Download p p 1 p p 1 p p p

Math 025 Sections 1 and 2
Properties of Logarithms (the rst four properties hold for logs with any base, including ln)
1. log(AB) = log A + log B
2. log BA = log A log B
3. log B1 = log B
4. log(AC ) = C log A
ln A
5. logb A = ln
B
Values of the Trigonometric Functions
x
0 =6 p=4 p=3
=2
3=2
sin x 0 p1/2 p2=2
3=2
1
0
1
cos x 1
3p=2
2=2 1/2
0
1
0
p
tan x 0 1= 3
1
3 undened 0 undened
Trigonometric Identities
1. csc x = sin1 x ; sec x = cos1 x ; cot x = tan1 x
sin x
cos x
2. tan x = cos
x2; cot x = sin
x
2
3. sin x + cos x = 1; sec2 x = tan2 x + 1; csc2 x = cot2 x + 1
4. sin2 x = 1 cos 2x ; cos2 x = 1+cos 2x
2
Dierentiation Rules:
2
d
dx c = 0
d
n
n 1 for any real number n 6= 0. Note: Special cases of this
Power Rule:
dx x = nx
rule include the following:
d
d
Linear Functions:
dx (mx +pb) = m (in particular dx x = 1)
d
1
Square Root Function:
dx x = 2px
1
d 1
Reciprocal Function:
dx ( x ) = x2
d x
x
Exponential Function:
dx e = e
d
d
d
d
2
Trig Functions:
dx sin x = cos x; dx cos x = sin x; dx tan x = sec x; dx cot x =
d sec x = sec x tan x; d csc x = csc x cot x
csc2 x; dx
dx
d arcsin x = p 1 ; d arctan x = 1 2
Inverse Trig Functions:
dx
1+x
1 x2 dx
d (f g ) = g ( d f ) + f ( d g )
Product Rule:
dx
dx
dx
d f )g ( d g )f
( dx
f
d
dx
Quotient Rule:
dx ( g ) =
g2
dy
dy
du
0
0
0
Chain Rule:
dx = du dx i.e. (f g ) (x) = f (g (x)) g (x)
Integration Rules:
R
1. K dx = Kx + C
R
n+1
2. R xn dx = xn+1 + C for any rational number n 6= 1
3. R x1 dx = ln jxj + C
4. R sin x dx = cos x + C
5. R cos x dx = sin x + C
6. R sec2 x dx = tan x + C
7. R csc2 x dx = cot x + C
8. R sec x tan x dx = sec x + C
9. R csc x cot x dx = csc x + C
10. 1+1x2 dx = arctan x + C
Constant Functions:
Fall 2010
Math 025 Sections 1 and 2
11.
12.
R
R
Fall 2010
p 1 2 dx = arcsin x + C
1 x
ex dx = ex + C
Main Theorems of Dierential Calculus:
Intermediate Value Theorem: If f is continuous on [a; b], then for every y -value d
between f (a) and f (b) there is at least one x-value c such that f (c) = d.
Max-Min Existence Theorem (Extreme Value Theorem): If a function f is cts on a
closed interval [a; b], then f has an absolute maximum and an absolute minimum
on that interval.
First Derivative Theorem for Extrema: If a function f has a local extremum at an
interior point c of its domain, then either f 0 (c) = 0 or f 0 (c) DNE.
Critical Point Theorem: All local extrema of a function f must occur at endpoints
of the domain of f and/or critical points of f .
Mean Value Theorem: If f is cts on [a; b] and dierentiable on (a; b), then there is
at least one x-value c in (a; b) such that f 0 (c) = f (bb) af (a) .
Monotonicity Test (First Derivative Test for Monotone Functions): If f is cts on
[a; b] and dierentiable on (a; b), then
0
{ f (x) > 0 , f is increasing
0
{ f (x) < 0 , f is decreasing
First Derivative Test: Let c be a critical point for f . Suppose that f is dierentiable
in an open interval containing c, but is not necessarily dierentiable at c itself.
Then:
0
{ If f changes sign from + to
from left to right at c, then f has a local
maximum at c.
0
{ If f changes sign from
to + from left to right at c, then f has a local
minimum at c.
0
{ If there is no change in the sign of f at c, then f has no local extremum at
c.
00
Concavity Test: Let f (x) be a function so that f (x) exists on an interval. Then:
00
{ f (x) > 0 , f is concave up
00
{ f (x) < 0 , f is concave down
00
{ c is an inection point of f , sign of f changes at x = c.
00 is cts on an open interval containing x = c
Second Derivative Test: Suppose f
where c is a critical point of f . Then:
00
{ f (c) < 0 , f has a local max at x = c
00
{ f (c) > 0 , f has a local min at x = c
00
{ f (c) = 0 , test is inconclusive
L'Hopital's Rule: Let f and g be dierentiable functions such that either limx!a f (x) =
limx!a g (x) = 0 or limx!a f (x) = limx!a g (x) = 1. Then limx!a fg((xx)) =
limx!a fg ((xx)) .
Fundamental Theorem of Calculus Part I: Let f be continuous on [a; b]. Then the
R
function F (x) = ax f (t)dt is continuous and dierentiable on [a; b] and F 0 (x) =
f (x).
Rb
Fundamental Theorem of Calculus Part II: Let f be continuous on [a; b]. Then
a f (x)dx =
F (b) F (a) where F is any antiderivative of f on [a; b].
0
0