Formulas Review Sheet Answers
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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
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