Skeleton Calculus
Salamon, Michelsen
Skeleton Calculus
Peter Salamon
Professor, SDSU Mathematics
Eric L. Michelsen
Lecturer, UCSD Physics
Contents
I.
Overview .............................................................................................................1
Functions......................................................................................................................1
II.
Limits...................................................................................................................2
III.
f: R1 → R1............................................................................................................4
Differential Calculus .....................................................................................................4
Integral Calculus...........................................................................................................5
IV.
f: R1 → R2............................................................................................................7
Differential Calculus .....................................................................................................7
Integral Calculus...........................................................................................................7
V.
f: R2 → R1............................................................................................................9
Differential Calculus .....................................................................................................9
Integral Calculus.........................................................................................................10
VI.
f: Rn → Rm .........................................................................................................12
Differential Calculus ...................................................................................................12
VII.
Convergence of Infinite Series............................................................................13
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Copyright 2002-2011. All rights reserved.
Skeleton Calculus
I.
Salamon, Michelsen
Overview
Basic calculus relies on 4 major concepts:
1. Functions
2. Limits
3. Derivatives
4. Integrals
Functions
A function takes one or more real values as inputs, and produces one or more real values
as outputs. The inputs to a function are called the arguments. The simplest case is a
real-valued function of a real-valued argument (f : R1 → R1), e.g., f(x) = sin x. A function
which produces more than one output may be considered a vector-valued function.
There are 4 cases of interest:
Case
Example
1
1. f : R → R
1
y = f(x) = x2
x2
x
0

2. f : R1  R 2

( x, y )  f (t )  (t  cos t , sin t )
1
y
x
3. f : R2 → R1
z = f(x, y) = x2 + y2
z
x
y

4. f : R 2  R 2

y

(r , )  f ( x, y )   x 2  y 2 , arctan 
x

/2
θ = angle

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1 2 3 4 5
r = radius
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Salamon, Michelsen
II.
Skeleton Calculus
Limits
A. Definition (for a real-valued function of a single argument, f : R1 → R1):
L is the limit of f(x) as x approaches a, iff for every ε > 0, there exists a δ (> 0) such that
|f(x) – L| < ε whenever 0 < |x – a| < δ. In symbols:
L  lim f ( x ) iff   0,    f ( x )  L   whenever 0  x  a  
x a
This says that the value of the function at a doesn’t matter; in fact, most often the function
is not defined at a. However, the behavior of the function near a is important. If you can
make the function arbitrarily close to some number, L, by restricting the function’s
argument to a small neighborhood around a, then L is the limit of f as x approaches a.
2x2  2
 4 . We prove the existence of δ given any ε by
x 1
x 1
2x2  2
computing the necessary δ from ε. Note that for x  1,
 2( x  1) . The definition
x 1
of a limit requires that
Example: Show that lim
2x2  2
 4   whenever 0  x  1   
x 1

2( x  1)  4  
2 ( x  1)  2  


x 1 

2
So by setting δ = ε/2, we construct the required δ for any given ε. Hence, for every ε,
there exists a δ satisfying the definition of a limit.
B. Theorems which make the definition easy to apply (a is a constant; f, g, h functions):


lim  f ( x)  g ( x)  lim f ( x)  lim g ( x)

x a
x a

x a
lim  f ( x) g ( x )  lim f ( x) lim g ( x)
x a
xa
xa
f ( x)
 f ( x )  lim
x a
lim 

x a g ( x ) 
g ( x)

 lim
x a
L’Hôpital’s rule: If

if this fraction is defined
f ( x)
f ' ( x)
f ( a)

0
is indetermin ate  or , lim
 lim
x

a
x

a
g ( x)
g ' ( x)
g ( a)

0
1  cos
0
  , so
 0

0
1  cos
sin 
 lim
0
 0
 0

1
lim
Example: lim
The Squeeze Theorem: If
f ( x )  g ( x)  h( x ), and lim f ( x )  lim h( x)  L, then lim g ( x )  L.
x a
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xa
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xa
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Skeleton Calculus
Salamon, Michelsen
Example: Show that lim
 0
sin 
 1.

The diagram shows a section of the unit
circle. Comparing the areas of triangle
OAB with circular segment OAB, we see
sin   

sin 
1

.
Comparing segment OAB with triangle
OCB:
  tan  
sin 
cos
A

cos 
1u
nit
sin θ
O θ
C
tan θ
B
sin 
 .
Given that lim cos  1 , and noting that the inequalities apply for both small positive and
 0
negative θ, we apply the squeeze theorem:
cos 
sin 
 1, and lim cos  lim 1  1 
 0
 0

sin 
1
 0 
lim
C. Infinite Limits: Definitions:
L  lim f ( x ) iff   0,  M  f ( x )  L   whenever x  M
x 
lim f ( x)   iff  N ,    f ( x)  N whenever 0  x  a  
x a
lim f ( x )   iff  N ,  M  f ( x)  N whenever x  M
x 
Example: lim sin 
 
does not exist (is not finite), and is not infinite.
Derivatives and integrals are discussed below for each case separately.
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Salamon, Michelsen
III.
Skeleton Calculus
f: R1 → R1
Differential Calculus
f ( x  h)  f ( x )
.
h
A. Definition: f ' ( x )  lim
h 0
d ( x2 )
x 2  2 xh  h 2  x 2
( x  h) 2  x 2
 lim
 lim
 lim 2 x  h  2 x
h0
h 0
h 0
dx
h
h
Examples:
d (sin  )
sin(  h)  sin( )
sin  cos h  cos sin h  sin( )
 lim
 lim
h0
h 0
d
h
h
 lim cos
h 0
sin h
 cos
h
All trigonometric derivative formulas follow from that for sin θ.
B. Theorems which make the definition easy to apply (a, b constants; f(x), g(x)
functions):
(af + bg)’ = af’ + bg’
(fg)’ = f ·g’ + f’·g
(product rule)

f
f 'g  f  g '
  
g2
g
(quotient rule)
 f g ( x)    f ' g ( x)   g ' ( x)
(chain rule)
Example: Chain rule: f(x) = sin x
f’(x) = cos x
g(x) = x2
g’(x) = 2x
f(g(x)) = sin x2
[f(g(x))]’ = f’(g(x))·g’(x) = (cos x2)(2x)
C. The derivative approximates the change in the value of a function as a linear function
of the change in its argument (the differential).
f(x) = x2 => Δf ≈ 2x Δx
Δf ≈ f’(a) Δx
E.g.,
near x = 3:
f(3) = 9, f’(3) = 6,
f(x) - 9 ≈ 6(x - 3)
D. Taylor’s Theorem: using higher derivatives, one can construct better (quadratic, cubic,
etc.) approximations to a function at a point. Expanded about a:
f ( x)  f ( a) 
Rn a, x  
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f ' a 
f ' ' a 
f ( n ) a 
( x  a) 
( x  a) 2  ... 
( x  a ) n  Rn a, x 
1!
2!
n!
f ( n1) c 
( x  a) n1 ,
n  1 !
acx
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Skeleton Calculus
Salamon, Michelsen
Example: Expanding ex about x = 0:
x 2 x3
  ...  Rn 0, x 
2! 3!
f ( x)  e x  1  x 
Rn 0, x  
ec
x n 1
x n 1  e c
n  1 !
n  1 !
Integral Calculus
A. Definition:

b
a
f ( x) dx  lim
 0
N
 f ( )  x
i
i
i 1
  x0  a, x1 , x2 , x3 ,...x N  b,
xi   i  xi 1
 i x  xi 1  xi ,
B. Theorems which make the definition easy to apply:
F ( x)  
x
a
f ( x) dx

F ' ( x)  f ( x)
f(x)
(Fundamental Theorem of Calculus)
dF  f ( x) dx 

b
a
f  g ( x) g ' ( x) dx  
g (b )
F '
f ( y ) dy
g (a )
(change of variable)
dF
 f ( x)
dx
dx
a
x
Example: Change of variable:
4
3/ 4
3/ 4
7/4
 0 1  cos  sin  d  0  1  cos  d 1  cos    1  cos 


7
C. Integral = limit of a sum of pieces which approximate
the quantity of interest. The limit is taken as the pieces get
smaller and more numerous. To get a useful result, the
approximation must be perfect (the error must go to zero)
for infinitely many infinitesimal pieces.
Example:
x3
x
dx

0
3
1
1

2
0
D. Advanced techniques of evaluation:


0
4 7 / 4 215 / 4
2 
7
7
x2
1
3
dx
0
1
1. Trigonometric substitution:
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Salamon, Michelsen

Skeleton Calculus
1  x 2 dx, let x  sin  ,
 dx  cos d
1
dx
x  2x  6

2
1
dx
x 1
2. Partial fractions: 
2
3. Integration by parts (product rule in reverse):  U dV  UV   V dU
Example:


1
0
1
0
Let U  x  dU  dx,
dx  xe    e dx  xe  e 
x e x dx
x ex
x 1
0
1
0
x
x
x 1
0
dV  e x dx  V  e x

1
E. Improper Integrals: Definition: If f is not continuous at a,

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b
a
b
f ( x) dx  lim  f ( x) dx
xa
x
(and similarly if f discontinuous at b, or both)
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Skeleton Calculus
Salamon, Michelsen
f: R1 → R2

f (t )   x (t ), y (t )  , in other words, f is a collection (vector) of two functions, x(t) and y(t).
IV.
Differential Calculus



f (t  h)  f (t )
A. Definition: f ' (t )  lim
h 0
h

 dx dy 
B.
f ' (t )   ,    x, y 
 dt dt 
= velocity vector = tangent to curve

f '  x 2  y 2  speed
unit tangent 
y(t)
velocity 
x
y
,

 x 2  y 2
speed
x 2  y 2

f’(t)
f(t)
x(t)




f ' ' (t )  x, y  acceleration vector
Chain Rule (change of parameter) :



df dt  dx dt dy dt 
 
,
f (t ( s))'  f ' (t (s))  t ' ( s) 

dt ds  dt ds dt ds 
C. The derivative approximates the change in the value of a function as a linear function
of the change in its argument (the differential).


f  x, y   f ' (a )t
D. Taylor’s Theorem: using higher derivatives, one can construct better (quadratic, cubic,
etc.) approximations to a function at a point. Expanded about a:




f ' a 
f ' ' a 
f (t )  f (t ) 
(t  a ) 
(t  a) 2  ...
1!
2!
x(t ), y(t )  x(a), y (a)   x(a), y(a)  (t  a)  x(a), y(a)  (t  a) 2  ...


1!


2!
Integral Calculus
A. s = arc length =


df  lim
 0
N

i 1


f (ti 1 )  f (ti )
y(t)
f(t)
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x(t)
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Salamon, Michelsen
Skeleton Calculus
dy 2
 dx 
 dx 
B. s   x 2  y 2 dt  time integral of speed   1  

2
 dx 
1    dy
 dy 
C. Exercises:
1.
(x, y, z) = (t, t2, 1)
velocity(t = 1) =
speed =
distance traveled t=0 to t=1 ?
unit tangent at t=1 ?
2.
8 of 14
Find length of y = x2 between x = 0 and x = 1.
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Skeleton Calculus
V.
Salamon, Michelsen
f: R2 → R1
z = f(x, y)
Differential Calculus
A. Definition:
f ( x  h, y )  f ( x, y )
f
 f x  lim
h 0
h
x
f ( x, y  h)  f ( x, y )
f
 f y  lim
h 0
h
y
 f f 
f '  Df  f  gradient  grad f   , 
 x y 
2 f 

y x 

2 f 
y 2 

 2 f

2
 x
f ' '  D2 f   2
  f
 x y


f is perpendicular to level curves (curves such that f(x, y) = constant)
f points in the direction of fastest (steepest) increase of f.
B.
Da f = directional derivative of f

= rate of change of f per unit distance in the direction a .



 v
 f  a
where a  1, i.e., a  
v
Maximum value of Da f  f 
2
fx  f y
2
Chain rule:

z  f  x(t ), y (t )   f  g (t ) 
dz
f dx f dy
 f ( x, y ) ( x , y ) 

dt
x dt y dt


 f '  g (t )   g ' (t )
C. The derivative approximates the change in the value of a function as a linear function
of the change in its argument (the differential).
f  f  x, y  
D.
f
f
x  y
x
y
Tangent plane:
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Salamon, Michelsen
Skeleton Calculus

N  X  X 0   0
f , f
x
y
,1   x  a, y  b, z  f (a, b)  0
Normal line:

X  X 0  tN
x, y, z   a, b, f (a, b)   t  f x , f y ,1
E. Taylor’s Theorem, expanded about (a,b):


[ x  a, y  b] D 2 f
f a, b 
 ( x  a, y  b) 
 ( x  a, y  b)  ...
1!
2!

1  2 f
f
f
2 f
f
( x  a )( y  b)  ( y  b) 2   ...
 f (a, b)  ( x  a )  ( y  b)   2 ( x  a) 2  2
2!  x
x
y
x y
y

f ( x, y )  f ( a, b) 
F. Exercises:
1.
Find D(1,0,0)f(x, y, z)
2.
(xy – z2 cos x) =
3.
Tangent plane to x2 + y2 = z, at (1, 1, 2)
4.
Maximum value of f(x, y) = 9 – x2 + 2x – y2 + 6y
5.
Minimum of x2 + y2 , given x2 – y = 1
Integral Calculus
A. Definition, integral over a region F:
 f ( x, y) dA  lim  f (i ,i ) A(ri ),
 0
F
B.

F
f dA  
( i ,i )  (ri )
i
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
F
...
ri
 f dx  dy   f dy  dx
F  ( x, y ); 1  x  2, x  y  2,
Exercise:
r1 r2
x ln y dA 
f ( x, y )  x ln y
(do both ways)
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Skeleton Calculus
Salamon, Michelsen
C. Change of variable (the “anyway you slice it” theorem):
g ( x, y )  (u , v)
( x, y )  g 1 (u , v)

F
f (u , v) du dv  
 
1
g (F )
g 1 ( F )
f  g ( x, y ) 
 (u , v)
dx dy
 ( x, y )
f  g ( x, y )  Dg dx dy  
1
g (F )
f  g ( x, y ) 
1
dx dy
Dg 1
Exercises:
1.

y

g ( x, y )  (r , )   x 2  y 2 , arctan 
x

( x, y )  g 1 (r , )  (r cos , r sin  )
2.

h ( x, y, z )  (r , , z ) 
( x, y , z )  h 1 (r , , z ) 
3.
dx dy  r dr d ??
dx dy dz 

j ( x, y , z )  (  , ,  ) 
( x, y , z )  j 1 (r , , z ) 

dx dy dz 
x 2  y 2 dx dy , where F  (r , ); r   , 0     
4.
Find
5.
Find the center of mass of the cone
F
( x, y, z);

x2  y 2  z  1 ,
  x2  y 2  z 2  1
6.
Find the volume of the paraboloid z = x2 + 2y2 , below z = 1.
7.
Find the volume of R  ( x, y , z ); 0  x  1, 1  y  z, 1  z  2
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Copyright 2002-2011. All rights reserved.
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Salamon, Michelsen
VI.
Skeleton Calculus
f: Rn → Rm
f(x, y) = (u, v) = (u(x, y), v(x, y))
Differential Calculus
A. Definition:
 u



  (u , v )  x
f '  Df  Derivative of f  Jacobian of f 

( x, y )  v
 x


B.
f : R3 → R3, g : R3 → R3
Chain rule, Case 1:
Example:
u 

y 

v 
y 

 f g( x) '  f ' g( x)  g ' ( x)
 (  , ,  )  (  ,  ,  )  ( r , , z )


 ( x, y , z ) (r , , z ) ( x, y, z )
f : R3 → R1, g : R3 → R3
Chain rule, Case 2:

g ( r , s, t )  ( x , y , z )
 f g (r , s, t )  '  f ' g (r, s, t )  g ' (r , s, t )

 x

 r

 f f f   f f f  y
  , ,    , , 
 r s t   x y z  r

 z
 r


x
s
y
s
z
s

x 

t 

y 
t 

z 
t 

  f x xr  f y yr  f z zr , f x xs  f y ys  f z zs , f x xt  f y yt  f z zt 
Exercise:
f ( x, y )  x e y ,
x, y   s 2  t , cos t ,
Find
f
f
and
s
t

C. f  u, v   f ' (a, b)x, y 
D. Taylor’s Theorem, expanded about (a,b):


Df a, b 
f ( x, y )  f (a, b) 
 ( x  a, y  b)  ...
1!
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Copyright 2002-2011. All rights reserved.
2/25/2011 9:45
Skeleton Calculus
Salamon, Michelsen
VII. Convergence of Infinite Series

Arbitrary:
 un ; or power series:
n 0
A. Ratio test: Arbitrary:   lim
n 
If


n 0
n 0
 u n   an x n
un1
a
 Power series:   lim n1  x
n  a
un
n
ρ < 1: Converges absolutely
ρ > 1: Diverges absolutely
ρ = 1: No information
Which implies a power series converges for x  lim
n
an

an 1
B. Comparison tests:
(1) Compare un to a known convergent series, vn. If there exist m and n such that
um+i ≤ vn+i for all i ≥ 0, then Σun converges.
(2) Compare un to a known divergent series, vn. If there exist m and n such that
um+i ≥ vn+i for all i ≥ 0, then Σun diverges.
C. Integral test: If un can be written as a
decreasing function on the reals, f(x), such that
f(n) = un , for all n ≥ m, then the series Σun has
the same convergence or divergence as the
integral


m
f ( x) dx .
∞
∞
∞
Σ u < ∫m f(x) dx
n=m+1 n
D. Alternating series test: For a series whose
terms alternate in sign, the series converges iff
u
lim n 1  1 .
n  u
n
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∞
Σ u > ∫m f(x) dx
n=m n
Copyright 2002-2011. All rights reserved.
m m+1
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Salamon, Michelsen
Skeleton Calculus
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