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Calculus Review

Calculus Review. GLY-4822. Slope. Slope = rise/run = D y/ D x = ( y 2 – y 1 )/( x 2 – x 1 ) Order of points 1 and 2 not critical, but keeping them together is Points may lie in any quadrant: slope will work out

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Calculus Review

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  1. Calculus Review GLY-4822

  2. Slope • Slope = rise/run • = Dy/Dx • = (y2 – y1)/(x2 – x1) • Order of points 1 and 2 not critical, but keeping them together is • 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 0bx-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: C(x,y) = x4 + y3 + xy

  10. Partial Derivatives • Partial derivative of h with respect to x at a y location y0 • Notation h/x|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: • C(x,y) = x4 + y3 + xy • C/x|y=y0 = 4x3 + y0

  12. WHY?

  13. Gradients • del h (or grad h) • Flow (Darcy’s Law):

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

  15. Basic MATLAB

  16. Matlab Programming environment Post-processer Graphics Analytical solution comparisons Use File/Preferences/Font to adjust interface font size

  17. Vectors >> a=[1 2 3 4] a = 1 2 3 4 >> a' ans = 1 2 3 4

  18. Autofilling and addressing Vectors > a=[1:0.2:3]' a = 1.0000 1.2000 1.4000 1.6000 1.8000 2.0000 2.2000 2.4000 2.6000 2.8000 3.0000 >> a(2:3) ans = 1.2000 1.4000

  19. xy Plots >> x=[1 3 6 8 10]; >> y=[0 2 1 3 1]; >> plot(x,y)

  20. Matrices >> b=[1 2 3 4;5 6 7 8] b = 1 2 3 4 5 6 7 8 >> b' ans = 1 5 2 6 3 7 4 8

  21. Matrices >> b=2.2*ones(4,4) b = 2.2000 2.2000 2.2000 2.2000 2.2000 2.2000 2.2000 2.2000 2.2000 2.2000 2.2000 2.2000 2.2000 2.2000 2.2000 2.2000

  22. Reshape >> a=[1:9] a = 1 2 3 4 5 6 7 8 9 >> bsquare=reshape(a,3,3) bsquare = 1 4 7 2 5 8 3 6 9 >>

  23. Load a = load(‘filename’); (semicolon suppresses echo)

  24. If if(1) … else … end

  25. For for i = 1:10 … end

  26. BMP Output bsq=rand(100,100); %bmp1 output e(:,:,1)=1-bsq; %r e(:,:,2)=1-bsq; %g e(:,:,3)=ones(100,100); %b imwrite(e, 'junk.bmp','bmp'); image(imread('junk.bmp')) axis('equal')

  27. Quiver (vector plots) >> scale=10; >> d=rand(100,4); >> quiver(d(:,1),d(:,2),d(:,3),d(:,4),scale)

  28. Contours h=[…]; Contour(h) Or Contour(x,y,h)

  29. Contours w/labels C=[…]; [c,d]=contour(C); clabel(c,d), colorbar

  30. Numerical Partial Derivatives slope between points MATLAB h=[]; (order assumed to be low y on top to high y on bottom!) [dhdx,dhdy]=gradient(h,spacing) contour(x,y,h) hold quiver(x,y,-dhdx,-dhdy)

  31. Gradient Function and Streamlines [dhdx,dhdy]=gradient(h); [Stream]= stream2(X,Y,U,V,STARTX,STARTY); [Stream]= stream2(-dhdx,-dhdy,[51:100],50*ones(50,1)); streamline(Stream) (This is for streamlines starting at y = 50 from x = 51 to 100 along the x axis. Different geometries will require different starting points.)

  32. Stagnation Points

  33. Integral Calculus

  34. Integral Calculus: Special Case

  35. Integral Calculus: Special Case

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