Title: Calculus Review
1Calculus Review
2Slope
- Slope rise/run
- Dy/Dx
- y2 y1/x2 x1
- Order of points 1 and 2 not critical
- Points may lie in any quadrant slope will work
out
- Leibniz notation for derivative based on Dy/Dx
the derivative is written dy/dx
3Exponents
4Derivative of a line
- y mx b
- slope m and y axis intercept b
- derivative of y axn b with respect to x
- dy/dx a n xn1
- Because b is a constant think of it as bx0
its derivative is 0b1 0
- For a straight line a m and n 1 so
- dy/dx m 1 x0 or because x0 1
- dy/dx m
5Derivative of a polynomial
- In differential Calculus we consider the slopes
of curves rather than straight lines
- For polynomial y axn bxp cxq
- derivative with respect to x is
- dy/dx a n xn1 b p xp1 c q xq1
6Example
y axn bxp cxq
dy/dx a n xn1 b p xp1 c q xq1
7Numerical Derivatives
8Derivative of Sine and Cosine
- sin0 0
- period of both sine and cosine is 2p
- dsinx/dx cosx
- dcosx/dx sinx
9Partial Derivatives
- Functions of more than one variable
- Example hxy x4 y3 xy
10Partial Derivatives
- Partial derivative of h with respect to x at a y
location y0
- Notation dh/dxyy0
- Treat ys as constants
- If these constants stand alone they drop out of
the result
- If they are in multiplicative terms involving x
they are retained as constants
11Partial Derivatives
- Example
- hxy x4 y3 xy
- dh/dxyy0 4x3 y0
12WHY?
13Gradients
- del C or grad C
- Diffusion Ficks 1st Law
14Numerical Derivatives
- slope between points
- MATLAB
- c[]
- [dcdxdcdy]gradientc
- contour[120][120]c
- hold
- quiver[120][120]dcdxdcdy
15Mathematica
16Mathematica
