1. CALCULUS REVIEW BME 3510 C PO-WEI CHEN 8/19/2014

2. Po-Wei Chen Your TA pchen41@gatech.edu Regular office hour: TBA soon First week office hour: By appointment Tuesday before/after class Monday before class

3. Calculus Engineering mathematics (ODEs) Numerical methods Problems in real world

4. What is calculus Mathematical study of change - Dynamics Differential and integral calculus Prerequisites: Concepts of function and limit

5. Function Relation between inputs and outputs Domain: XRange: Y A B C D E F G Domain: X1Range: Y1 A B C D E F G Mathematical representation of function

6. Function Important properties of functions Domain and Range Boundedness Continuity Extrema (local & absolute) Asymptotic behavior

7. X Y Range Domain Bounded function Unbounded function Are these two functions well defined?

8. Global maximum Global minimum X Y Local minima Local maxima Is this function well defined?

9. X Y Does Y approach to some value when X goes to infinity? Asymptotic behavior Is this function well defined?

10. X Y Is this function in domain 0<X<a well defined? Is this function in domain 0<X<a well behaved?

11. Linear function X Y

12. Logarithmic & exponential function

13. Power function

14. Power function (cont.)

15. Trigonometric function

16. Differential Calculus 1 st order derivatives Slope = 1 Why? Slope?

17. Differential Calculus Slope of this straight line is an estimation of the slope at x = 80, any better idea?

18. Differential Calculus Slope of this straight line is an estimation of the slope at x = 80, any better idea?

19. Derivative of a function is called the derivative of at. We write: “The derivative of f with respect to x is …”

20. Example Shortcut

21. Table of derivatives More shortcuts http://www.globalspec.com/reference/62245/203279/chapter-3-calculus-and-differential-equations

22. Chain Rule Shortcut:Chain rule: By definition: http://mypaper.pchome.com.tw/show/article/gb615/A1267631488

23. Differentiability vs Continuity Not differentiable Not continuous Not differentiable Continuous Differentiable Continuous Differentiability includes continuity, but continuity does not include differentiability

24. Partial Derivatives Partial derivative of a function with more than one variable can be defined as its derivative with respect to one of the variables Total derivative

25. Approximation What is approximation? 0.98 is an approximation of 0.985324376056 X Y Red line is an approximation of blue curve around (X,Y)

26. Taylor series A representation of a function* Infinite terms calculated from derivatives of this function at certain point* (operation point) Taylor approximation * Requirement: The function is infinitely differentiable in the operation point http://en.wikipedia.org/wiki/Taylor_series Infinitely differentiable?

27. Taylor approximation example Find the Taylor approximation of (0,1)

28. 1 st order approximation

29. 2 rd order approximation

30. 4 th order approximation

31. 6 th order approximation

32. Ordinary Differential Equations General definition of ordinary differential equations (ODEs) Why ODEs, or why differential equations

33. Differential Equations Differential calculusIntegral calculus

34. Differential Equations Amount flowing into the tank/unit time Amount flowing out of the tank/unit time Directly solving ODEs Numerical methods

35. Differential Equations Integral calculus Known

36. Solving ODEs Analytical solutions

37. Solving ODEs Analytical solutions Rare in the real world Numerical solutions needed!

38. Solving ODEs Numerical solutions Euler method X Y Initial condition First solution Second solution Numerical vs analytical solutions Errors in Euler method Local error Global error Rounding error

39. Solving ODEs Numerical solutions Euler method X Y Numerical vs analytical solutions Errors in Euler method Local error – Error occurred in single step Global error – Accumulated error in certain x Rounding error - # digits allowed after the decimal points

40. Solving ODEs Numerical methods Runge-Kutta method Euler method RK2 RK works better than Euler method Matlab ODE45 uses RK4

41. Differential Equations Differential calculusIntegral calculus This graph is incomplete…

42. Differential Equations Differential calculusIntegral calculus Numerical methods (Numerical integration) Approx.

43. QUESTIONS?