AP CALC

Integrals

100

∫3x+3 dx

3/2x^2+3x+C 

200

∫(√x + 1/x^2) dx

(2/3 x^3)/(-1/x )+ c

300

∫cosxsinx dx

1/2 sin^2(x)+c

400

a=0         b=1     

∫(x^3-√x) dx

-5/12

500

 a=0     b=pi 

∫ (4sinx) dx 

8

Derivatives

100

d/dx(6x+7)

6

200

d/dx(23x^4-7x+2)

92x^3-7

300

d/dx(e^2x*sinx) 

e^2x( 2sin(x)+cos(x)

400

d/dx( e^x/x^2+1)

(3e^3x)(6x^7+9)-(e^3x)(42x^6)/(6x^7+9)^2  or 

3e^3x(6x^7-14x^6+9)/(6x^7+9)^2

500

d/dx(ln(cos(10x))

-10tan(10x)

Limits

100

lim x-2   ( 3x^2+2x+9)

25

200

lim x-4  x^2-16/x-4  

8

300

lim x-3 √x+1 -2/x-3    

1/4

400

lim x- ∞ 9x^3-4x-1/3x^3+4x^2

9/3  or 3 

500

lim x-∞ 10x^4-9x^3+3x^2+2x+2/ 2x^10

0

Definitions

100

IVT

if F(x) is continuous on a closed interval [a,b], f(x) must take on every y value between f(a) and f(b) 

200

MVT

if f(x) is differentiable on the interval (a,b), then there must be a value c on (a,b) where  f'(c)=f(b)-f(a)/b-a

300

EVT

if f(x) is closed and continuous on [a,b] f(x) will have both an absolute maximum and minimum on the interval [a,b]. 

400

Rolle's

1)  f(x) is continuous on [a,b]

2) f(x) is differentiable on (a,b)

3) f(a)= f(b)  

Then there exist at least on c in (a,b) such that f '(c)=0

500

FTC and 2nd FTC

FTC 1 Derivative of an integral equal to the original function. 

FTC 2nd  Integrals are calculated by subtracting anti-derivative endpoints (f(b)-f(a))

Trig

100

limx-0 sinx/x 

1

200

lim x-0 (1-cosx)/x

0

300

d/dx (tanx)

sec^2x

400

∫cosx dx 

sinx+c

500

d/dx ( csc x) 

- cscxcot x