Formulas Review Sheet Answers

FORMULAS REVIEW SHEETANSWERS

1) SURFACE AREA FOR A PARAMETRIC FUNCTION

2 2

2f

i

t

t

dx dyS dt

dt dt

2) TRAPEZOIDAL APPROXIMATION OF THE AREA UNDER A CURVE (BOTH FORMS)

• Recall that Tn was all about approximating the area under a curve. If you subdivide an interval [a, b] into equal sized subintervals, then you can imagine a string of inputs or points c0, c1, c2, c3, …, cn-1, cn between a and b, and you can write the trapezoidal sum as Ln is the left-hand approximation and Rn is the right hand approximation for the area under a curve.

3) THE MACLAUREN SERIES FOR….

• ;

• ;

1 1ln(1 ), , ln(1 ), &

1 1x x

x x

4) LIMIT DEFINITION OF THE DERIVATIVE (BOTH FORMS)

0

( ) ( ) ( ) ( )lim limh x a

f x h f x f x f a

h x a

5) THE VOLUME OF TWO FUNCTIONS

2 2: ( ) ( ) ;

: 2 ( ( ) ( ))

b

a

b

a

x axis V f x g x dx

y axis V x f x g x dx

6) THE COORDINATE WHERE THE POINT OF INFLECTION OCCURS FOR A LOGISTIC FUNCTION

• If the general form of a logistic is given by ( ) ,1 Mkt

MP t

ae

then the coordinate of the point of inflection is ln, .

2

a M

Mk

7) A HARMONIC SERIES

• Notice the request was for a harmonic series. There are many and they all diverge:

1 1 1 2

1 2 2

1 1 1 1 1, ; ;

2 2 1

1 1 1 1; ; ...

2 1 3 3 3 1

n n n n

n n n

n n n n

and so onn n n

8) DISPLACEMENT IF GIVEN A VECTOR-VALUED FUNCTION

( ) , ( ) ( ) ( ), ( ) ( )f f

i i

t t

f i f i

t t

x t dt y t dt x t x t y t y t

9) MVT (BOTH FORMS)

• If a function is continuous, differentiable and integrable, then

( ) ( ) 1( ) ; ( ) ( ) .

b

avg

a

f b f af c OR f c f x dx

b a b a

Think about it, they really are the same formula

10) ARC LENGTH FOR A RECTANGULAR FUNCTION

21 ( )

b

a

l f x dx

11) THE DERIVATIVE AND ANTIDERVIATIVE OF LN(AX)

1ln( ) ;

ln( ) ln( )

d aax

dx ax x

ax dx x ax ax C

12) LAGRANGE ERROR BOUND

13) THE PRODUCT RULE

( ) ( )

( ) ( ) ( ) ( )

df x g x

dxf x g x f x g x

14) THE SOLUTION TO THE FOLLOWING DE: DP/DT = .05P(500-P), & IVP: P(0) = 50

• See #6 above because that logistic function is the general solution to this specific logistic DE (differential equation) where k = 0.05 & M = 500. Now use the initial condition to find a:

25 25 0

25

500 500 500( ) (0) 50 50

1 1 1500

1 10 9, , ( ) .1 9

t

t

P t Pae ae a

a a so P te

15) VOLUME OF A SINGLE FUNCTION SPUN ‘ROUND Y-AXIS

2 ( )b

a

V xf x dx

16) HOOKE’S LAW FUNCTION AND THE GENERAL FORM OF THE INTEGRAL THAT COMPUTES WORK DONE ON A SPRING

• F(x) = kx where k is the spring constant and x is the distance the spring is stretched/compressed as a result of F force can be integrated to get work: where a = initial spring position and b = final spring position.

17) AVERAGE RATE OF CHANGE

( ) ( )f b f am slope

b a

18) ALL LOG RULES

log( ) log log ;

log log log ;

log logn

a b a b

aa b

b

a n a

19) DISTANCE TRAVELED BY A BODY MOVING ALONG A VECTOR-VALUED FUNCTION

2 2f

i

t

t

dx dyd dt

dt dt

20) A LEAST TWO LIMIT TRUTHS (YOU KNOW AT LEAST EIGHT)

0 0 0

0

sin sin 1 coslim 1; lim 1; lim 0;

1 cos 1 cos 1 coslim 0; lim 0; lim 0;

sin sinlim 0; lim 0

x x x

x x x

x x

x ax x

x ax x

ax x ax

ax x ax

x ax

x ax

21) CONVERSION FORMULAS: POLAR VS. RECTANGUALR

2 2 2; tan ;

cos ; sin

yr x y

xx r y r

22) AREA OF A TRAPEZOID

1 22

hA b b

23) IF GIVEN POSITION FUNCTION IN RECTANGULAR FORM: SPEED

( ) ( )x t v t

24) THE FOLLOWING ANTIDERIVATIVE

ln[f (x)] + C ( )

( )

f xdx

f x

25) VOLUME OF TWO FUNCTIONS (AS ABOVE) SPUN AROUND AN AXIS TO THE LEFT OF THE GIVEN REGION

Assuming that the axis is something of the form x = q, 2 ( ) ( ) ( ) .

b

a

V x q f x g x dx

26) THE QUADRATIC THEOREM (NOT JUST THE FORMULA)

If given an equation of the form ax2 + bx + c = 0, then the solutions to this quadratic can be found by using 2 4

.2

b b acx

a

27) SIMPSON’S RULE FOR THE APPROXIMATION OF THE AREA UNDER THE CURVE

If you apply what was said above for the Trapezoidal approximation (#2 above) with an even number of subintervals, then the Simpson’s approximation is given by

0 1 2 3 2 1( ) 4 ( ) 2 ( ) 4 ( ) ... 2 ( ) 4 ( ) ( ) .3n n n n

hS f c f c f c f c f c f c f c

28) GENERAL FORMULA FOR A CIRCLE CENTERED ANYWHERE

where (a, b) is the center2 2 2( ) ( )x a y b r

29) TAYLOR’S THEOREM

If you want to approximate the value of a function, like sinx, you need some process or formula to do it. Taylor decided that a polynomial could approximate the value of a function if you make sure it has the requisite juicy tidbits: the same value at a center point (x = a), the same slope at that point, the same concavity at that point, the same jerk at that point, and so on. The led him to create the following formula:

( ) ( )2

0

( ) ( ) ( )( ) ( )( ) ( ) ... ( ) ... ( ) .

2! ! !

n kn k

k

f a f a f af a f a x a x a x a x a

n k

And centered at x = 0 (Maclaurin),

( ) ( )2

0

(0) (0) (0)(0) (0) ... ... .

2! ! !

n kn k

k

f f ff f x x x x

n k

The remainder

formula later

led to the

LaGrange Error

Bound Formula,

given in #12 above.

•He also pointed out that if you truncate thepolynomial to n terms, then the part you cut off (the “tail”), Rn, represents the error in

doing the cutting.

. and between somefor )()!1(

)()( where

)()(!

)(...)(

!2

)())(()()(

1)1(

)(2

xacaxn

cfxR

xRaxn

afax

afaxafafxf

nn

n

nn

n

30) THE CHAIN RULE

( ( )) ( ( )) ( )dg f x g f x f x

dx

31) VOLUME OF A SINGLE FUNCTION SPUN ROUND THE X-AXIS

2( )

b

a

V f x dx

32) ALTERNATING SERIES ERROR BOUND

error next term

33) THE THREE PYTHAGOREAN IDENTITIES

2 2

2 2

2 2

sin cos 1;

tan 1 sec ;

1 cot csc

x x

x x

x x

34) FIRST DERIVATIVE OF A PARAMETRIC FUNCTION

/( ), ( ) ;

/

dy dtx t y t slope

dx dt

35) VOLUME OF TWO FUNCTIONS SPUN ‘ROUND AN AXIS THAT IS ABOVE THE GIVEN REGION

If y = q is above the function f (x), then the volume is given by 2 2

( ) ( ) .b

a

V q g x q f x dx

36) VOO DOO

This is also known as the Integration by Parts process:

udv uv vdu

37) ARC LENGTH FOR A POLAR FUNCTION

22

f

i

drl r d

d

38) FTC (BOTH PARTS)

If a function is continuous, then (part I)

and if F(x) is an antiderivative of f (x),

then (part II)

( ) ( ),x

a

df t dt f x

dx

( ) ( ) ( )b

a

f x dx F b F a

39) ANTIDERVIATIVE OF A FUNCTION

ln cos ln secx C x C

40) AVERAGE VALUE OF A FUNCTION

1( )

b

a

f x dxb a

41) THE DE THAT IS SOLVED BY Y=PE^N

dyry

dt

42) THE GENERAL LOGISTIC FUNCTION

where M = the Max value of the population (or where the population is heading), k = the constant of proportionality, and a = a coefficient found with an initial value.

( ) ,1 Mkt

MP t

ae

43) VOLUME OF TWO FUNCTIONS (AS ABOVE) SPUN ‘ROUND AN AXIS THAT IS BELOW THE GIVEN REGION

If y = q is below the given region, then the volume is given by

2 2( ) ( ) .

b

a

V q f x q g x dx

44) AREA OF A EQUILATERAL TRIANGLE IN TERMS OF ITS BASE

23( )

4A base

45) SECOND DERIVATIVE FOR A PARAMETRIC FUNCTION

2

2

//

/

d dy dtd y dt dx dtdx dx dt

46) IF GIVEN A POSITION VECTOR-VALUED FUNCTION: SPEED

2 2dx dy

dt dt

47) ARC LENGTH FOR A PARAMETRIC FUNCTION

2 2f

i

t

t

dx dyl dt

dt dt

48) AN ALTERNATING HARMONIC SERIES

Again, note that the prompt requests an alternating series. There are many:

And all alternating harmonics are convergent by the AST (alternating series test).

1

1 1 1 2

1

1 2 2

( 1) ( 1) 1 ( 1) ( 1), ; ;

2 2 1

( 1) ( 1) 1 ( 1); ; ...

2 1 3 3 3 1

n n n n

n n n n

n n n

n n n

n n n n

and so onn n n

49) DERIVATIVE OF THE FOLLOWING FUNCTION Y= B’

lnxdyb b

dx

50) THE X-COORDINATE OF THE VERTEX OF ANY QUADRATIC FUNCTION

2

2 2 2

,2 2 2

, ,2 4 2 2 4

b b ba b c

a a a

b b b b bc c

a a a a a

51) MAGNITUDE OF A VECTOR

2 2,a b a b

52) AT LEAST ONE LIMIT EXPRESSION THAT GIVES YOU THE VALUE OF E

1

0

1lim 1 lim 1

n

xn x

x en

53) THE MACLAUREN SERIES FOR SIN(X), COS(X), AND E^X

54) SLOPE OF AN INVERSE FUNCTION AT THE INVERTED COORDINATE

1

( )

1

f a

x a

dfdfdxdx

55) THE QUOTIENT RULE

2

( ) ( ) ( ) ( ) ( )

( ) ( )

d t x t x b x t x b x

dx b x b x

56) SOH-CAH-TOA WITH A RIGHT TRIANGLE DRAWING

sin ;

cos ;

tan

b

ca

cb

a

57) GENERAL GEOMETRIC SERIES AND ITS SUM

1

0

11

n

n

aar if r

r

58) SLOPE OF A LINE NORMAL TO A CURVE

If m = f’(x) represents the slope of the tangent line to a curve or the instantaneous rate of change of f (x), then the slope of the line normal to the curve is given by

1 1

( )m f x

59) DISTANCE TRAVELED BY A RECTANGULAR FUNCTION

This is the same as arc length:

21 ( ) .

b

a

l f x dx

60) VOLUME OF 2 FUNCTION (AS ABOVE) SPUN ‘ROUND AN AXIS THAT IS TO THE RIGHT OF THE GIVEN REGION

If x = q is to the right of the given region, then the volume is given by

2 ( ) ( ) ( ) .b

a

V q x f x g x dx

61) SURFACE AREA FOR PARAMETRIC FUNCTIONS SPUN ‘ROUND BOTH THE X AND Y-AXES

• For spinning around the x-axis

• For spinning around the y-axis:

2 22 ( ) ( ) ( ) ;

f

i

t

t

SA y t x t y t dt

2 22 ( ) ( ) ( ) ;

f

i

t

t

SA x t x t y t dt

62) NEWTON’S LAW OF COOLING DE AND GENERAL SOLUTION FUNCTION

( );

( ) kt

dyk T y

dt

y t T Ae

63) AREA BETWEEN TWO FUNCTIONS (AS ABOVE)

( ) ( )b

a

A f x g x dx

64) CHANGE OF BASE FOR LOGS (18???)

lnlog

lnb

aa

b

65) DERIVATIVE FOR AS MANY INVERSE TRIGONOMETRIC FUNCTIONS AS YOU CAN REMEMBER

1 12 2

1 1

2 2

1 122

1 1tan ; sin ;

1 1

1 1cos ; sec ;

1 1

1 1csc ; cot

11

d dx x

dx x dx x

d dx x

dx dxx x x

d dx x

dx dx xx x