http://apod.nasa.gov/apod/ap070819.html

Overview of Calculus

• Derivatives

• Indefinite integrals

• Definite integrals

Derivative is the rate at which something is changing

Velocity: rate at which position changes with timeVdt

dx

adt

dV Acceleration: rate at which velocity changes with time

Force: rate at which potential energy changes with position

Fdx

dU

x

xfxxf

dx

xdfx

)()(lim

)(0

Derivatives

t

txttx

dt

tdxt

)()(lim

)(0

or

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

,)( 1 nn

ntdt

td

Derivatives

t

txttx

dt

tdxt

)()(lim

)(0

0)(

dt

Constd

ttdt

dcossin

ttdt

dsincos

tt

dt

d 1ln

tt eedt

d

dt

dg

dt

df

dt

gfd

)(

dt

dgf

dt

dfg

dt

fgd

)(

dt

dg

dg

df

dt

tgfd

))(((2

''

g

fggf

g

f

dt

d

Differentiation techniques

?Cfdt

d

?sin tdt

d

?xedx

d

?xxdx

d

?log10 xdx

d

?gfdx

d?

1 3

2

x

x

dx

d

Derivatives

?xadx

d

?ln xdx

d

Applications of derivatives

• Maxima and minima

• Differentialsarea of a ringvolume of a spherical shell

• Taylor’s series

Indefinite integral(anti-derivative)

A function F is an “anti-derivative” or an indefinite integral of the function f

dxxfF )(

if )(xfdx

dF

Also a machine: you plug in function f(x) and get function F(x)

Indefinite integral(anti-derivative)

)()(

xfdxdx

xdf Const

dx

xdfConstxf

dx

d )())((

Constkxn

dxkxdxxf nn

1

1

1)()(

nkxxf )(

)1

1( 1 Constkxndx

d n

)()1

1( 1 Const

dx

dkx

ndx

d n

01

1 nkxn

n nkx 1n

Integrals of elementary functions

Definite integral

Definite integral

)()()( AFBFFdxxfB

A

BA

F is any indefinite integral of f(x) (antiderivative)

11

1

1

1

1

nn

B

A

n kAn

kBn

dxkx

Indefinite integral is a function, definite integral is a number (unless integration limits are variables)

The fundamental theorem of calculus (Leibniz)

“Proof” of the fundamental theorem of calculus

3/3/)(

3/

3/)(

3/)()(

300

3

300

3000

3

ktxkttx

ktxC

Ckttxx

CktCdttvtx

Example

002 )(,)( xtxkttv Given:

Solve for x(t) using indefinite integral:

3/3/)( 30

3

0

ktktdttvt

t

002 )(,)( xtxkttv Given:

Solve for x(t) using definite integral

Using the fundamental theorem of calculus,

3/3/)( 300

3 ktxkttx

00 )()()()(0

xtxtxtxdttvt

t

On the other hand, since ,)( 2kttv

3/3/)( 30

30 ktktxtx Therefore,

or

Integration techniques

Change of variable

Integration by parts

Gottfried Leibniz

dxxfF )(

)(xfdx

dF

These are Leibniz’ notations: Integral sign as an elongated S from “Summa” and d as a differential (infinitely small increment).

1646-1716

Leibniz-Newton calculus priority dispute

“Moscow Papirus” (~ 1800 BC), 18 feet long

Problem 14: Volume of the truncated pyramid.The first documented use of calculus?

Leonhard Euler 1707-1783

“Read Euler, read Euler, he is the master of us all”Pierre-Simon Laplace

f(x), complex numbers, trigonometric and exponential functions, logarithms, power series, calculus of variations, origin of analytic number theory, origin of topology, graph theory, analytical mechanics, …• 80 volumes of papers!• Integrated Leibniz’ and Newton’s calculus• Three of the top five “most beautiful formulas” are Euler’s

01ie “Most beautiful formula ever”

“the beam equation”: a cornerstone of mechanical engineering