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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<at<a 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)tavave=s(t)s(a)ta
constant multiple law for limits
the limit law limxacf(x)=c·limxaf(x)=cLlimxacf(x)=c·limxaf(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) limxaf(x)limxaf(x) exists, and (3) limxaf(x)=f(a)limxaf(x)=f(a)
continuity from the left
A function is continuous from the left at b if limxbf(x)=f(b)limxbf(x)=f(b)
continuity from the right
A function is continuous from the right at a if limxa+f(x)=f(a)limxa+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 limxa(f(x)g(x))=limxaf(x)limxag(x)=LMlimxa(f(x)g(x))=limxaf(x)limxag(x)=LM
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
limxaf(x)=Llimxaf(x)=L if for every ε>0,ε>0, there exists a δ>0δ>0 such that if 0<|xa|<δ,0<|xa|<δ, then |f(x)L|<ε|f(x)L|<ε
infinite discontinuity
An infinite discontinuity occurs at a point a if limxaf(x)=±limxaf(x)=± or limxa+f(x)=±limxa+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 limxaf(x)limxaf(x) and limxa+f(x)limxa+f(x) both exist, but limxaf(x)limxa+f(x)limxaf(x)limxa+f(x)
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 xaxa over some open interval containing a; assume that L and M are real numbers so that limxaf(x)=Llimxaf(x)=L and limxag(x)=M;limxag(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 limxa(f(x))n=(limxaf(x))n=Lnlimxa(f(x))n=(limxaf(x))n=Ln for every positive integer n
product law for limits
the limit law limxa(f(x)·g(x))=limxaf(x)·limxag(x)=L·Mlimxa(f(x)·g(x))=limxaf(x)·limxag(x)=L·M
quotient law for limits
the limit law limxaf(x)g(x)=limxaf(x)limxag(x)=LMlimxaf(x)g(x)=limxaf(x)limxag(x)=LM for M0M0
removable discontinuity
A removable discontinuity occurs at a point a if f(x)f(x) is discontinuous at a, but limxaf(x)limxaf(x) exists
root law for limits
the limit law limxaf(x)n=limxaf(x)n=Lnlimxaf(x)n=limxaf(x)n=Ln for all L if n is odd and for L0L0 if n is even
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)xamsec=f(x)f(a)xa
squeeze theorem
states that if f(x)g(x)h(x)f(x)g(x)h(x) for all xaxa over an open interval containing a and limxaf(x)=L=limxah(x)limxaf(x)=L=limxah(x) where L is a real number, then limxag(x)=Llimxag(x)=L
sum law for limits
The limit law limxa(f(x)+g(x))=limxaf(x)+limxag(x)=L+Mlimxa(f(x)+g(x))=limxaf(x)+limxag(x)=L+M
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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