Transcript Indefinite integral

Physics 218: Mechanics
Instructor: Dr. Tatiana Erukhimova
Lecture 2
Overview of Today’s Class
•Derivatives
•Indefinite integrals
•Definite integrals
•Examples
Quiz
If f(x)=2x3-5 what is the derivative of f(x) with respect to x?
•6x
•8x2
•I don’t know how to start
•6x2
1. If f(x)=2x3 evaluate
•2x4
 f ( x)dx
•I don’t know how to start
•0.5x4+Const
•2x2/3
1. If f(x)=2x evaluate
3
 f ( x)dx 
2
•2x3/3
•I don’t know how to start
•5
•x2+Const
Derivatives
A derivative of a function at a point is a slope
of a tangent of this function at this point.
Derivatives
df
f ( x  x)  f ( x)
 lim
dx x0
x
n
df d (kx )

 nkxn 1
dx
dx
d ( f  g ) df dg


dx
dx dx
Derivatives
dx(t )
x(t  t )  x(t )
 lim
t 0
dt
t
df ( x)
f ( x  x)  f ( x)
 lim
or
x 0
dx
x
Function x(t) is a machine: you plug in the value of
argument t and it spits out the value of function x(t).
Derivative d/dt is another machine: you plug in the
function x(t) and it spits out another function V(t) = dx/dt
n
dx d (kt )

 nkt n 1 ,
dt
dt
d ( f  g ) df dg


dt
dt dt
d (Const )
0
dt
d ( fg )
df
dg
g
f
dt
dt
dt
Derivative is the rate at which something is
changing
dx
V
dt
Velocity: rate at which position changes with time
dV
 a Acceleration: rate at which velocity changes with time
dt
dU

 F Force: rate at which potential energy changes with
position
dx
Derivative is the rate at which something is
changing
-Size of pizza with respect to the price
-Population of dolphins with respect to the sea
temperature
…………………
GDP per capita
d (GDP )
dt
Quiz
If f(x)=2x3+5x what is the derivative of f(x) with respect to x?
1 2
If x(t )  at what is the derivative of x(t) with respect to t?
2
Indefinite integral
(anti-derivative)
A function F is an “anti-derivative” or an
indefinite integral of the function f
F   f ( x)dx
if
dF
 f (x)
dx
Indefinite integral
(anti-derivative)
df ( x)
dx

f
(
x
)
 Const
 dx
d
df ( x)
( f ( x)  Const ) 
dx
dx
f ( x)  kx
n

1
n 1
f ( x)dx   (kx )dx 
kx  Const
n 1
n
n – integer except n= -1
d
1
d
d
1
n 1
n 1
(
kx )  (Const ) 
(
kx  Const ) 
dx n  1
dx
dx n  1
n 1 n
kx  0 
n 1
n
kx
òx
òx
2
òx
2
cosq = ?
cosq dx = ?
2
cosq dq = ?
Lev Landau
1962, Nobel Prize
“An integral without dx is like a man
without pants”
Definite integral
B
 f ( x)dx  F
B
A
 F ( B )  F ( A)
A
F is indefinite integral
Definite integral
Integrals
Indefinite integral:
1
n 1
(
kx
)
dx

kx
 Const

n 1
n
n – any number except -1
Definite integral:
B
1
1
n 1
n 1
kx
dx

kB

kA
A
n 1
n 1
n
Gottfried Leibniz
F   f ( x)dx
dF
 f (x)
dx
1646-1716
These are Leibniz’ notations: Integral sign as an
elongated S from “Summa” and d as a differential
(infinitely small increment).
Leibniz-Newton calculus priority dispute
Have a great day!
Reading: Chapter 2