Calculus Volume 1

# Key Terms

absolute extremum
if $ff$ has an absolute maximum or absolute minimum at $c,c,$ we say $ff$ has an absolute extremum at $cc$
absolute maximum
if $f(c)≥f(x)f(c)≥f(x)$ for all $xx$ in the domain of $f,f,$ we say $ff$ has an absolute maximum at $cc$
absolute minimum
if $f(c)≤f(x)f(c)≤f(x)$ for all $xx$ in the domain of $f,f,$ we say $ff$ has an absolute minimum at $cc$
antiderivative
a function $FF$ such that $F′(x)=f(x)F′(x)=f(x)$ for all $xx$ in the domain of $ff$ is an antiderivative of $ff$
concave down
if $ff$ is differentiable over an interval $II$ and $f′f′$ is decreasing over $I,I,$ then $ff$ is concave down over $II$
concave up
if $ff$ is differentiable over an interval $II$ and $f′f′$ is increasing over $I,I,$ then $ff$ is concave up over $II$
concavity
the upward or downward curve of the graph of a function
concavity test
suppose $ff$ is twice differentiable over an interval $I;I;$ if $f″>0f″>0$ over $I,I,$ then $ff$ is concave up over $I;I;$ if $f″<0f″<0$ over $I,I,$ then $ff$ is concave down over $II$
critical point
if $f′(c)=0f′(c)=0$ or $f′(c)f′(c)$ is undefined, we say that $cc$ is a critical point of $ff$
differential
the differential $dxdx$ is an independent variable that can be assigned any nonzero real number; the differential $dydy$ is defined to be $dy=f′(x)dxdy=f′(x)dx$
differential form
given a differentiable function $y=f′(x),y=f′(x),$ the equation $dy=f′(x)dxdy=f′(x)dx$ is the differential form of the derivative of $yy$ with respect to $xx$
end behavior
the behavior of a function as $x→∞x→∞$ and $x→−∞x→−∞$
extreme value theorem
if $ff$ is a continuous function over a finite, closed interval, then $ff$ has an absolute maximum and an absolute minimum
Fermat’s theorem
if $ff$ has a local extremum at $c,c,$ then $cc$ is a critical point of $ff$
first derivative test
let $ff$ be a continuous function over an interval $II$ containing a critical point $cc$ such that $ff$ is differentiable over $II$ except possibly at $c;c;$ if $f′f′$ changes sign from positive to negative as $xx$ increases through $c,c,$ then $ff$ has a local maximum at $c;c;$ if $f′f′$ changes sign from negative to positive as $xx$ increases through $c,c,$ then $ff$ has a local minimum at $c;c;$ if $f′f′$ does not change sign as $xx$ increases through $c,c,$ then $ff$ does not have a local extremum at $cc$
horizontal asymptote
if $limx→∞f(x)=Llimx→∞f(x)=L$ or $limx→−∞f(x)=L,limx→−∞f(x)=L,$ then $y=Ly=L$ is a horizontal asymptote of $ff$
indefinite integral
the most general antiderivative of $f(x)f(x)$ is the indefinite integral of $f;f;$ we use the notation $∫f(x)dx∫f(x)dx$ to denote the indefinite integral of $ff$
indeterminate forms
when evaluating a limit, the forms $00,00,$ $∞/∞,∞/∞,$ $0·∞,0·∞,$ $∞−∞,∞−∞,$ $00,00,$ $∞0,∞0,$ and $1∞1∞$ 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 $ff$ is continuous at $cc$ and $ff$ changes concavity at $c,c,$ the point $(c,f(c))(c,f(c))$ is an inflection point of $ff$
initial value problem
a problem that requires finding a function $yy$ that satisfies the differential equation $dydx=f(x)dydx=f(x)$ together with the initial condition $y(x0)=y0y(x0)=y0$
iterative process
process in which a list of numbers $x0,x1,x2,x3…x0,x1,x2,x3…$ is generated by starting with a number $x0x0$ and defining $xn=F(xn−1)xn=F(xn−1)$ for $n≥1n≥1$
L’Hôpital’s rule
if $ff$ and $gg$ are differentiable functions over an interval $a,a,$ except possibly at $a,a,$ and $limx→af(x)=0=limx→ag(x)limx→af(x)=0=limx→ag(x)$ or $limx→af(x)limx→af(x)$ and $limx→ag(x)limx→ag(x)$ are infinite, then $limx→af(x)g(x)=limx→af′(x)g′(x),limx→af(x)g(x)=limx→af′(x)g′(x),$ assuming the limit on the right exists or is $∞∞$ or $−∞−∞$
limit at infinity
the limiting value, if it exists, of a function as $x→∞x→∞$ or $x→−∞x→−∞$
linear approximation
the linear function $L(x)=f(a)+f′(a)(x−a)L(x)=f(a)+f′(a)(x−a)$ is the linear approximation of $ff$ at $x=ax=a$
local extremum
if $ff$ has a local maximum or local minimum at $c,c,$ we say $ff$ has a local extremum at $cc$
local maximum
if there exists an interval $II$ such that $f(c)≥f(x)f(c)≥f(x)$ for all $x∈I,x∈I,$ we say $ff$ has a local maximum at $cc$
local minimum
if there exists an interval $II$ such that $f(c)≤f(x)f(c)≤f(x)$ for all $x∈I,x∈I,$ we say $ff$ has a local minimum at $cc$
mean value theorem
if $ff$ is continuous over $[a,b][a,b]$ and differentiable over $(a,b),(a,b),$ then there exists $c∈(a,b)c∈(a,b)$ such that
$f′(c)=f(b)−f(a)b−af′(c)=f(b)−f(a)b−a$
Newton’s method
method for approximating roots of $f(x)=0;f(x)=0;$ using an initial guess $x0;x0;$ each subsequent approximation is defined by the equation $xn=xn−1−f(xn−1)f′(xn−1)xn=xn−1−f(xn−1)f′(xn−1)$
oblique asymptote
the line $y=mx+by=mx+b$ if $f(x)f(x)$ approaches it as $x→∞x→∞$ or $x→−∞x→−∞$
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 $f(x)f(x)$ 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 $ΔqΔq$ for a particular quantity, $ΔqqΔqq$ is the relative error.
rolle’s theorem
if $ff$ is continuous over $[a,b][a,b]$ and differentiable over $(a,b),(a,b),$ and if $f(a)=f(b),f(a)=f(b),$ then there exists $c∈(a,b)c∈(a,b)$ such that $f′(c)=0f′(c)=0$
second derivative test
suppose $f′(c)=0f′(c)=0$ and $f″f″$ is continuous over an interval containing $c;c;$ if $f″(c)>0,f″(c)>0,$ then $ff$ has a local minimum at $c;c;$ if $f″(c)<0,f″(c)<0,$ then $ff$ has a local maximum at $c;c;$ if $f″(c)=0,f″(c)=0,$ then the test is inconclusive
tangent line approximation (linearization)
since the linear approximation of $ff$ at $x=ax=a$ is defined using the equation of the tangent line, the linear approximation of $ff$ at $x=ax=a$ is also known as the tangent line approximation to $ff$ at $x=ax=a$