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

Po-Wei Chen Your TA Regular office hour: TBA soon First week office hour: By appointment Tuesday before/after class Monday before class

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

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

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

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

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

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

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

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

Linear function X Y

Logarithmic & exponential function

Power function

Power function (cont.)

Trigonometric function

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

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

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

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

Example Shortcut

Table of derivatives More shortcuts

Chain Rule Shortcut:Chain rule: By definition:

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

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

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

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 Infinitely differentiable?

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

1 st order approximation

2 rd order approximation

4 th order approximation

6 th order approximation

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

Differential Equations Differential calculusIntegral calculus

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

Differential Equations Integral calculus Known

Solving ODEs Analytical solutions

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

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

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

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

Differential Equations Differential calculusIntegral calculus This graph is incomplete…

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

QUESTIONS?