FORMULAS REVIEW SHEET
ANSWERS
1) SURFACE AREA FOR A PARAMETRIC
FUNCTION
tf
2
2
 dx   dy 
S  2        dt
 dt   dt 
ti
2) TRAPEZOIDAL APPROXIMATION OF THE AREA
UNDER A CURVE (BOTH FORMS)
•
Recall that Tn was all about approximating the area under a
curve. If you subdivide an interval [a, b] into equal sized
subintervals, then you can imagine a string of inputs or points
c0, c1, c2, c3, …, cn-1, cn between a and b, and you can write the
trapezoidal sum as Ln is the left-hand approximation and Rn is
the right hand approximation for the area under a curve.
3) THE MACLAUREN SERIES FOR….
1
1
ln(1  x),
, ln(1  x), &
1 x
1 x
•  ln 1 − 𝑥 = − 𝑥 +
1
1−x
𝑥2
2
+
𝑥3
3
+
𝑥4
4
+⋯=−
= 1 + x + x2 + x3 + ⋯ =
•  ln 1 + 𝑥 = 𝑥 −
𝑥2
2
+
𝑥3
3
−
𝑥4
4
+⋯=
1
= 1 − x + x2 − x3 + ⋯ =
1+x
∞
;
∞
n
n=0 x
−1 𝑛 𝑥 𝑛+1
∞
;
𝑛=0
𝑛+1
−x
n=0
𝑛
∞ 𝑥
𝑛=1 𝑛
n
4) LIMIT DEFINITION OF THE DERIVATIVE (BOTH
FORMS)
f ( x  h)  f ( x )
f ( x )  f (a )
lim
 lim
h 0
x a
h
xa
5) THE VOLUME OF TWO FUNCTIONS
b
x  axis : V     f ( x)    g ( x)  dx;
2
2
a
b
y  axis : V  2  x( f ( x)  g ( x))dx
a
6) THE COORDINATE WHERE THE POINT OF
INFLECTION OCCURS FOR A LOGISTIC
FUNCTION
•
If the general form of a logistic is given by P(t ) 
M
,
 Mkt
1  ae
then the coordinate of the point of inflection is  ln a M 
, .

 Mk 2 
7) A HARMONIC SERIES
• Notice the request was for a harmonic series.
There are many and they all diverge:


1 1  1
1
  ; 
;

2 n 1 n n 2 n  1
n 1 2n


1
1
1  1
; 
 
; and so on...

3 n2 n  1
n 1 2n  1
n  2 3n  3
1
,

n 1 n

8) DISPLACEMENT IF GIVEN A VECTOR-VALUED
FUNCTION
tf
tf
 x(t )dt ,  y(t )dt
ti
ti
 x(t f )  x(ti ), y (t f )  y (ti )
9) MVT (BOTH FORMS)
• If a function is continuous, differentiable and
integrable, then
f (b)  f (a )
f (c) 
; OR
ba
b
1
f avg (c) 
f ( x)dx.

ba a
Think about it, they really are the same formula
10) ARC LENGTH FOR A RECTANGULAR
FUNCTION
b
l   1   f ( x)  dx
2
a
11) THE DERIVATIVE AND ANTIDERVIATIVE OF
LN(AX)
d
a 1
ln(ax) 
 ;
dx
ax x
ln(
ax
)
dx

x
ln(
ax
)

ax

C

12) LAGRANGE ERROR BOUND
 𝑅𝑛 (𝑥) ≤
𝑀𝑛+1 𝑥−𝑎 𝑛+1
𝑛+1 !
13) THE PRODUCT RULE
d
 f ( x)  g ( x)  
dx
f ( x)  g ( x)  f ( x)  g ( x)
14) THE SOLUTION TO THE FOLLOWING DE:
DP/DT = .05P(500-P), & IVP: P(0) = 50
• See #6 above because that logistic function is the general
solution to this specific logistic DE (differential equation) where k
= 0.05 & M = 500. Now use the initial condition to find a:
500
500
500
P(t ) 
 P(0) 
 50  50 

25t
250
1  ae
1  ae
1 a
500
1  a  10  a  9, so, P(t ) 
.
25t
1  9e
15) VOLUME OF A SINGLE FUNCTION SPUN
‘ROUND Y-AXIS
b
V  2  xf ( x)dx
a
16) HOOKE’S LAW FUNCTION AND THE GENERAL
FORM OF THE INTEGRAL THAT COMPUTES
WORK DONE ON A SPRING
• F(x) = kx where k is the spring constant and x is
the distance the spring is stretched/compressed
as a result of F force can be integrated to get
work: where a = initial spring position and b =
final spring position.
17) AVERAGE RATE OF CHANGE
f (b)  f (a )
m  slope 
ba
18) ALL LOG RULES
log(a  b)  log a  log b;
a
log    log a  log b;
b
n
log a  n log a
19) DISTANCE TRAVELED BY A BODY MOVING
ALONG A VECTOR-VALUED FUNCTION
tf
d

ti
2
2
 dx   dy 
     dt
 dt   dt 
20) A LEAST TWO LIMIT TRUTHS (YOU KNOW AT
LEAST EIGHT)
sin x
sin ax
1  cos x
lim
 1; lim
 1; lim
 0;
x 0
x

0
x

0
x
ax
x
1  cos ax
1  cos x
1  cos ax
lim
 0; lim
 0; lim
 0;
x 0
x

x

ax
x
ax
sin x
sin ax
lim
 0; lim
0
x 
x 
x
ax
21) CONVERSION FORMULAS: POLAR VS.
RECTANGUALR
y
r  x  y ; tan   ;
x
x  r cos  ; y  r sin 
2
2
2
22) AREA OF A TRAPEZOID
h
A   b1  b2 
2
23) IF GIVEN POSITION FUNCTION IN
RECTANGULAR FORM: SPEED
x(t )  v(t )
24) THE FOLLOWING ANTIDERIVATIVE

f ( x)
dx  ln[f (x)] + C
f ( x)
25) VOLUME OF TWO FUNCTIONS (AS ABOVE)
SPUN AROUND AN AXIS TO THE LEFT OF THE
GIVEN REGION
Assuming that the axis
is something of
b
the form x = q, V  2  ( x  q)  f ( x)  g ( x) dx.
a
26) THE QUADRATIC THEOREM (NOT JUST THE
FORMULA)
If given an equation of the form ax2 + bx + c = 0,
then the solutions to this quadratic can be found by
2
using
b  b  4ac
x
2a
.
27) SIMPSON’S RULE FOR THE APPROXIMATION
OF THE AREA UNDER THE CURVE
If you apply what was said above for the Trapezoidal
approximation (#2 above) with an even number of
subintervals, then the Simpson’s approximation is given
by
h
S n   f (c0 )  4 f (c1 )  2 f (c2 )  4 f (c3 )  ...  2 f (cn  2 )  4 f (cn 1 )  f (cn ) .
3
28) GENERAL FORMULA FOR A CIRCLE
CENTERED ANYWHERE
( x  a)  ( y  b)  r where (a, b) is the center
2
2
2
29) TAYLOR’S THEOREM
If you want to approximate the value of a function, like sinx, you need some
process or formula to do it. Taylor decided that a polynomial could approximate
the value of a function if you make sure it has the requisite juicy tidbits: the same
value at a center point (x = a), the same slope at that point, the same concavity
at that point, the same jerk at that point, and so on. The led him to create the
following formula:

f (a )
f ( n ) (a)
f ( k ) (a)
f (a )  f (a )( x  a ) 
( x  a ) 2  ... 
( x  a ) n  ...  
( x  a) k .
2!
n!
k!
k 0
f (0)  f (0) x 

f (0) 2
f (0) n
x  ... 
x  ...  
2!
n!
k 0
(n)
And centered at x = 0
(Maclaurin),
•He also pointed out that if you truncate the
f (0) k polynomial to n terms, then the part you cut
x .
off (the “tail”), Rn, represents the error in
k!
doing the cutting.
(k )
 The remainder

formula later
f (a)
f ( n ) (a)
2
n
f ( x)  f (a )  f (a )( x  a) 
( x  a )  ... 
( x  a )  Rn ( x) 
led to the
2!
n!

f ( n 1) (c)
 LaGrange Error
where Rn ( x) 
( x  a ) n 1 for some c between a and x.
 Bound Formula,
(n  1)!

 given in #12 above.









30) THE CHAIN RULE
d
g ( f ( x))  g ( f ( x))  f ( x)
dx
31) VOLUME OF A SINGLE FUNCTION SPUN
ROUND THE X-AXIS
b
V     f ( x)  dx
2
a
32) ALTERNATING SERIES ERROR BOUND
error  next term
33) THE THREE PYTHAGOREAN IDENTITIES
sin x  cos x  1;
2
2
tan 2 x  1  sec 2 x;
1  cot x  csc x
2
2
34) FIRST DERIVATIVE OF A PARAMETRIC
FUNCTION
dy / dt
 x(t ), y(t )  ; slope 
dx / dt
35) VOLUME OF TWO FUNCTIONS SPUN ‘ROUND
AN AXIS THAT IS ABOVE THE GIVEN REGION
If y = q is aboveb the function f (x), then the volume is
given by V    q  g ( x)2   q  f ( x)2 dx.

a
36) VOO DOO
This is also known as the Integration by Parts
process:
udv

uv

vdu


37) ARC LENGTH FOR A POLAR FUNCTION
f
l 

i
2
 dr 
r 
 d
 d 
2
38) FTC (BOTH PARTS)
If a function is continuous, then (part I)
d
f (t )dt  f ( x),

dx a
and if F(x) is an antiderivative of f (x),
b
then (part II)
 f ( x)dx  F (b)  F (a)
a
x
39) ANTIDERVIATIVE OF A FUNCTION
 ln cos x  C  ln sec x  C
40) AVERAGE VALUE OF A FUNCTION
b
1
f
(
x
)
dx

ba a
41) THE DE THAT IS SOLVED BY Y=PE^N
dy
 ry
dt
42) THE GENERAL LOGISTIC FUNCTION
M
P(t ) 
,
 Mkt
1  ae
where M = the Max value of the population (or where the
population is heading), k = the constant of proportionality,
and a = a coefficient found with an initial value.
43) VOLUME OF TWO FUNCTIONS (AS ABOVE)
SPUN ‘ROUND AN AXIS THAT IS BELOW THE
GIVEN REGION
If y = q is below the given region, then the volume is given
by
b
V     q  f ( x)    q  g ( x)  dx.
2
a
2
44) AREA OF A EQUILATERAL TRIANGLE IN
TERMS OF ITS BASE
3
2
A
(base)
4
45) SECOND DERIVATIVE FOR A PARAMETRIC
FUNCTION
d  dy / dt 


2
d y dt  dx / dt 

2
dx
dx / dt
46) IF GIVEN A POSITION VECTOR-VALUED
FUNCTION: SPEED
2
 dx 
 dy 

 

 dt 
 dt 
2
47) ARC LENGTH FOR A PARAMETRIC FUNCTION
tf
l

ti
2
2
 dx 
 dy 
 dt    dt  dt
48) AN ALTERNATING HARMONIC SERIES
Again, note that the prompt requests an alternating series. There
are many:
(1)
,

n
n 1

n
(1)
1 ( 1)
 
;

2 n 1 n
n 1 2n
(1) n 1
;

n 1 2n  1


n

n
n 1
( 1)
;

n2 n  1

(1) n 1  ( 1) n
 
; and so on...

3 n2 n  1
n  2 3n  3

And all alternating harmonics are convergent by the AST
(alternating series test).
49) DERIVATIVE OF THE FOLLOWING FUNCTION
Y= B’
dy
x
 b ln b
dx
50) THE X-COORDINATE OF THE VERTEX OF ANY
QUADRATIC FUNCTION
 b
 ,
 2a
 b
 ,
 2a
 b 
 b  
a   b   c 
 2a 
 2a  
2
2



b
b
b b

 c   ,
 c
4a 2a
  2a 4a

2
2
51) MAGNITUDE OF A VECTOR
a, b  a  b
2
2
52) AT LEAST ONE LIMIT EXPRESSION THAT
GIVES YOU THE VALUE OF E
n
1
1


lim 1    lim 1  x  x  e
n 
 n  x 0
53) THE MACLAUREN SERIES FOR SIN(X),
COS(X), AND E^X
sin 𝑥 = 𝑥 −
𝑥3
3!
2
+
𝑥5
5!
4
−
𝑥7
7!
6
+
𝑥9
9!
∞
+⋯=
𝑥
𝑥
𝑥
cos 𝑥 = 1 −
+
−
+⋯=
2! 4! 6!
−1
𝑛
𝑛=0
∞
−1
𝑛=0
2
3
𝑥
𝑥
𝑒𝑥 = 1 + 𝑥 +
+
+⋯=
2! 3!
𝑛
𝑥 2𝑛
;
2𝑛 !
∞
𝑛=0
𝑥𝑛
𝑛!
𝑥 2𝑛+1
2𝑛 + 1 !
54) SLOPE OF AN INVERSE FUNCTION AT THE
INVERTED COORDINATE
df 1
dx

f (a)
1
df
dx
xa
55) THE QUOTIENT RULE
d t ( x) t ( x)  b( x)  t ( x)  b( x)

2
dx b( x)
 b( x ) 
56) SOH-CAH-TOA WITH A RIGHT TRIANGLE
DRAWING
b
sin   ;
c
a
cos   ;
c
b
tan  
a
57) GENERAL GEOMETRIC SERIES AND ITS SUM

a1
ar 
if r  1

1 r
n 0
n
58) SLOPE OF A LINE NORMAL TO A CURVE
If m = f’(x) represents the slope of the tangent line to a
curve or the instantaneous rate of change of f (x), then the
slope of the line normal to the curve is given by
1
1

m f ( x)
59) DISTANCE TRAVELED BY A RECTANGULAR
FUNCTION
This is the same as arc length:
b
l   1   f ( x)  dx.
2
a
60) VOLUME OF 2 FUNCTION (AS ABOVE) SPUN
‘ROUND AN AXIS THAT IS TO THE RIGHT OF THE
GIVEN REGION
If x = q is to the right of the given region, then
the volume is given by
b
V  2  (q  x)  f ( x)  g ( x)  dx.
a
61) SURFACE AREA FOR PARAMETRIC
FUNCTIONS SPUN ‘ROUND BOTH THE X AND YAXES
• For spinning around the x-axis
tf
SA  2  y (t )
 x(t )   y(t )
2
2
dt ;
ti
• For spinning around the y-axis:
tf
SA  2  x(t )
ti
 x(t )   y(t )
2
2
dt ;
62) NEWTON’S LAW OF COOLING DE AND
GENERAL SOLUTION FUNCTION
dy
 k (T  y );
dt
y (t )  T  Ae
kt
63) AREA BETWEEN TWO FUNCTIONS (AS
ABOVE)
b
A    f ( x)  g ( x)  dx
a
64) CHANGE OF BASE FOR LOGS (18???)
ln a
log b a 
ln b
65) DERIVATIVE FOR AS MANY INVERSE
TRIGONOMETRIC FUNCTIONS AS YOU CAN
REMEMBER
d
1
1
tan x 
;
2
dx
1 x
d
1
1
sin x 
;
2
dx
1 x
d
1
d
1
cos x 
;
sec 1 x 
dx
dx
1  x2
x
1
x2 1
d
1
d
1
1
1
csc x 
;
cot x 
2
2
dx
dx
1

x
x x 1
;