of a function $f(x)f(x)$ as x approaches a is the value that $f(x)f(x)$ approaches as x approaches a
limit laws
the individual properties of limits; for each of the individual laws, let $f(x)f(x)$ and $g(x)g(x)$ be defined for all $x≠ax≠a$ over some open interval containing a; assume that L and M are real numbers so that $limx→af(x)=Llimx→af(x)=L$ and $limx→ag(x)=M;limx→ag(x)=M;$ let c be a constant
multivariable calculus
the study of the calculus of functions of two or more variables
one-sided limit
A one-sided limit of a function is a limit taken from either the left or the right
power law for limits
the limit law $limx→a(f(x))n=(limx→af(x))n=Lnlimx→a(f(x))n=(limx→af(x))n=Ln$ for every positive integer n
product law for limits
the limit law $limx→a(f(x)·g(x))=limx→af(x)·limx→ag(x)=L·Mlimx→a(f(x)·g(x))=limx→af(x)·limx→ag(x)=L·M$
quotient law for limits
the limit law $limx→af(x)g(x)=limx→af(x)limx→ag(x)=LMlimx→af(x)g(x)=limx→af(x)limx→ag(x)=LM$ for $M≠0M≠0$
removable discontinuity
A removable discontinuity occurs at a point a if $f(x)f(x)$ is discontinuous at a, but $limx→af(x)limx→af(x)$ exists
root law for limits
the limit law $limx→af(x)n=limx→af(x)n=Lnlimx→af(x)n=limx→af(x)n=Ln$ for all L if n is odd and for $L≥0L≥0$ if n is even
secant
A secant line to a function $f(x)f(x)$ at a is a line through the point $(a,f(a))(a,f(a))$ and another point on the function; the slope of the secant line is given by $msec=f(x)−f(a)x−amsec=f(x)−f(a)x−a$
squeeze theorem
states that if $f(x)≤g(x)≤h(x)f(x)≤g(x)≤h(x)$ for all $x≠ax≠a$ over an open interval containing a and $limx→af(x)=L=limx→ah(x)limx→af(x)=L=limx→ah(x)$ where L is a real number, then $limx→ag(x)=Llimx→ag(x)=L$
sum law for limits
The limit law $limx→a(f(x)+g(x))=limx→af(x)+limx→ag(x)=L+Mlimx→a(f(x)+g(x))=limx→af(x)+limx→ag(x)=L+M$
tangent
A tangent line to the graph of a function at a point $(a,f(a))(a,f(a))$ is the line that secant lines through $(a,f(a))(a,f(a))$ approach as they are taken through points on the function with x-values that approach a; the slope of the tangent line to a graph at a measures the rate of change of the function at a
triangle inequality
If a and b are any real numbers, then $|a+b|≤|a|+|b||a+b|≤|a|+|b|$
vertical asymptote
A function has a vertical asymptote at $x=ax=a$ if the limit as x approaches a from the right or left is infinite
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