Calculus 2 Ultimate Summary

Review

Derivative Rules
·

(f(x) = g(x))) =
f(x) =
g(x) power rule -
(f(x)g(x))) =
f'(x)g(x) + f(x)g'(x)

F

quotient rule
-

( =
g

Trig Derivatives Derivative Rules


positive negative power role-(ax") = anx-
- -




one Sin X = COSX =


(a*)' Ina-a

tereared
COS X - sin X
exponent rue
- =

-




+anx = co+ x=


Secx-cs2X chain rule -
(f(g(x)))) = f ' (g(x))g'(x)

See X = CSCX =
two
term SecXtax-cscXcotX




Integration Rules


·
limit definition :

Sf(x)dx na =
+
x)Ax


g 0
b


f(x)dx =
f(x)x =

-Six Saf(x)dx a) f(x)dx
=




rule-Saxdx =
**
x +
power



exponential rule -

Sa" =
ia +

Chain rule -

Sf' (g(x)g'(x)dx f(g(x)) = +C

, Functions

in this class !
Algebraic +
Trig-covered
AlgebraicFunctions




in
&
S




-
- 3xn




Definition of Functions
function -
a
curve/graph that is both :

continuous (no jumps/breaks)
·
differentiable (can take a derivative)

repeating x-values (Vertical
no Line Test


One-to-one function -
no
repeating y-values (Horizontal Line Test

function can be flipped over
y =X




one number in, one number out

used to determine if the inverse of a function can be taken

If
-
a function is not one-to-one, an interval can be used to make it one-to-one .




ex .. f(x) = Sin(x)
One-to-one on


es [0,]

, FTC


Definition -
Iff(x) is continuous on the interval [2 b]
, ,
then

f(x) has a derivative.
-
used to find the derivative of the
integral



Formula :
F(Adt =
f(x) + Co-
3 + (A)d+ =
f(g(x)) g((x)
- + c


an tent




Use when :

there is an X in the top bound of the integral
may need to flip the integral using integral properties