Calculus Volume 1

# Key Terms

absolute value function
$f(x)={−x,x<0x,x≥0f(x)={−x,x<0x,x≥0$
algebraic function
a function involving any combination of only the basic operations of addition, subtraction, multiplication, division, powers, and roots applied to an input variable $xx$
base
the number $bb$ in the exponential function $f(x)=bxf(x)=bx$ and the logarithmic function $f(x)=logbxf(x)=logbx$
composite function
given two functions $ff$ and $g,g,$ a new function, denoted $g∘f,g∘f,$ such that $(g∘f)(x)=g(f(x))(g∘f)(x)=g(f(x))$
cubic function
a polynomial of degree 3; that is, a function of the form $f(x)=ax3+bx2+cx+d,f(x)=ax3+bx2+cx+d,$ where $a≠0a≠0$
decreasing on the interval $II$
a function decreasing on the interval $II$ if, for all $x1,x2∈I,f(x1)≥f(x2)x1,x2∈I,f(x1)≥f(x2)$ if $x1
degree
for a polynomial function, the value of the largest exponent of any term
dependent variable
the output variable for a function
domain
the set of inputs for a function
even function
a function is even if $f(−x)=f(x)f(−x)=f(x)$ for all $xx$ in the domain of $ff$
exponent
the value $xx$ in the expression $bxbx$
function
a set of inputs, a set of outputs, and a rule for mapping each input to exactly one output
graph of a function
the set of points $(x,y)(x,y)$ such that $xx$ is in the domain of $ff$ and $y=f(x)y=f(x)$
horizontal line test
a function $ff$ is one-to-one if and only if every horizontal line intersects the graph of $f,f,$ at most, once
hyperbolic functions
the functions denoted $sinh,cosh,tanh,csch,sech,sinh,cosh,tanh,csch,sech,$ and $coth,coth,$ which involve certain combinations of $exex$ and $e−xe−x$
increasing on the interval $II$
a function increasing on the interval $II$ if for all $x1,x2∈I,f(x1)≤f(x2)x1,x2∈I,f(x1)≤f(x2)$ if $x1
independent variable
the input variable for a function
inverse function
for a function $f,f,$ the inverse function $f−1f−1$ satisfies $f−1(y)=xf−1(y)=x$ if $f(x)=yf(x)=y$
inverse hyperbolic functions
the inverses of the hyperbolic functions where $coshcosh$ and $sechsech$ are restricted to the domain $[0,∞);[0,∞);$ each of these functions can be expressed in terms of a composition of the natural logarithm function and an algebraic function
inverse trigonometric functions
the inverses of the trigonometric functions are defined on restricted domains where they are one-to-one functions
linear function
a function that can be written in the form $f(x)=mx+bf(x)=mx+b$
logarithmic function
a function of the form $f(x)=logb(x)f(x)=logb(x)$ for some base $b>0,b≠1b>0,b≠1$ such that $y=logb(x)y=logb(x)$ if and only if $by=xby=x$
mathematical model
A method of simulating real-life situations with mathematical equations
natural exponential function
the function $f(x)=exf(x)=ex$
natural logarithm
the function $lnx=logexlnx=logex$
number e
as $mm$ gets larger, the quantity $(1+(1/m)m(1+(1/m)m$ gets closer to some real number; we define that real number to be $e;e;$ the value of $ee$ is approximately $2.7182822.718282$
odd function
a function is odd if $f(−x)=−f(x)f(−x)=−f(x)$ for all $xx$ in the domain of $ff$
one-to-one function
a function $ff$ is one-to-one if $f(x1)≠f(x2)f(x1)≠f(x2)$ if $x1≠x2x1≠x2$
periodic function
a function is periodic if it has a repeating pattern as the values of $xx$ move from left to right
piecewise-defined function
a function that is defined differently on different parts of its domain
point-slope equation
equation of a linear function indicating its slope and a point on the graph of the function
polynomial function
a function of the form $f(x)=anxn+an−1xn−1+…+a1x+a0f(x)=anxn+an−1xn−1+…+a1x+a0$
power function
a function of the form $f(x)=xnf(x)=xn$ for any positive integer $n≥1n≥1$
a polynomial of degree 2; that is, a function of the form $f(x)=ax2+bx+cf(x)=ax2+bx+c$ where $a≠0a≠0$
for a circular arc of length $ss$ on a circle of radius 1, the radian measure of the associated angle $θθ$ is $ss$
range
the set of outputs for a function
rational function
a function of the form $f(x)=p(x)/q(x),f(x)=p(x)/q(x),$ where $p(x)p(x)$ and $q(x)q(x)$ are polynomials
restricted domain
a subset of the domain of a function $ff$
root function
a function of the form $f(x)=x1/nf(x)=x1/n$ for any integer $n≥2n≥2$
slope
the change in y for each unit change in x
slope-intercept form
equation of a linear function indicating its slope and y-intercept
the graph of a function $ff$ is symmetric about the origin if $(−x,−y)(−x,−y)$ is on the graph of $ff$ whenever $(x,y)(x,y)$ is on the graph
the graph of a function $ff$ is symmetric about the $yy$-axis if $(−x,y)(−x,y)$ is on the graph of $ff$ whenever $(x,y)(x,y)$ is on the graph
table of values
a table containing a list of inputs and their corresponding outputs
transcendental function
a function that cannot be expressed by a combination of basic arithmetic operations
transformation of a function
a shift, scaling, or reflection of a function
trigonometric functions
functions of an angle defined as ratios of the lengths of the sides of a right triangle
trigonometric identity
an equation involving trigonometric functions that is true for all angles $θθ$ for which the functions in the equation are defined
vertical line test
given the graph of a function, every vertical line intersects the graph, at most, once
zeros of a function
when a real number $xx$ is a zero of a function $f,f(x)=0f,f(x)=0$