Calculus I Hughes- Hallett Math 131 Br. Joel Baumeyer Christian Brothers University
Calculus I Hughes-Hallett
Math 131Br. Joel Baumeyer
Christian Brothers University
Function: (Data Point of View) One quantity H, is a function of
another, t, if each value of t has a unique value of H associated with it. In symbols: H = f(t).
We say H is the value of the function or the dependent variable or output; and
t is the argument or independent variable or input.
Working Definition of Function: H = f(t)
A function is a rule (equation) which assigns to each element of the domain (independent variable) one and only one element of the range (dependent variable).
Working definition of function continued:
Domain is the set of all possible values of the independent variable (t).
Range is the corresponding set of values of the dependent variable (H).
Questions?
General Types of Functions (Examples):
Linear: y = m(x) + b; proportion: y = kx Polynomial: Quadratic: y =x2 ; Cubic: y= x3 ;
etc Power Functions: y = kxp
Trigonometric: y = sin x, y = Arctan x Exponential: y = aebx ; Logarithmic: y = ln x
Graph of a Function:
The graph of a function is all the points in the Cartesian plane whose coordinates make the rule (equation) of the function a true statement.
Slope
• m - slope :
b: y-intercept• a: x-intercept
• .
run
rise
x
y
xx
yym
12
12
sintpoareyxandyx 2211 ,,
5 Forms of the Linear Equation
• Slope-intercept: y = f(x) = b + mx• Slope-point:• Two point:
• Two intercept:
• General Form: Ax + By = C
)( 11 xxmyy
)( 112
121 xx
xx
yyyy
1b
y
a
x
Exponential Functions: If a > 1, growth; a<1, decay
• If r is the growth rate then a = 1 + r, and
• If r is the decay rate then a = 1 - r, and
taPP 0
tt rPaPP )1(00 0P
tt rPaPP )1(00
Definitions and Rules of Exponentiation:• D1: • D2:• R1:
• R2:
• R3:
0,,,1 1110 aaandaa xa
xa
evennforaaaandaa nn 0;1
21
txtx aaa
txt
x
aa
a
xttx aa
Inverse Functions:
• Two functions z = f(x) and z = g(x) are inverse functions if the following four statements are true:
• Domain of f equals the range of g.• Range of f equals the domain of g.• f(g(x)) = x for all x in the domain of g.• g(f(y)) = y for all y in the domain of f.
)()( 1 xfxg
A logarithm is an exponent.
.
bameanscb
generalinand
xemeanscxx
xmeanscx
ca
ce
c
log
:
,logln
10log10
General Rules of Logarithms:
log(a•b) = log(a) + log(b) log(a/b) = log(a) - log(b)
b
aaalso
ccbecause
xcandxc
apa
c
cb
c
xxc
p
c
log
loglog
0,101log
log
)log()log(
0
log
e = 2.718281828459045...
• Any exponential function
can be written in terms of e by using the fact that
So that
kxaby beb ln
kxbkx ea yaby )(becomes ln
Making New Functions from Old
Given y = f(x):
(y - b) =k f(x - a) stretches f(x) if |k| > 1
shrinks f(x) if |k| < 1
reverses y values if k is negative
a moves graph right or left, a + or a -
b moves graph up or down, b + or b -
If f(-x) = f(x) then f is an “even” function.
If f(-x) = -f(x) then f is an “odd” function.
Polynomials:
• A polynomial of the nth degree has n roots if complex numbers a allowed.
• Zeros of the function are roots of the equation.
• The graph can have at most n - 1 bends.• The leading coefficient determines the
position of the graph for |x| very large.
nn
kn
kk xaxaxaxay
10
00
na
Rational Function: y = f(x) = p(x)/q(x)where p(x) and q(x) are polynomials.• Any value of x that makes q(x) = 0 is called a
vertical asymptote of f(x).• If f(x) approaches a finite value a as x gets larger
and larger in absolute value without stopping, then a is horizontal asymptote of f(x) and we write:
• An asymptote is a “line” that a curve approaches but never reaches.
axfx
)(lim
Asymptote Tests y = h(x) =f(x)/g(x)
• Vertical Asymptotes: Solve: g(x) = 0If y as x K, where g(K) = 0,
then x = K is a vertical asymptote.• Horizontal Asymptotes:
If f(x) L as x then y = L is a vertical asymptote. Write h(x) as:
, where n is the highest power of x in f(x) or g(x).
n
n
x1
x1
)x(g
)x(f)x(h
Basic Trig
• radian measure: = s/r and thus s = r , • Know triangle and circle definitions of the
trig functions.• y = A sin (Bx + C) + k
• A amplitude; • B - period factor; period, p = 2/B - phase shift; -C/B• k (raise or lower graph factor)
Continuity of y = f(x)
• A function is said to be continuous if there are no “breaks” in its graph.
• A function is continuous at a point x = a if the value of f(x) L, a number, as x a for values of x either greater or less than a.
Intermediate Value Theorem
• Suppose f is continuous on a closed interval [a,b]. If k is any number between f(a) and f(b) then there is at least one number c in [a,b] such that f(x) = k.