AP Calculus Notecards Flashcards

1
Q

Derivative of tanx

A

sec^2x

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2
Q

Derivative of secx

A

secxtanx

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3
Q

Derivative of lnx

A

1/x

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4
Q

Derivative of a^x

A

a^xlna

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5
Q

Derivative of sin^-1 (stuff)

A

1/sqrt(1-(stuff)^2)

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6
Q

Derivative of tan^-1

A

1/(1+(stuff)^2)dstuff

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7
Q

Total Distance (rectangular)

A

S|v(t)|dt

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8
Q

“average”

A

1/(b-a)Sa-b(f(x))dx
(y-y)/(x-x)

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9
Q

Definition of Derivative

A

lim f(a+h)-f(a)
h->0 /h
lim f(x)-f(a)
x->a /x-a

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10
Q

Speed (increasing or decreasing)

A

increasing - a+v same sign
decreasing - a+v different sign

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11
Q

S1/sqrt(a^2-u^2)du

A

sin^-1(a/u)+c

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12
Q

S1/(a^2+u^2)du

A

1/atan^-1(u/a)+c

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13
Q

lim sinx
x->0 /x

A

1

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14
Q

d/dxSa-f(x)f(t)dt

A

(plug in stuff) dstuff f(x) is the stuff

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15
Q

f(x) = Sg(t)dt

A

f’(x) = g(x)
f’‘(x) = g’(x)

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16
Q

f’(x) [first derivative]

A

increasing/decreasing
maximum/minimum

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17
Q

f’‘(x) [second derivative]

A

concavity, POI’s

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18
Q

Critical #’s

A

derivative = 0 or DNE

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19
Q

Absolute Maximum or Minimum

A

x/f(x)
endpoints+
critical #’s

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20
Q

d/dxf^-1(x)

A

1/(f’(f^-1(x)))

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21
Q

Derivative to a Graph

A

Look at Tangent Lines

22
Q

Integral of Graph

A

Look at Area

23
Q

Derivative to Table

A

(y-y)/(x-x)

24
Q

a) Sa->brate
b) # + Sa->brate

A

a) Amount from a to b only
b) final amount

25
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!-...
26
Taylor Formula
f(a) + f'(a)(x-a)/1! + f''(a)(x-a)/2!...
27
Radius of Convergence
Ratio Test |x-a|< R x must have coeefficient of 1
28
Interval of Convergence
Ratio Test, |x-a|< R a+/-R, test endpoints
29
Lagrange Error
Next Term in Taylor Formula
30
Parametric a) dy/dx b) d^2y/dx^2
a)y'/x' b) derivative of (dy/dx)/x'
31
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
32
Speed a) Vector b) Function
a) sqrt((x')^2+(y')^2) b) |v(t)|
33
dy/dx (polar)
(rsin(theta))/(rcos(theta)) plug in r first
34
Polar Area
.5Sa-br^2dtheta
35
Polar Formulas x+y
x=rcos(theta) y=rsin(theta)
36
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)
37
Ratio Test
lim |an+1| n->infinity | / an | <1 convergent >1 divergent
38
Comparison Test
< goes w/ convergent > goes w/ divergent (compare to eyeball)
39
Integral Test
Sa-infinity -> if integral converges then so does the series
40
Limit Comparison
lim original/compared = L>0 both converge or both diverge
41
Root Test
lim nth root < 1 converges lim nth root >1 diverges
42
Antiderivative of tanx
-ln|cosx|+c ln|secx|+c
43
King
n^nFEPL King on top -> infinity King on bottom -> 0
44
anti of lnx
xlnx-x+c
45
Integration by parts
ILATE to determine u
46
Particle Motion
P original V 1st derivative A 2nd Derivative at rest, left, right, up, down, change direction v(t)=0 (sign chart)
47
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 pK * p=K/(1+Ae^-kt)
48
L'Hopital's
Direct sub on limit 0/0 or infinity/infinity take derivative of top and bottom seperately and direct sub again.
49
IVT/MVT
* IVT - y values only (must be continuous) * MVT - derivatives (must be differentiable + continuous) * MVT connects 8th grade slope to derivatives
50
Swith stuff
use usub