Calculus Review

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Calculus Review
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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

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Exponents

• x0 1

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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

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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)

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Example
y axn bxp cxq
dy/dx a n x(n-1) b p x(p-1) c q x(q-1)

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Numerical Derivatives

• slope between points

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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)

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Partial Derivatives

• Functions of more than one variable
• Example h(x,y) x4 y3 xy

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Partial 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

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Partial Derivatives

• Example
• h(x,y) x4 y3 xy
• dh/dxyy0 4x3 y0

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WHY?
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Gradients

• del C (or grad C)
• Diffusion (Ficks 1st Law)

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Numerical Derivatives

• slope between points
• MATLAB
• c
• dcdx,dcdygradient(c)
• contour(120,120,c)
• hold
• quiver(120,120,-dcdx,-dcdy)

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Mathematica
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Mathematica