AP Calculus Notecards Flashcards
Derivative of tanx
sec^2x
Derivative of secx
secxtanx
Derivative of lnx
1/x
Derivative of a^x
a^xlna
Derivative of sin^-1 (stuff)
1/sqrt(1-(stuff)^2)
Derivative of tan^-1
1/(1+(stuff)^2)dstuff
Total Distance (rectangular)
S|v(t)|dt
“average”
1/(b-a)Sa-b(f(x))dx
(y-y)/(x-x)
Definition of Derivative
lim f(a+h)-f(a)
h->0 /h
lim f(x)-f(a)
x->a /x-a
Speed (increasing or decreasing)
increasing - a+v same sign
decreasing - a+v different sign
S1/sqrt(a^2-u^2)du
sin^-1(a/u)+c
S1/(a^2+u^2)du
1/atan^-1(u/a)+c
lim sinx
x->0 /x
1
d/dxSa-f(x)f(t)dt
(plug in stuff) dstuff f(x) is the stuff
f(x) = Sg(t)dt
f’(x) = g(x)
f’‘(x) = g’(x)
f’(x) [first derivative]
increasing/decreasing
maximum/minimum
f’‘(x) [second derivative]
concavity, POI’s
Critical #’s
derivative = 0 or DNE
Absolute Maximum or Minimum
x/f(x)
endpoints+
critical #’s
d/dxf^-1(x)
1/(f’(f^-1(x)))
Derivative to a Graph
Look at Tangent Lines
Integral of Graph
Look at Area
Derivative to Table
(y-y)/(x-x)
a) Sa->brate
b) # + Sa->brate
a) Amount from a to b only
b) final amount
a) 1/(1-x)
b) e^x
c) sinx
d) cosx
a) 1+x+x^2+x^3…
b) 1+x+x^2/2!+x^3/3!…
c) x-x^3/3!+x^5/5!-…
d) 1-x^2/2!+x^4/4!-…
Taylor Formula
f(a) + f’(a)(x-a)/1! + f’‘(a)(x-a)/2!…
Radius of Convergence
Ratio Test |x-a|< R
x must have coeefficient of 1
Interval of Convergence
Ratio Test, |x-a|< R
a+/-R, test endpoints
Lagrange Error
Next Term in Taylor Formula
Parametric
a) dy/dx
b) d^2y/dx^2
a)y’/x’
b) derivative of (dy/dx)/x’
Arc Length
a) Parametric
b) Polar
c) Function
a) Sa-bsqrt((x’)^2+(y’)^2)dt
b) Sa-bsqrt(r^2+(r’)^2)d(theta)
c) Sa-bsqrt(1+(y’)^2)dx
Speed
a) Vector
b) Function
a) sqrt((x’)^2+(y’)^2)
b) |v(t)|
dy/dx
(polar)
(rsin(theta))/(rcos(theta))
plug in r first
Polar Area
.5Sa-br^2dtheta
Polar Formulas
x+y
x=rcos(theta)
y=rsin(theta)
Eyeballs
a) harmonic 1/n divergent
b) alternating harmonic (-1)^n/n convergent
c) test for divergence limit is not 0 (including DNE) divergent
d) P-series 1/n^p p>1 convergent p< or =1 divergent
e) Geometric |r|<1 convergent |r|>1 divergent a/(1-r)
Ratio Test
lim |an+1|
n->infinity | / an |
<1 convergent >1 divergent
Comparison Test
< goes w/ convergent
> goes w/ divergent
(compare to eyeball)
Integral Test
Sa-infinity -> if integral converges then so does the series
Limit Comparison
lim original/compared = L>0
both converge or both diverge
Root Test
lim nth root < 1 converges
lim nth root >1 diverges
Antiderivative of tanx
-ln|cosx|+c
ln|secx|+c
King
n^nFEPL
King on top -> infinity
King on bottom -> 0
anti of lnx
xlnx-x+c
Integration by parts
ILATE to determine u
Particle Motion
P original
V 1st derivative
A 2nd Derivative
at rest, left, right, up, down, change direction v(t)=0 (sign chart)
Logistic
dP/dt=kP(1-P/K) or kP((K-P)/K)
* growing fastest at k/2
* dP/dt=0 when p=K or p=0
* p increasing when p<K>K
* p=K/(1+Ae^-kt)</K>
L’Hopital’s
Direct sub on limit
0/0 or infinity/infinity
take derivative of top and bottom seperately and direct sub again.
IVT/MVT
- IVT - y values only (must be continuous)
- MVT - derivatives (must be differentiable + continuous)
- MVT connects 8th grade slope to derivatives
Swith stuff
use usub
