ESSENTIAL FORMULAS IN MATHEMATICS

7/25/2019 ESSENTIAL FORMULAS IN MATHEMATICS

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MATH 150 Review

Lines

y = mx + b (slope-intercept equation of a line)

y = b is the equation of a horizontal line

The slope of a

nonvertical line is

given by the formulam

x

y=

Secant Lines

To calculate f(x) using a secant line approximation of the tangent line

1. alculate the difference quotient h

xfhxf )()( +

.

!. "et h approach zero.

#. The quantity h

xfhxf )()( +

$ill approach f(x).

f(a) = h

afhaf

h

)()(lim

%

+


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Limit Theorems

&uppose that

)(lim xfax an'

)(lim xgax both exist. Then $e hae the

follo$ing theorems.

Limit of a Polynomial Function

"et p(x) be a polynomial function an' a any number. Then

)()(lim apxpax

=

Limit of a Rational Function

"et r(x) = p(x)q(x) be a rational function $here p(x) an' q(x) arepolynomials. "et a be a number such that q(a) 'oes not equal zero. Then*

)()(lim arxrax

=

Derivative via Limits

The basic rules of 'ifferentiation are obtaine' from the limit 'efinition of the

'eriatie. There are three main steps for calculating the 'eriatie of a

function f(x) at x = a.

1. rite the difference quotient h

afhaf )()( +

.

!. &implify the 'ifference quotient.#. ,in' the limit as h approaches zero.

ontinuity an! Differentia"ility

e say that a function is continuous at x = a if its graph has no breas or gapsas it passes through the point (a* f(a)). f a function f(x) is continuousat x =

a* it shoul' be possible to setch its graph $ithout lifting the pencil from the

paper at the point (a* f(a)).

If f(x) is differentiable at x = a, then f(x) is continuous at x = a.

A function f(x) is continuous at x = a provided the following limit relation

holds:

)()(lim afxfax

=

In order for this to hold, three conditions must be fulfilled.

. f(x) must be defined at x = a

!.

)(lim xfax must exist

". #he limit

)(lim xfax must have the value f(a)

!


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#asic Rules for Differentiation

f $e 'ifferentiate the function f(x)* $e obtain $hat is calle' the secon''eriatie of f(x)* 'enote' f(x). &o* $e hae

)(//)(/ xfxfdx

d=

Rates of han$e

The aerage rate of change of f(x) oer this interal is the change in f(x)'ii'e' by the length of the interal.

ab

afbf

x

y

bxaervaltheover

xfofchangeofrateaverage

=

=

)()(

int

)(

n the special case $here b is a+h* the alue of b 0 a is (a + h) 0 a or h* an' the

aerage rate of change of the function oer the interal is the 'ifference

quotient

h

afhaf )()( +

The 'eriatie f(a) measures the (instantaneous) rate of change of f(x) at x =a.

Motion% &elocity an! Acceleration

The average velocit$from t = ! to t = ! + h is

h

shs

elapsedtime

traveledcedis )!()!(tan +=

f s(t) 'enotes the position function of an obect moing in a straight line* then

the elocity (t) of an obect at time t is gien by

(t) = s(t)

f $e tae the 'eriatie of the elocity function (t)* $e get $hat is calle' the

acceleration function

a(t) = (t) = s(t)

A''ro(imatin$ the han$e in a Function

onsi'er the function f(x) near x = a. e no$ that

)(/)()(

afh

afhaf

+

hen h is small* hf(a) is a goo' approximation to the change in f(x). napplications* hf(a) is calculate' an' use' to estimate f(a+h) 0 f(a).

Mar$inal ost

2conomists often use the a'ectie marginalto 'enote a 'eriatie.

f (x) is a cost function* then the alue of the 'eriatie (a) is calle' the

marginal costat pro'uction leel a. The number (a) gies the rate at $hich

costs are increasing $ith respect to the leel of pro'uction $hen the pro'uction is

currently at leel a.

)ra'hin$

3 relatie maximum point is a point at $hich the graph changes from increasing

to 'ecreasing.

3 relatie minimum point is a point at $hich the graph changes from 'ecreasing

to increasing.

The maximum alue of a function is the largest alue that the function assumes onits 'omain.

The minimum alue of a function is the smallest alue that the function assumes

on its 'omain.

4ote5 ,unctions might or might not hae maximum an'or minimum alues.

e say that a function is concae up at x = a if there is an open interal on the x-

axis containing a throughout $hich the graph of f(x)lies aboe its tangent line.

e say that a function is concae 'o$n at x = a if there is an open interal on the

x-axis containing a throughout $hich the graph of f(x)lies belo$ its tangent line.

3n inflection point is a point on the graph of a function at $hich the function is

continuous an' the concaity of the graph changes* i.e.* goes from concae up toconcae 'o$n* or concae 'o$n to concae up.

The x-intercept is a point at $hich a graph intersects the x-axis. (x*%)

The y-intercept is a point at $hich the graph intersects the y-axis. (%*y)

6raphs sometimes straighten out an' approach some straight line as x increases

(or 'ecreases). Theses straight lines are calle' asymptotes.

3symptotes of a graph may be horizontal* ertical or 'iagonal.

First Derivative Rule

f f(a) 7 % then f(x) is increasing at x = a.

f f(a) 8 % then f(x) is 'ecreasing at x = a.

Secon! Derivative Rule

f f(a) 7 %* then f(x) is concae up at x = a.

f f(9(a) 8 %* then f(x) is concae 'o$n at x = a.

#


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%oo& for possible relative extreme points of f(x) b$ setting f'(x) = and

solving for x.

s the point a relatie maximum point or a relatie minimum point: ;o$ can

$e tell:

hec concaity at relatie extreme point using secon' 'eriatie.

2xamine slope of nearby points on either si'e using the first

'eriatie.

%oo& for possible points of inflection b$ setting f''(x) = and solving for x.

ost% Revenue an! Profit

(x) = cost of pro'ucing x units of a pro'uct


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Relate! Rates

There are some applications $herexan'yare relate' by an equation* an'both ariables are functions of a thir' ariable t. @ften the formulas forxan'

yas functions of tare not no$n.

hen $e 'ifferentiate such an equation $ith respect to t* $e 'erie a

relationship bet$een rates of change dt

dy

an' dt

dx

. e say that theses

'eriaties are related rates.

The equation relating the rates may be use' to fin' one of the rates $hen theother is no$n.

Laws of e('onents

Differentiatin$ ,('onential Functions

Differential ,-uations

&uppose thaty = f(x)satisfies the 'ifferential equation

Theyis an exponential function of the form

Pro'erties of ,('onential Functions

ures of the formy = ekx* $here kis positie* hae seeral properties in

common5

ures of the formy = ekx* $here kis negatie* hae seeral properties in

common5

Pro'erties of +atural Lo$arithms

1. The point (1*%) is on the graph of y = ln x Abecause (%*1) is on thegraph of y = exB. &o*ln 1 = %.

!. ln x is 'efine' only for positie alues of x.

#. ln x is negatie for x bet$een % an' 1.

?. ln x is positie for x greater than 1.

C. ln x is an increasing function an' concae 'o$n.

Differentiatin$ +atural Lo$arithmic Functions

xx

dx

d 1)(ln =

More Pro'erties of Lo$arithms

More on Differential ,-uations

The function y . e/tsatisfies the 'ifferential equation

y . /y

onersely* if y . ft2satisfies the 'ifferential equation* then

y . e/t

for some constant .

t is important to note that if ft2 . e/t* then by setting t . 0* $e hae

f02 . e0.

&o* is the alue of f(t) at t = %.

C


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Functions for com'oun! interest

hen interest is compoun'e' continuously* the compoun' amount 3(t) is an

exponential function of the number of years t that interest is earne'

3(t) = ert

an' 3(t) satisfies the 'ifferential equation

3(t) = r 3(t)

Relative Rates of han$e


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Area an! the Definite 3nte$ral

&uppose that f(x) is continuous on the interal bxa . Then

b

adxxf )(

is equal to the area aboe the x-axis boun'e' by the graph of y = f(x) from

x = a to x = b minus the area belo$ the x-axis.

Fun!amental Theorem of alculus

&uppose that f(x) is continuous on the interal bxa * an' let ,(x) bean anti'eriatie of f(x). Then

)()()( abdxxfb

a=

This theorem connects the t$o ey concepts of calculus 0 the integral an' the

'eriatie.

,(b) 0 ,(a) is calle' the net change of ,(x) from a to b. t is represente'

symbolically by

Area #etween Two urves

f y = f(x) lies aboe y = g(x) from x = a to x = b* then the area of the region

bet$een f(x) an' g(x) from x = a to x = b is

The Avera$e &alue of a Function

"et f(x) be a continuous function on the interal bxa . The 'efiniteintegral may be use' to 'efine the aerage alue of f(x) on this interal.

The aerage alue of a continuous function f(x) oer the interal

bxa is 'efine' as the quantity

b

adxxf

ab)(

1

onsumers Sur'lus

The consumers surplus for a commo'ity haing 'eman' cure p = f(x) is

!

dx"xf%

B)(A

$here the quantity 'eman'e' is 3 an' the price is H = f(3).

I


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Future &alue of an 3ncome Stream

The future alue* of a continuous income stream* of J 'ollars per year for 4

years at interest rate r compoun'e' continuously is

#t#r dt$e

%

)(

$here is any constant.

K

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ESSENTIAL FORMULAS IN MATHEMATICS