AP Calculus Notecards

Question 1

Derivative of tanx

Answer 1

sec^2x

Question 2

Derivative of secx

Answer 2

secxtanx

Question 3

Derivative of lnx

Answer 3

1/x

Question 4

Derivative of a^x

Answer 4

a^xlna

Question 5

Derivative of sin^-1 (stuff)

Answer 5

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

Question 6

Derivative of tan^-1

Answer 6

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

Question 7

Total Distance (rectangular)

Answer 7

S|v(t)|dt

Question 8

“average”

Answer 8

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

Question 9

Definition of Derivative

Answer 9

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

Question 10

Speed (increasing or decreasing)

Answer 10

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

Question 11

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

Answer 11

sin^-1(a/u)+c

Question 12

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

Answer 12

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

Question 13

lim sinx
x->0 /x

Answer 13

1

Question 14

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

Answer 14

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

Question 15

f(x) = Sg(t)dt

Answer 15

f’(x) = g(x)
f’‘(x) = g’(x)

Question 16

f’(x) [first derivative]

Answer 16

increasing/decreasing
maximum/minimum

Question 17

f’‘(x) [second derivative]

Answer 17

concavity, POI’s

Question 18

Critical #’s

Answer 18

derivative = 0 or DNE

Question 19

Absolute Maximum or Minimum

Answer 19

x/f(x)
endpoints+
critical #’s

Question 20

d/dxf^-1(x)

Answer 20

1/(f’(f^-1(x)))

Question 21

Derivative to a Graph

Answer 21

Look at Tangent Lines

Question 22

Integral of Graph

Answer 22

Look at Area

Question 23

Derivative to Table

Answer 23

(y-y)/(x-x)

Question 24

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

Answer 24

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

Question 25

a) 1/(1-x)
b) e^x
c) sinx
d) cosx

Answer 25

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!-…

Question 26

Taylor Formula

Answer 26

f(a) + f’(a)(x-a)/1! + f’‘(a)(x-a)/2!…

Question 27

Radius of Convergence

Answer 27

Ratio Test |x-a|< R
x must have coeefficient of 1

Question 28

Interval of Convergence

Answer 28

Ratio Test, |x-a|< R
a+/-R, test endpoints

Question 29

Lagrange Error

Answer 29

Next Term in Taylor Formula

Question 30

Parametric
a) dy/dx
b) d^2y/dx^2

Answer 30

a)y’/x’
b) derivative of (dy/dx)/x’

Question 31

Arc Length
a) Parametric
b) Polar
c) Function

Answer 31

a) Sa-bsqrt((x’)^2+(y’)^2)dt
b) Sa-bsqrt(r^2+(r’)^2)d(theta)
c) Sa-bsqrt(1+(y’)^2)dx

Question 32

Speed
a) Vector
b) Function

Answer 32

a) sqrt((x’)^2+(y’)^2)
b) |v(t)|

Question 33

dy/dx
(polar)

Answer 33

(rsin(theta))/(rcos(theta))
plug in r first

Question 34

Polar Area

Answer 34

.5Sa-br^2dtheta

Question 35

Polar Formulas
x+y

Answer 35

x=rcos(theta)
y=rsin(theta)

Question 36

Eyeballs

Answer 36

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)

Question 37

Ratio Test

Answer 37

lim |an+1|
n->infinity | / an |
<1 convergent >1 divergent

Question 38

Comparison Test

Answer 38

< goes w/ convergent
> goes w/ divergent
(compare to eyeball)

Question 39

Integral Test

Answer 39

Sa-infinity -> if integral converges then so does the series

Question 40

Limit Comparison

Answer 40

lim original/compared = L>0
both converge or both diverge

Question 41

Root Test

Answer 41

lim nth root < 1 converges
lim nth root >1 diverges

Question 42

Antiderivative of tanx

Answer 42

-ln|cosx|+c
ln|secx|+c

Question 43

King

Answer 43

n^nFEPL
King on top -> infinity
King on bottom -> 0

Question 44

anti of lnx

Answer 44

xlnx-x+c

Question 45

Integration by parts

Answer 45

ILATE to determine u

Question 46

Particle Motion

Answer 46

P original
V 1st derivative
A 2nd Derivative
at rest, left, right, up, down, change direction v(t)=0 (sign chart)

Question 47

Logistic

Answer 47

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>

Question 48

L’Hopital’s

Answer 48

Direct sub on limit
0/0 or infinity/infinity
take derivative of top and bottom seperately and direct sub again.

Question 49

IVT/MVT

Answer 49

• IVT - y values only (must be continuous)
• MVT - derivatives (must be differentiable + continuous)
• MVT connects 8th grade slope to derivatives

Question 50

Swith stuff

Answer 50

use usub