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Calculus

  1. Some useful derivatives:

    1. d (c xn)/dx = c n xn - 1
    2. d (sin x)/dx = cos x
      d (cos x)/dx = - sin x
    3. d (ef(x))/dx = ef(x) df/dx
      d (ln x)/dx = 1/x
    4. d (f(x) + g(x))/dx = df/dx + dg/dx
    5. d (f(x) g(x))/dx = (df/dx) g(x) + f(x) (dg/dx)
    6. d f(g(x))/dx = df(g)/dg * dg(x)/dx
  2. d (∫ f(x) dx)/dx = f(x)

    ∫ df(x)/dx dx = f(x)

  3. Let f and g be functions related by an equation. Then the fractional change in g
    Δ g / g = (1 / g) Δ f / (df/dg).
  4. Any quantity f defined by integrating over contributions from infinitesimal pieces is additive:
    ftotal = f1 + f2.
  5. If u is a function only of v, (du/dv) * dv = du.
  6. Some useful integrals:

    1. ∫ √ (x2 + a) dx = (x √ (x2 + a) + a ln (x + √ (x2 + a))) / 2
    2. ∫ dx / √ (x2 + a) = ln (x + √ (x2 + a))
    3. ∫ dx / √ (x2 + a)3 = x / (a √ (x2 + a))
    4. ∫ cos2 (a x) dx = ∫ sin2 (a x) dx = x / 2 + sin (2 a x) / (4 a)
  7. To integrate around a circle of radius r centered at the origin, use
    dx = -r sin θ dθ and
    dy = r cos θ dθ
  8. To solve the differential equation dx/dt = f(x) g(t), use separation of variables:
    dx / f(x) = g(t) dt
    and integrate the left side with respect to x and the right side with respect to t. Be sure to add a constant of integration.
  9. We will be interested in the following types of quantities:

    1. infinity
    2. finite numbers
    3. infinitesimal numbers
    4. zero


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©2010, Kenneth R. Koehler. All Rights Reserved. This document may be freely reproduced provided that this copyright notice is included.

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