Limits&Continuity
limx→3(x2−1)
8
d/dx(sinx)
cos(x)
∫x^2dx
1/3x^3+C
If f′(x)>0, f(x) is doing this.
Increasing
The derivative of Position
velocity
limx→0xsinx
1
d/dx(lnx)
1/x
∫e^xdx
ex+C
The value of ∫f(x)dx.
0
The derivative of Velocity
Acceleration
Limit definition of a derivative.
limh→0hf(x+h)−f(x)
Rule used for f(x)⋅g(x).
Product Rule
∫1/xdx
ln|x|+C
A point where f′′(x) changes sign
Point of inflection
Speed is the absolute value of this
Velocity
limx→∞x2+1003x2
3
d/dx(e^x^2)
2xe^x^2
∫cos(2x)dx
1/2sin(2x)+C
d/dx∫1xtdt
x^-2
Average value" formula of f(x) on [a,b]
b−a1∫abf(x)dx
L'Hôpital's Rule requirement.
00 or ∞∞
Derivatives
Derivative of arctan(x)
1/1+x^2
Use u-sub for ∫x(x2+1)5dx
1/12(x^2+1)^6+C
If f′(c)=0 and f′′(c)<0, then c is a...
x
Rate at which a radius changes if A=πr2
dtdA=2πrdtdr