Essential Calculus Facts - CSU, Chico

Essential Calculus Facts: DERIVATIVES: 1).

Definition of

f ′( x) :

_____________________________________________ d2 y

2)

19). d x 2 cosθ __________________________________ dt 20). Linearization of f at x=a _______________________________________

means ________________________

dx 2 d 3) e x ________________________________ dx

INTEGRALS: b

21).

Definition of

∫ f (t ) dt : a

_______________________________________

4) df ( x ) ______________________________ it means:______________________________

5)

d dx

( f ⋅g ) ________________________________

6)

d dx

( f /g ) ________________________________

7)

f ( g ( x )) _____________________________

d dx

d dx

10)

11)

22) ∫ x n dx ______________________ 1 23) ∫ dx _______________________ x 24) ∫ sin x dx _____________________ 25) ∫ cos x dx _____________________

d 8) sin x ________________________________ dx

9)

antiderivatives:

26) ∫ sec 2 x dx _____________________

cos x ________________________________

27) ∫ sec x tan x dx __________________

d sec x ________________________________ dx

d dx

28) ∫ e x dx ________________________

tan x ________________________________

1

29) ∫ 12)

d dx

cot x ________________________________

d 13) dx

csc x ________________________________

14)

d dx

ln x _________________________________

15)

d sin −1 x ______________________________ dx

d dx

cos−1 x ______________________________

d 17) dx

tan −1 x ______________________________

16)

18).

1 − x2

d dx

( 3x2 y3 ) ___________________________

30) ∫

dx ____________________

1 dx ______________________ 1 + x2

FUNDAMENTAL THEOREMS: x

d 31). f ( t ) dt = ___________________ dx ∫a d 32). dx 33). b

If

g( x )

∫ f (t ) dt = __________________ a

F′ ( x) = f ( x) ,

then:

∫ f (t ) dt = ________________________________ a