Physics 218: Mechanics Instructor: Dr. Tatiana Erukhimova Lecture 2

Derivatives Indefinite integrals Definite integrals Examples Overview of Today’s Class

Quiz If f(x)=2x 3 -5 what is the derivative of f(x) with respect to x? 6x 8x 2 I don’t know how to start 6x 2

1. If f(x)=2x 3 evaluate 2x 4 I don’t know how to start 0.5x 4 +Const 2x 2 /3

1. If f(x)=2x evaluate 2x 3 /3 I don’t know how to start 5 x 2 +Const

Derivatives A derivative of a function at a point is a slope of a tangent of this function at this point.

Derivatives

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

Derivative is the rate at which something is changing Velocity: rate at which position changes with time Acceleration: rate at which velocity changes with time Force: rate at which potential energy changes with position

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

Quiz If what is the derivative of x(t) with respect to t? If f(x)=2x 3 +5x what is the derivative of f(x) with respect to x?

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

Indefinite integral (anti-derivative)

n – integer except n= -1

Lev Landau 1962, Nobel Prize “An integral without dx is like a man without pants”

Definite integral F is indefinite integral

Definite integral

Integrals Indefinite integral: n – any number except -1 Definite integral:

Gottfried Leibniz 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