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Introduction to Calculus: Fundamentals and Applications

Learn essential calculus concepts such as derivatives, partial derivatives, gradients, and MATLAB programming for analytical solutions and graphical representations. Understand numerical derivatives, Sine and Cosine derivatives, and explore integral calculus applications.

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Introduction to Calculus: Fundamentals and Applications

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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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