Calculus 1A, College of the Desert, Chapter 3.txt

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Calculus 1A, College of the Desert, Chapter 3.txt
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2011-06-23 18:25:14
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Chapter3 Calculus 1A
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  1. d/dx(ax) =


    d/dx(ax)= ax ln a
  2. Steps in Logarithmic Differentiation
    • 1) Take natural logarithms of both sides of an equation y = f(x) and use the laws of Logarithms to simplify.
    • 2) Differentiate implicitly with respect to x.
    • 3) Solve the resulting equation for y '.
  3. The Power Rule
    F'(x) = nxn-1
  4. e =
    • e = lim(1+x)1/x
    • x-->0
  5. e =
    • e = lim(1+1/n)n
    • x-->inf
  6. Compressibility
    • is defined by introducing a minus sign and dividing this derivitive by the volume V:
    • isothermal compressibility = B = - 1/V dV/dP
  7. V =
    V = 5.3/P
  8. Half-Life
    • broken into 3 parts:
    • A) m(t) =
    • 100e-(ln2)t/1590

    • OR
    • m(t) = 100 • 2-t/1590
    • B) mass after 1000 years
    • m(1000) = 100e-(ln2)1000/1590 = aprox 65 mg
    • C) 100e-(ln2)t/1590 = 30
    • or
    • e-(ln2)t/1590 = 0.3
  9. Continuosly compounded interest
    Ao(1+r/n)nt
  10. Hyperbolic Identities
    sinh(-x) =
    sinh(-x) = -sinh(x)
  11. Hyperbolic identities
    cosh(-x) =
    cosh(-x) = cosh(x)
  12. Hyperbolic identities
    cosh2(x)-sinh2(x)
    cosh2(x) - sinh2(x) = 1
  13. Hyperbolic identities
    1 - tanh2(x) =
    1 - tanh2(x) = sinh2(x)
  14. Hyperbolic identities
    sinh(x+y) =
    sinh x cosh y + coshx sinhy
  15. Hyperbolic identities
    cosh(x+y) =
    • cosh(x+y) =
    • cosh x cosh y + sinh x sinh y
  16. Derivitives of hyperbolic Id
    d/ex(sinh x) =
    d/dx(sinh x) = cosh x
  17. Derivitives of hyperbolic iden
    d/dx(cosh x) =
    d/dx(cosh x) = sinh x
  18. Derivitives of hyperbolic ident
    d/dx(tanh x) =
    d/dx(tanh x) = sech2 x
  19. Derivitives of hyperbolic ident
    d/dx(csch x) =
    d/dx(csch x) = -csch x coth x
  20. Derivitives of hyperbolic ident
    d/dx(sech x) =
    d/dx(sech x) = -sech x tanh x
  21. Derivitives of hyperbolic ident
    d/dx(coth x) =
    d/dx(coth x) = -csch^2 x
  22. y = sinh-1 x =
    • y = sinh-1 x <---->
    • x = sinh y
  23. y = cosh-1 (x) =
    (y = cosh-1 (x) )<==> (cosh y = x) and y >_0
  24. y = tanh-1 (x) =
    y = tanh-1 (x) <==> tanh y = x
  25. sinh-1 (x) =
    • sinh-1 (x) =
    • ln(x+ (x2+1)1/2
    • X € R
  26. cosh-1 (x) =
    • cosh-1 (x) =
    • ln(x + (x2-1)1/2)
    • X >_1
  27. tanh-1 (x) =
    • tanh-1 (x) =
    • (1/2)ln[(1+x)/(1-x)]

    -1< x < 1
  28. F(x) = cos x
    F'(x)=
    • F'(x)= -sin x
    • F''(x) = - cos x
    • F'''(x) = sin x
    • F''''(x) = cos x
    • F^5(x) = - sin x

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