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 B(x - ) + k

• A amplitude; • B - period factor; period, p = 2/B - phase shift• 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.