Physics 218: Mechanics Instructor: Dr. Tatiana Erukhimova Lecture 2 Derivatives Indefinite integrals Definite integrals Examples Overview of Todays Class Quiz If f(x)=2x 3 -5 what is the derivative of f(x) with respect to x? 6x 8x 2 I dont know how to start 6x 2 1. If f(x)=2x 3 evaluate 2x 4 I dont know how to start 0.5x 4 +Const 2x 2 /3 1. If f(x)=2x evaluate 2x 3 /3 I dont 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