Calculus Differentiation: Exploring sin(x) Function Graphically

1. Calculus Differentiation

2. Calculus Differentiation dy/dx = y

3. Calculus Differentiation dy/dx = y y = ex

4. Calculus Differentiation dy/dx = y y = ex eix = cosx + isinx

5. dsinx/dx = cosx

6. dsinx/dx = cosx and dcosx/dx = - sinx

7. dsinx/dx = cosx and dcosx/dx = - sinx thus d2sinx/dx2 = -sinx

8. Problem 3 Plot on graph paper the function y = sinx from x = 0 to x = 360o y 0 x -y 0 15 30 45 60 75 900 ……………… 3600x

9. Problem 3 Plot on graph paper the function y = sinx from x = 0 to x = 360o y 0 x -y 0 15 30 45 60 75 900 ……………… 3600x

11. The simplest operation in differential calculus

14. When you see e …….think WAVES ex at: thestewscope.wordpress.com/2009/07/

15. or dy/dx = ay or d2y/dx2 = ay

16. at: zaksiddons.wordpress.com/.../

17. dxn/dx = nxn-1 http://en.wikipedia.org/wiki/Taylor_series

19. at: www.windows.ucar.edu/.../tornado/waves.html

20. e at: thestewscope.wordpress.com/2009/07/

21. The Taylor series for the exponential function ex at a = 0 is • The above expansion holds because the derivative of ex with respect to x is also ex and e0 equals 1. This leaves the terms (x − 0)n in the numerator and n! in the denominator for each term in the infinite sum. dxn/dx = nxn-1 http://en.wikipedia.org/wiki/Taylor_series

22. Leibniz notation Main article: Leibniz's notation A common notation, introduced by Leibniz, for the derivative in the example above is In an approach based on limits, the symbol dy/dx is to be interpreted not as the quotient of two numbers but as a shorthand for the limit computed above. Leibniz, however, did intend it to represent the quotient of two infinitesimally small numbers, dy being the infinitesimally small change in y caused by an infinitesimally small change dx applied to x. We can also think of d/dx as a differentiation operator, which takes a function as an input and gives another function, the derivative, as the output. For example: In this usage, the dx in the denominator is read as "with respect to x". Even when calculus is developed using limits rather than infinitesimals, it is common to manipulate symbols like dx and dy as if they were real number

23. Main article: Leibniz's notation • A common notation, introduced by Leibniz, for the derivative in the example above is • In an approach based on limits, the symbol dy/dx is to be interpreted not as the quotient of two numbers but as a shorthand for the limit computed above. Leibniz, however, did intend it to represent the quotient of two infinitesimally small numbers, dy being the infinitesimally small change in y caused by an infinitesimally small change dx applied to x. We can also think of d/dx as a differentiation operator, which takes a function as an input and gives another function, the derivative, as the output. For example:

24. Problem 3 Plot on graph paper the function y = sinx from x = 0 to x = 360o y 0 x -y 0 15 30 45 60 75 900 ……………… 3600x

25. Problem 3 Plot on graph paper the function y = sinx from x = 0 to x = 360o y 0 x -y 0 15 30 45 60 75 900 ……………… 3600x

29. Symbols E = Electric field ρ = charge density i = electric current B = Magnetic field εo = permittivity J = current density D = Electric displacement μo = permeability c = speed of light H = Magnetic field strength M = Magnetization P = Polarization