LÍMITES Y DERIVADAS aplicados a ingenieria

Diapositivas de limites y derivadas, aplicados en ingenieria.

1. 1
4
Límites y
Derivadas

2. 2
Límites y la Derivada
•Límites
•Límites Laterales
•Continuidad
•La Derivada

3. 3
Introducción al Cálculo
Existen dos áreas principales de interés:
1. Encontrar la recta tangente a una curva en un
punto dado.()y f x
Recta tangente
11
,xy
2. Encontrar el área de una región plana acotada
por una curva.
Área
x
x
y
y

4. 4
Velocidad
PromedioDistancia recorrida
Tiempo transcurrido

Instantánea
Cuando el tiempo
transcurrido se
aproxima a cero
Sobre cualquier
intervalo de tiempo
Si viajo 200 millas en 5 horas, mi velocidad
promedio es 40 millas/hora.
Cuando veo al oficial de policia, mi velocidad
instantánea es 60 millas/hora.Distancia recorrida
Tiempo transcurrido


5. 5
Velocidad
Ej. Given the position function
2
10s t t t
where tis in seconds and s(t) is measured in
feet, find:
a.The average velocity for t= 1 to t = 3.
b.The instantaneous velocity at t= 1.ave
(3) (1) 39 11
Velocity 14 ft/sec
3 1 2
ss
  

Average velocityt( ) (1)
1
s t s
t


1.1 12.1
1.001 12.001
1.01 12.01
Answer: 12 ft/sec
Notice how
elapsed time
approaches zero

6. 6
Límite de una Funcción
The limit of f (x), as xapproaches a, equals L
written:
if we can make the value f (x) arbitrarily close
to L by taking xto be sufficiently close to a.lim ( )
xa
f x L


a
L()y f x
x
y

7. 7
Computing Limits
Ex.2
3 if 2
lim ( ) where ( )
1 if 2x
xx
f x f x
x
  


6
-222
lim ( ) = lim 3
xx
f x x
 
 2
3 lim
3( 2) 6
x
x


   
Note: f(-2) = 1
is not involved
x
y

8. 8
Properties of Limits
 
 
Suppose lim ( ) and lim ( )
Then,
1. lim ( ) , real number
2. lim ( ) lim ( ) , real number
3. lim ( ) ( )
4. lim ( ) ( )
lim ( )
()
5. lim
( ) lim
x a x a
r r
xa
x a x a
xa
xa
xa
xa
f x L g x M
f x L r a
cf x c f x cL c a
f x g x L M
f x g x LM
fx
fx
gx










  

 Provided that 0
()
xa
L
M
g x M



9. 9
Computing Limits
Ex.
Ex. 
2
3
lim 1
x
x

 2
33
lim lim1
xx
x

 
2
33
2
lim lim1
3 1 10
xx
x


   1
21
lim
35x
x
x

  
 
1
1
lim 2 1
lim 3 5
x
x
x
x




 11
11
2lim lim1
3lim lim5
xx
xx
x
x




 2 1 1
3 5 8




10. 10
Indeterminate Form:0
0 2
5
5
lim
25x
x
x


Ex. Notice form0
0 
5
5
lim
55x
x
xx


 
5
11
lim
5 10xx


Factor and cancel
common factors

11. 11
Limits at Infinity
For all n> 0,11
lim lim 0
nn
xxxx 

provided that is defined.1
n
x
Ex.2
2
3 5 1
lim
24
x
xx
x


 2
2
51
3
lim
2
4
x
xx
x



 3 0 0 3
0 4 4

  

Divide
by2
x 2
2
51
lim 3 lim lim
2
lim lim 4
x x x
xx
x x
x
  
 
   

   
   






12. 12
One-Sided Limit of a Function
The right-hand limit of f (x), as xapproaches a,
equals L
written:
if we can make the value f (x) arbitrarily close
to L by taking xto be sufficiently close to the
right of a.lim ( )
xa
f x L



a
L()y f x

13. 13
One-Sided Limit of a Function
The left-hand limit of f (x), as xapproaches a,
equals M
written:
if we can make the value f (x) arbitrarily close
to M by taking xto be sufficiently close to the
left of a.lim ( )
xa
f x M



a
M()y f x
x
y

14. 14
One-Sided Limit of a Function2
if 3
()
2 if 3
xx
fx
xx
 


Ex. Given3
lim ( )
x
fx

 33
lim ( ) lim 2 6
xx
f x x


 2
33
lim ( ) lim 9
xx
f x x



Find
Find 3
lim ( )
x
fx



15. 15
Continuity of a Function
A function fis continuousat the point x= aif
the following are true:) ( ) is definedi f a ) lim ( ) exists
xa
ii f x
 )lim ( ) ( )
xa
iii f x f a


a
f(a)
y
x

16. 16
Properties of Continuous Functions
The constant function f (x) is continuous everywhere.
Ex. f (x) = 10 is continuous everywhere.
The identity function f (x) = x is continuous
everywhere.

17. 17
Properties of Continuous Functions
A polynomial functiony= P(x) is continuous at
everywhere.
A rational function is continuous
at all x valuesin its domain.()
()
()
px
Rx
qx

If fand gare continuous at x = a, then , , and ( ) 0 are continuous
at .
f
f g fg g a
g
xa



18. 18
Intermediate Value Theorem
If fis a continuous function on a closed interval [a, b]
and Lis any number between f (a) and f (b), then there
is at least one number cin [a, b] such that f(c) = L. ()y f x
a b
f (a)
f (b)
L
c
f (c) =
x
y

19. 19
Intermediate Value Theorem
2
Given ( ) 3 2 5. Show that ( ) 0
has at least one solution on 1, 2 .
f x x x f x   
Ex. (1) 4 0 and (2) 3 0ff    
f (x) is continuous for all values of xand since
f (1) < 0 and f (2) > 0, by the Intermediate Value
Theorem, there exists a con (1, 2) such that
f (c) = 0.

20. 20
Existence of Zeros of a Continuous
Function
If fis a continuous function on a closed interval [a, b],
and f(a) and f(b) have opposite signs, then there is at least
one solution of the equation f(x) = 0 in the interval (a, b).
f(b)
f(a)
a
b
x
y

21. 21
Example
(Existence of zeros of a continuous function)
1.Show that f(x) is a continuous function everywhere.
The function is a polynomial function and is
therefore continuous everywhere.
2.Show that f(x) = 0 has at least one solution on the
interval (0, 2)Since (0) 5 and (2) 5 have opposite signs,
there must be at least one number with
0 2 such that ( ) 0.
ff
xc
c f c
  

   2
Let ( ) 3 5.f x x x  

22. 22
Rates of Change
Averagerate of change of f over the interval
[x, x+h]( ) ( )f x h f x
h


Instantaneousrate of change of fat x
Slope of the
Tangent Line
Slope of Secant Line0
( ) ( )
lim
h
f x h f x
h



23. 23
The Derivative0
( ) ( )
( ) lim
h
f x h f x
fx
h


The derivative of a function fwith respect to xis
the function ,f given by
It is read “f prime of x.”

24. 24
The Derivative0
( ) ( )
( ) lim
h
f x h f x
fx
h


Four-step process for finding:f ( ) ( )f x h f x ( ) ( )f x h f x
h
 ()f x h
1. Compute
2. Find
3. Find
4. Compute

25. 25
The Derivative 
2
00
0
( ) ( ) 4 2
lim lim
lim 4 2 4
hh
h
f x h f x xh h
hh
x h x


  

   
Given 2 2 2
( ) ( ) 2 4 2 1 (2 1)f x h f x x xh h x        2
( ) ( ) 4 2f x h f x xh h
hh
  
 
2 22
( ) 2 1 2 4 2 1f x h x h x xh h       
1.
2.
3.
4. 2
( ) 2 1, find ( ).f x x f x  2
42xh h ( ) 4f x x

26. 26
Example
Find the slope of the tangent line to the
graph of at any
point (x, f(x)).
Step 1.
Step 2.
Step 3.
Step 4. ( ) 2( ) 3 2 2 3f x h x h x h         ( ) ( ) 2
2
f x h f x h
hh
  
   ( ) 2 3f x x   ( ) ( ) ( 2 2 3) ( 2 3) 2f x h f x x h x h           
00
( ) ( )
( ) lim lim 2 2
hh
f x h f x
fx
h


    

27. 27
Differentiability and Continuity
If a function is differentiable at x= a, then it is
continuous at x = a.
Not
Differentiable
Not
Continuous
Still
Continuous
x
y

28. 28
Example
The function is not
differentiable at x= 0 but it is continuous
everywhere. ()f x x
x
y
O()f x x

29. 29
Axiomas de Probabilidad
Dado un espacio muestral W, la probabilidad P de un
evento A es un número real no negativo P(A), que debe
satisfacer los tres axiomas siguientes:
•Límites
•Límites Laterales
•Continuidad
•La Derivada

30. 30