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Calculus Review

Calculus Review. Slope. Slope = rise/run = D y/ D x = (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 D y/ D x; the derivative is written dy/dx. Exponents. x 0 = 1.

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Calculus Review

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  1. Calculus Review

  2. Slope • 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

  3. Exponents • x0 = 1

  4. Derivative 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 x(n-1) • Because b is a constant -- think of it as bx0 -- its derivative is 0b-1 = 0 • For a straight line, a = m and n = 1 so • dy/dx = m 1 x(0), or because x0 = 1, • dy/dx = m

  5. Derivative 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 x(n-1) + b p x(p-1) + c q x(q-1) + …

  6. Example y = axn + bxp + cxq + … dy/dx = a n x(n-1) + b p x(p-1) + c q x(q-1) + …

  7. Numerical Derivatives • slope between points

  8. Derivative of Sine and Cosine • sin(0) = 0 • period of both sine and cosine is 2p • d(sin(x))/dx = cos(x) • d(cos(x))/dx = -sin(x)

  9. Partial Derivatives • Functions of more than one variable • Example: h(x,y) = x4 + y3 + xy

  10. Partial Derivatives • Partial derivative of h with respect to x at a y location y0 • Notation dh/dx|y=y0 • 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

  11. Partial Derivatives • Example: • h(x,y) = x4 + y3 + xy • dh/dx|y=y0 = 4x3 + y0

  12. WHY?

  13. Gradients • del C (or grad C) • Diffusion (Fick’s 1st Law):

  14. Numerical Derivatives • slope between points • MATLAB • c=[]; • [dcdx,dcdy]=gradient(c) • contour([1:20],[1:20],c) • hold • quiver([1:20],[1:20],-dcdx,-dcdy)

  15. Mathematica

  16. Mathematica

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