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

2. Overview of Calculus Derivatives Indefinite integrals Definite integrals

3. 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

4. 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

5. Derivatives

6. Differentiation techniques

7. Derivatives

8. Applications of derivatives Maxima and minima Differentials area of a ring volume of a spherical shell Taylor’s series

9. Indefinite integral (anti-derivative) A function F is an “anti-derivative” or an indefinite integral of the function f if Also a machine: you plug in function f(x) and get function F(x)

10. Indefinite integral (anti-derivative)

11. Integrals of elementary functions

12. Definite integral

13. F is any indefinite integral of f(x) (antiderivative) Indefinite integral is a function, definite integral is a number (unless integration limits are variables) The fundamental theorem of calculus (Leibniz)

14. “Proof” of the fundamental theorem of calculus

15. Example Given: Solve for x(t) using indefinite integral:

16. Given: Solve for x(t) using definite integral Using the fundamental theorem of calculus, On the other hand, since Therefore, or

17. Integration techniques Change of variable Integration by parts

18. Gottfried Leibniz These are Leibniz’ notations: Integral sign as an elongated S from “Summa” and d as a differential (infinitely small increment). 1646-1716

19. Leibniz-Newton calculus priority dispute

20. “Moscow Papirus” (~ 1800 BC), 18 feet long Problem 14: Volume of the truncated pyramid. The first documented use of calculus?

21. 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 “Most beautiful formula ever” “the beam equation”: a cornerstone of mechanical engineering