MAT 220- Calculus 1

MAT 231- Calculus 1 Review- Prof. Santilli
BACKGROUND INFORMATION FOR CALCULUS 2
INVERSE HYPERBOLIC TRIG FUNCTIONS:
(
)
1.)
sinh x = ln x + x + 1
2.)
cosh x = ln x + x − 1
3.)
1 ⎛ 1+ x ⎞
tanh x = ln ⎜
⎟
⎝
2
1− x ⎠
-1
-1
(
2
2
)
-1
LIMITS:
f (x) = ∞ − ∞ = indeterminate .
1.) lim
x→ a
2.)
lim f (x) =
1
= ±∞, DNE .
0
3.)
lim f (x) =
0
= indeterminate .
0
4.)
lim f (x) = ∞ ∞ = ∞ .
5.)
lim f (x) = 1∞ = indeterminate .
6.)
lim f (x) = 0 0 = indeterminate.
7.)
lim f (x) =
1
=0
∞
8.)
lim f (x) =
∞
= indeterminate
∞
x→ a
x→ a
x→ a
x→ a
x→ a
x→ a
x→ a
1
9.)
lim(1+ x )x = e
x→0
sin x
lim
=1
10.) x→0
x
11.) lim
x→0
1− cos x
=0
x
⎛ e x − 1⎞
⎜
⎟ =1
12.) lim
x→0 ⎝
x ⎠
Axm +1 + ...
=∞
m
13.) lim
x→∞
Bx + ...
Ax m + ...
=0
m +1
14.) lim
x→∞
Bx + ...
Axm + ... A
=
m
15.) lim
x→∞
Bx + ... B
f (x) = 0 = 0
16.) lim
x→ a
∞
f (x) = (0 )(∞) = indeterminate
17.) lim
x→ a
f (x) = (−∞ − ∞ ) = −∞
18.) lim
x→ a
f (x) = ∞
19.) lim
x→ a
−∞
=
1
1
=
=0
∞∞ ∞
f (x) = ∞ = indeterminate
20.) lim
x→ a
0
f (x) = ∞ + ∞ = ∞
21.) lim
x→ a
DERIVATIVES:
1.)
The product rule is the first times the derivative to the second plus the
second times the derivative of the first, i.e.,
d (f(x)g(x))
dg(x)
df (x)
= f (x)
+ g(x)
dx
dx
dx
2.)
The chain rule is the derivative of the outer function times the derivative of
3.)
The quotient rule is the bottom times the derivative of the top minus the top
times the derivative of the bottom all over the bottom squared, i.e.,
d (f(g(x))) df dg
=
the inner function, i.e.,
dx
dg dx
⎛ f(x) ⎞
⎟⎟ g(x) df (x) − f (x) dg(x)
d ⎜⎜
⎝ g(x) ⎠
dx
dx
=
2
dx
[g(x)]
4.)
d (u a )
a-1 du
= au
dx
dx
d (a u )
5.)
dx
d (e u )
6.)
7.)
8.)
dx
= a u ln a
= eu
du
dx
du
dx
d (lnu ) 1 du
=
dx
u dx
d (log au )
1 du
=
dx
uln a dx
9.)
du
u du
=
dx
u dx
d (sinu)
du
=
cosu
10.)
dx
dx
d (secu )
du
= secutan u
11.)
dx
dx
d (cosu)
du
=
−
sinu
12.)
dx
dx
d (cot u )
du
2
= − csc u
13.)
dx
dx
d (csc u)
du
=
−
csc
ucot
u
14.)
dx
dx
d (tan u )
du
2
=
sec
u
15.)
dx
dx
d (sin-1u )
1
du
=
16.)
dx
1 − u 2 dx
d (sec -1u )
du
1
=
17.)
dx
u u2 − 1 dx
d (csc -1u)
du
−1
=
18.)
dx
u u 2 − 1 dx
d (tan -1u )
19.)
dx
1 du
=
2
1 + u dx
d (cos-1u)
−1 du
=
20.)
dx
1− u 2 dx
d (cot -1u )
21.)
dx
=
−1 du
2
1 + u dx
⎛ d( variable) ⎞
d (constant variable )
variable
= (constant
)ln(constant)⎜⎝ dx ⎟⎠
22.)
dx
⎛ d( variable) ⎞
d (variable constant )
constant−1
= (constant )(variable
)⎜⎝ dx ⎟⎠
23.)
dx
d (variable variable )
24.)
dx
→ use logarithmic differentiation
d (constant constant )
25.)
dx
=0
d (sinhu)
du
= cosh u
26.)
dx
dx
d (coshu)
du
=
sinh
u
27.)
dx
dx
d (tanhu)
du
2
=
sec
h
u
28.)
dx
dx
d (cothu)
du
2
=
−
csch
u
29.)
dx
dx
d (sechu)
du
=
−
sec
hutanhu
30.)
dx
dx
d (cschu )
du
=
−
csc
hucothu
31.)
dx
dx
d (sinh-1u )
=
32.)
dx
d (tanh -1u )
33.)
dx
1
du
u2 + 1 dx
1 du
=
2
1 − u dx
d (cosh -1u)
1
du
=
34.)
dx
u 2 − 1 dx
d (coth -1u)
35.)
dx
=
1 du
2
1− u dx
d (sech -1u)
−1 du
=
36.)
dx
u 1 − u2 dx
d (csch -1u)
du
−1
=
37.)
dx
u 1+ u 2 dx