AP Calc BC Study Guide/Formula Sheet

8/13/2019 AP Calc BC Study Guide/Formula Sheet

1/15


2/15

Intermediate &a%ue Theorem

!ffis continuous on closed interval "a,# and cis an$ number betweenf(a) andf(), then

there is at least one number!on the interval such thatf(!) % c&

Difference "uotient:

3.3 Ta'in "eri#ati#es

Power Rule:

Product Rule:

"uotient Rule:

3. *otion +%on a Line'peeding up when v(t) and a(t) are the same sign& 'lowing down when v(t) and a(t) are

opposite signs&

Limit Identities

3., omposite !unctions and hain -u%e

S%opes of Parametrized ur#es

((t), $(t)) is differential at tif and $ are differential at t

ddt is the derivative of d$d

dd is the actual second derivative

3. Imp%icit "ifferentiation

*ifferentiate the whole thing with d$d&

3./ "eri#ati#e of In#erse f(0)

!ffis a one+to+one, differential function with inverseg,

"eri#ati#es


3/15

Loarithmic "ifferentiation

a-e the log or natural log of both sides and differentiate& .suall$ when there arefractions or a variable in the eponent&

.1+$so%ute0treme &a%ueshere can be more than one absolute mamin because the$ have the sameybut different

!values& here can be a point that is both ma and min if the line is a constant hori/ontal

no slope&

Asolute #a!:Asolute #in$

-e%ati#eLoca% &a%ues

Relati%e #a!$at cif for all!on an interval

Relati%e #in$at cif for all!on an interval

0treme &a%ue Theorem

!ffis continuous on "a,b# thenfhas an absolute ma and an absolute min on the interval&

.2 *ean &a%ue Theorem for "eri#ati#es

!ffis continuous and "a,b# and differential on (a,b) then there eists at least one point csuch that:

(the slope of the tangent % the slope of the secant line through a,b)

"efinitions

0etfbe defined on an interval with!&and!'as an$ points on the interval where!&!'&

ncreasing: !f!&!'thenf(!')f(!&)for all of (!& !')Decreasing: !f!&!'thenf(!')f(!&)

!ff*(!)% at each point in interval, thenf(!)% c

!ff*(!)%g*(!)at each point on an interval, thenf(!)%g(!)4 c

.31st"eri#ati#e

f(!)increases whenf*(!) and decreases whenf*(!) f(!)has a ma whenf*(!)changes from 4 to 5

f(!)has a min whenf*(!)changes from 5 to 4


4/15

2nd"eri#ati#e

6henf**(!) , f() is concave up

6henf**(!) , f() is concave down7oint of inflection is where the 2ndderivative changes sign

'nd

Deri%ati%e +est8ind 97 (where f() % or does not eist)

;valuate the second derivative& !f , then concave down and maimum& !f , then

concave up and minimum&

.*ode%inptimization (*a0*in)

average cost is minimum when marginal cost (9()) e=uals the

average cost&

7rofit 7() % R() 5 9()

R() % 9()maimum profit when marginal revenue % marginal cost

R() % p() where p() is the price function (demand)?rea- even point is when revenue % cost&

.4 Linearization

@iven f(), derive to find f(), then with given value a, substitute into 0()& hen

approimate a new value using 0()&

"ifferentia%s

5ewton6s *ethod

.sed to approimate /eros and roots&

Calculator


5/15

,!ampleis a solution of 25 & f() is 25 & 7ic- 1as 2 or 3 and approimate&

., -e%ated -ates

*iagram, then write -nown and un-nowns using 0eibni/ notation&

ie& Bou have given information, draw a diagram& @iven rates of change, write them downas d$dt or ddt, then use given info to come up with an e=uation, derive, substitute and

solve to find the un-nown rate&

4.1 -iemann Sum

4.2 "efinition of "efinite Intera%

4.3 Properties of "efinite Intera%

1)

2)

3)

)

)

C)

D)

A%erage Value of a -unction


6/15

!undamenta% Theorem of a%cu%us

Part &

Part '

+ppro0imation *ethods

+rapezoidal Rule

Simpson*s Rule

.ses a series of parabolic arcs to approimate the curve&

where nmust be even

,.1 +ntideri#ati#es*ifferential ;=uation 5 an e=uation that contains a derivative

Separation of Variales


7/15

0ewton*s 1aw of Cooling

he rate at which an obEect temperature is changing at an$ given time is approimatel$

proportional to the difference between its temperature and the surrounding temperature&

,.4

Population .rowt

(rate of relative growth)

1ogistic Differential ,2uation

,., 5umerica% *ethods,uler*s #etod

'ndDegree +aylor ,2uation

"efinition of Intera%

.1 Intera%s as 5et hane

Rate otalFet 9hange

(*isplacement)


8/15

Few 7osition % *isplacement (Fet 9hange) 4 !ntitial 9ondition C

.2 +rea 7etween Two ur#es

!ff(!)andg(!)intersect, find the points of intersection and use them as limits for each

section of the integral&

.3 &o%ume

?ased on G % H2rh:

Dis3 #etod

4aser #etod

!5a!is rotation

wheref(!)is the outer radius andg(!)is the inner radius&

?ased on G % 2HrhI!:

Sell #etod

y5a!is rotation

r%!when rotated around the ais

r% c5!when rotated around line!% c

Volume wit 6nown Cross5Section

A % area of given cross sectional shape in terms ofsie& A %s2

, the distance between the graphs that mar- the cross section


9/15

. Lenth of a ur#e

if JsmoothK (continuous)

in terms ofyif there is a vertical tangent

7iecewise function if there is a cusp& (absolute value)

+rea of Surface -e#o%ution

?ased on 'A % 2Hrl

!5a!is rotation

.4 8or'

Constant -orce

8 % ma when m is in -g (a is L&M)!f alread$ a force (weight) is given, ie& pounds, then $ou Eust multipl$ 8d to get wor-&

Variale -orce

where the function given to define the variable force isf(!)

Hoo3e*s 1aw

6ith springs when there is a variable force where!is units compressed or etended

be$ond natural length& o find wor-, $ou would integrate the function&

Pumpin 8or' Pro$%em

6 % 8orce distance

8orce % weight % volume densit$6 % volume densit$ distance

!f $ou have height $ou need to pump it to: distance % 5y

*ensit$ is usuall$ given, if its water its C2& lbft3

Golume is Hr2where r%!2and is Iy& .suall$ $ou epress!in terms ofyb$ finding

the e=uation of the line that includes the points on the outside of the tan-&


10/15

where d% densit$ and ris!in terms ofy

!%uid Pressure

8orce % pressure area7ressure % densit$ depth

8orce % densit$ depth area

Pressure on a Vertical Surface

N % densit$% total depth of water if the ais is the bottom of the tan-

0 % length aty, usuall$ solve for!in terms ofyusing the e=uation of the side of the tan-

when $ou set the ais at the top of the water level, $ou can use7yfor the depth but

usuall$ onl$ when the water level is e=ual to tan- height

/.1 L69:pita%6s -u%e

+eorem &

+eorem '

ndeterminate -orms

Special ndeterminate -orms

6hen the function has an eponent!:

8ind so that is e=uals a number 0& (9an then bring down eponent so

that its multiplication and rearrange into a fraction, appl$ 0OPpitals Rule from there&)

hen find limf(!)using: (where 0 % lim lnf(!))


11/15

/.2 -e%ati#e -ates of ;rowth

1& !ffgrows faster thang(gis slower)

2& !ffandggrow at the same rate

+ransiti%ity!ffgrows at the same rate asgandg grows the same as thenfis the same as

8 0otationf% o(g) iffgrows slower thang

f% (g) iffis the same or faster thang

/.3 Improper Intera%s

R if there is a discontinuit$

!f the limit is finite: con#eres&

!f the limit is infinite: di#eres&


12/15

2)

1imit Comparison +est

!ffandgare positive continuous functions on ato infinit$ and then:

8ter Stuff

!f $ou have

!f $ou have

/. Tri Su$stitutions


13/15


14/15

Alternating series has (+1)nin general term (but write out a couple of terms Eust in case to

chec-)&

!f it converges:

*ifference between sum to infinit$ and sum at term nis the remainder or error& 6hich ise=ual to the net term (n4 1)&

Ratio +est (for asolute con%ergence)'um anhas to be a series with non/ero terms


15/15

Asolute Con%ergence

!f series an converges then anconverges&

1& anis absolutel$ convergent if an converges2& anis conditionall$ convergent if an diverges

Inter#a% of on#erence9an be a point (radius % ), interval, or all reals (r is infinite)

.se ratio test and set less than 1, then solve and chec- endpoints&

rror

u%er6s !ormu%a

AP Calc BC Study Guide/Formula Sheet

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