AP Calculus AB Memorization List

y=e^x points

(0,1) (1,e)

y=lnx points

(1,0) (e,1)

Approximation of e

2.72

d/dx [lnu]

u'/u

d/dx [e^u]

e^u u'

d/dx [a^u]

a^u * lna * u'

∫a^xdx

a^x*1/lna+c

∫e^xdx

e^x+c

∫1/xdx

ln|x|+c

If f and g are inverses with (a,b) on f and (b,a) on g, f'(a)=?

1/g'(b)

Change of base log_ab

lnb/lna

Direct Variation

y=kx

Inverse Variation

y=k/x

Exponential Growth

y=Ce^(kt)

∫sinx dx

-cosx+c

∫cosx dx

sinx+c

∫csc^2x dx

-cotx+c

∫sec^2x dx

tanx+c

∫secx tanx dx

secx+c

∫cscx cotx dx

-cscx+c

ln1

0

e^0

1

ln0

dne

∫cotx dx

ln|sinx|+c

∫secx dx

ln|secx+tanx|+c

∫cscx dx

-ln|cscx+cotx|+c

If the prompt says "f(x) is diff." you write…

f(x) is diff., and since diff implies cont., f(x) is cont.

d/dx sinx

cosx

d/dx cosx

-sinx

d/dx tanx

sec^2x

d/dx secx

secxtanx

d/dx cscx

-cscxcotx

d/dx cotx

-csc^2x

v(t) for a particle

particle is moving down

v(t)<0

v(t) for a particle

particle is moving up

v(t)>0

v(t) for a particle

particle is stopped

v(t)=0

v(t) for a particle

particle is slowing down

v(t)&v'(t) have different signs

v(t) for a particle

particle is speeding up

v(t)&v'(t) have same sign

v(t) for a particle

acceleration is negative

a<0

v'(t)<0

v(t) for a particle

acceleration is positive

a>0

v'(t)>0

v(t) for a particle

particle changes direction

v(t) changes signs

Average velocity

Average acceleration

If given f'(x)

f(x) is increasing

f'(x)>0

If given f'(x)

f(x) is decreasing

f'(x)<0

If given f'(x)

f(x) has a relative max

f' changes from pos to neg

If given f'(x)

f(x) has a relative min

f' changes from neg to pos

If given f'(x)

f(x) has an extrema

f' changes signs

If given f'(x)

f(x) is concave up

f">0

slope of f'>0

d/dx[arcsin u]

u'/√(1-u^2)

d/dx[arccos u]

-u'/√(1-u^2)

d/dx[arctan u]

u'/(1+u^2)

d/dx[arccot u]

-u'/(1 + u^2)

d/dx[arcsec u]

u'/(|u|√(u^2-1))

d/dx[arccsc u]

-u'/(|u|√(u^2-1))

∫tanx dx

-ln|cosx|+c

∫du/√(a^2-u^2)

arcsin(u/a)+c

∫du/(a^2+u^2)

(1/a)arctan(u/a)+c

∫du/(u√u^2-a^2)dx

1/a arcsec |u|/a +c

Finding area by disc

π ∫ (radius)^2

Finding area by washer

π ∫ (outer)^2 - π ∫ (inner)^2

Square cross section

∫ (base)^2

Equalateral Triangle cross section

(√3)/4 ∫ (base)^2

Isosceles Right Triangle cross section

1/2 ∫ (base)^2

Semicircles cross section

π/8 ∫ (base)^2

Steps to solve Separable Differential Equations

Seperate, Integrate, Solve

Rate Problems Amt=

Initial + ∫ (Rate in) - ∫ (Rate out)

If given f'(x)

f(x) is concave down

f"<0

slope of f'<0

If given f'(x)

f(x) has a critical point

f'(x)=0 or DNE

If given f'(x)

f(x) has a POI-Point of inflection

f" changes signs

slope of f' changes signs

lim x -> ∞

Degree on top<degree on bottom

Limit=?

top<bottom

0

lim x -> ∞

Degree on top>degree on bottom

Limit=?

top>bottom

DNE or ∞