formulas review sheet answers
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Formulas Review Sheet Answers. 1) Surface area for a parametric function. 2) Trapezoidal approximation of the area under a curve (both forms). - PowerPoint PPT PresentationTRANSCRIPT
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FORMULAS REVIEW SHEETANSWERS
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1) SURFACE AREA FOR A PARAMETRIC FUNCTION
2 2
2f
i
t
t
dx dyS dtdt dt
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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.
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3) THE MACLAUREN SERIES FOR….
• ;
• ;
1 1ln(1 ), , ln(1 ), &1 1
x xx x
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4) LIMIT DEFINITION OF THE DERIVATIVE (BOTH FORMS)
0
( ) ( ) ( ) ( )lim limh x a
f x h f x f x f ah x a
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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
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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 tae
then the coordinate of the point of inflection is ln , .2
a MMk
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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
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8) DISPLACEMENT IF GIVEN A VECTOR-VALUED FUNCTION
( ) , ( ) ( ) ( ), ( ) ( )f f
i i
t t
f i f it t
x t dt y t dt x t x t y t y t
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9) MVT (BOTH FORMS)
• If a function is continuous, differentiable and integrable, then
( ) ( ) 1( ) ; ( ) ( ) .b
avga
f b f af c OR f c f x dxb a b a
Think about it, they really are the same formula
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10) ARC LENGTH FOR A RECTANGULAR FUNCTION
21 ( )b
a
l f x dx
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11) THE DERIVATIVE AND ANTIDERVIATIVE OF LN(AX)
1ln( ) ;
ln( ) ln( )
d aaxdx ax x
ax dx x ax ax C
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12) LAGRANGE ERROR BOUND
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13) THE PRODUCT RULE
( ) ( ) ( ) ( ) ( ) ( )d f x g x f x g x f x g xdx
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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) 50 501 1 1
5001 10 9, , ( ) .1 9
t
t
P t Pae ae a
a a so P te
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15) VOLUME OF A SINGLE FUNCTION SPUN ‘ROUND Y-AXIS
2 ( )b
a
V xf x dx
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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.
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17) AVERAGE RATE OF CHANGE
( ) ( )f b f am slopeb a
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18) ALL LOG RULES
log( ) log log ;
log log log ;
log logn
a b a ba a bba n a
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19) DISTANCE TRAVELED BY A BODY MOVING ALONG A VECTOR-VALUED FUNCTION
2 2f
i
t
t
dx dyd dtdt dt
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20) A LEAST TWO LIMIT TRUTHS (YOU KNOW AT LEAST EIGHT)
0 0 0 0
sin sin 1 cos 1 coslim 1; lim 1; lim 0; lim 0;
1 cos 1 cos sin sinlim 0; lim 0; lim 0; lim 0
x x x x
x x x x
x ax x axx ax x ax
x ax x axx ax x ax
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21) CONVERSION FORMULAS: POLAR VS. RECTANGUALR
2 2 2 ; tan ;
cos ; sin
yr x yx
x r y r
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22) AREA OF A TRAPEZOID
1 22hA b b
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23) IF GIVEN POSITION FUNCTION IN RECTANGULAR FORM: SPEED
( ) ( )x t v t
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24) THE FOLLOWING ANTIDERIVATIVE
ln[f (x)] + C ( )( )
f x dxf x
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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
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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 .
2b b acx
a
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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 nhS f c f c f c f c f c f c f c
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28) GENERAL FORMULA FOR A CIRCLE CENTERED ANYWHERE
where (a, b) is the center2 2 2( ) ( )x a y b r
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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 an 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 xn k
The remainder formula later led to theLaGrange ErrorBound 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
xRaxnafaxafaxafafxf
nn
n
nn
n
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30) THE CHAIN RULE
( ( )) ( ( )) ( )d g f x g f x f xdx
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31) VOLUME OF A SINGLE FUNCTION SPUN ROUND THE X-AXIS
2( )b
a
V f x dx
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32) ALTERNATING SERIES ERROR BOUND
error next term
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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
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34) FIRST DERIVATIVE OF A PARAMETRIC FUNCTION
/( ), ( ) ;/
dy dtx t y t slopedx dt
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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
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36) VOO DOO
This is also known as the Integration by Parts process:
udv uv vdu
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37) ARC LENGTH FOR A POLAR FUNCTION
22
f
i
drl r dd
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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
d f t dt f xdx
( ) ( ) ( )b
a
f x dx F b F a
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39) ANTIDERVIATIVE OF A FUNCTION
ln cos ln secx C x C
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40) AVERAGE VALUE OF A FUNCTION
1 ( )b
a
f x dxb a
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41) THE DE THAT IS SOLVED BY Y=PE^N
dy rydt
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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 tae
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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
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44) AREA OF A EQUILATERAL TRIANGLE IN TERMS OF ITS BASE
23 ( )4
A base
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45) SECOND DERIVATIVE FOR A PARAMETRIC FUNCTION
2
2
//
/
d dy dtd y dt dx dtdx dx dt
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46) IF GIVEN A POSITION VECTOR-VALUED FUNCTION: SPEED
2 2dx dydt dt
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47) ARC LENGTH FOR A PARAMETRIC FUNCTION
2 2f
i
t
t
dx dyl dtdt dt
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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
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49) DERIVATIVE OF THE FOLLOWING FUNCTION Y= B’
lnxdy b bdx
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50) THE X-COORDINATE OF THE VERTEX OF ANY QUADRATIC FUNCTION
2 2 2 2
, , ,2 2 2 2 4 2 2 4b b b b b b b ba b c c ca a a a a a a a
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51) MAGNITUDE OF A VECTOR
2 2,a b a b
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52) AT LEAST ONE LIMIT EXPRESSION THAT GIVES YOU THE VALUE OF E
1
0
1lim 1 lim 1n
xn x
x en
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53) THE MACLAUREN SERIES FOR SIN(X), COS(X), AND E^X
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54) SLOPE OF AN INVERSE FUNCTION AT THE INVERTED COORDINATE
1
( )
1
f a
x a
dfdfdxdx
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55) THE QUOTIENT RULE
2
( ) ( ) ( ) ( ) ( )( ) ( )
d t x t x b x t x b xdx b x b x
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56) SOH-CAH-TOA WITH A RIGHT TRIANGLE DRAWING
sin ; cos ; tanb a bc c a
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57) GENERAL GEOMETRIC SERIES AND ITS SUM
1
0
11
n
n
aar if rr
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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
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59) DISTANCE TRAVELED BY A RECTANGULAR FUNCTION
• This is the same as arc length:
21 ( ) .b
a
l f x dx
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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
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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
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62) NEWTON’S LAW OF COOLING DE AND GENERAL SOLUTION FUNCTION
( ); ( ) ktdy k T y y t T Aedt
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63) AREA BETWEEN TWO FUNCTIONS (AS ABOVE)
( ) ( )b
a
A f x g x dx
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64) CHANGE OF BASE FOR LOGS (18???)
lnloglnbaab
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65) DERIVATIVE FOR AS MANY INVERSE TRIGONOMETRIC FUNCTIONS AS YOU CAN REMEMBER
1 1 12 2 2
1 1 122 2
1 1 1tan ; sin ; cos ;1 1 1
1 1 1sec ; csc ; cot11 1
d d dx x xdx x dx dxx x
d d dx x xdx dx dx xx x x x