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Mathematics Dictionary
Dr. K. G. Shih
Calculus Index
A | B | C | D | E | F | G | H | I | J | K | L | M | N | O | P |
Q | R | S | T | U | V | W | X | Y | Z |
Keywords


C01. A

  • 08 01 Anti-differentiation
  • 17 00 Area : Integral method
  • 01 09 Area : Trapzoid method find area bounded by y = F(x) and x-axis
  • 09 00 Arc length : Integral method
  • 04 02 Arccos(x) : Derivative is -1/Sqr(1 - x^2)
  • 04 01 Arcsin(x) : Derivative is +1/Sqr(1 - x^2)
  • 04 03 Arctan(x) : Derivative is 1/(1 + x^2)

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C02. B

  • 01 08 Binomial theory
  • 18 00 Binomial theory : Find series

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C03. C

  • 02 07 Chain rule of derivative
  • 09 02 Circle : C = 2*pi*r
  • 02 11 Curve sketch instructions

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Q04. D
  • 02 01 Derivative : Definition and differentiation rules
  • 05 01 Derivative : Exponential family functions (e.g. exp(x), sinh(x), etc)
  • 04 01 Derivative : Inverse Trigonometric functions
    • 03 01 d/dx(arcsin(x)) = +1/Sqr(1 - x^2)
    • 03 02 d/dx(arccos(x)) = -1/Sqr(1 - x^2)
    • 03 03 d/dx(arctan(x)) = +1/(1 + x^2)
    • 03 04 d/dx(arccsc(x)) = -1/(x*Sqr(x^2 -1))
    • 03 05 d/dx(arcsec(x)) = +1/(x*Sqr(x^2 -1))
    • 03 06 d/dx(arccot(x)) = -1/(1 + x^2)
  • 06 01 Derivative : Logarithmic family functions (e.g. ln(x), arcsinh(x), etc)
  • 03 01 Derivative : Trigonometric functions
    • 03 01 d/dx(sin(x)) = +cos(x)
    • 03 02 d/dx(cos(x)) = -sin(x)
    • 03 03 d/dx(tan(x)) = +sec(x)^2
    • 03 04 d/dx(csc(x)) = -csc(x)*cot(x)
    • 03 05 d/dx(sec(x)) = +sec(x)*tan(x)
    • 03 06 d/dx(cot(x)) = -csc(x)^2

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Q05. E
  • 01 03 Exponent value e : Lim[(1 + x)^(1/x)] = 1 as x goes to zero
  • 01 04 Exponent value e : Lim[(1 + 1/x)^(x)] = 1 as x goes to infinite
  • 05 01 Expenont : Derivative of e^x = e^x

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Q06. F

  • 02 03 First derivative and curve : Critical points and slope

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Q07. G


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Q08. H
  • 05 02 Hyperbolic Function : d/x(sinh(x))
  • 05 03 Hyperbolic Function : d/x(cosh(x))
  • 05 04 Hyperbolic Function : d/x(tanh(x))
  • 05 05 Hyperbolic Function : d/x(csch(x))
  • 05 06 Hyperbolic Function : d/x(sech(x))
  • 05 07 Hyperbolic Function : d/x(coth(x))
  • 13 00 Hyperbolic Function : Integrals
  • 14 00 Hyperbolic Function : Integrals of inverse functions

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C09. I

  • 08 01 Integral by anti-derivative
  • 11 01 Integral by part
  • 08 05 Integral by substitution
  • 16 01 Integral of 1/(1 + x^2)
  • 16 04 Integral of 1/(a*x^2 + b*x + c) for b^2 - 4*a*c EQ 0
  • 16 05 Integral of 1/(a*x^2 + b*x + c) for b^2 - 4*a*c LT 0
  • 16 06 Integral of 1/(a*x^2 + b*x + c) for b^2 - 4*a*c GT 0
  • 10 01 Integral of sin(x)
  • 10 07 Integral of sin(x)^2
  • 10 13 Integral of (sin(x)^n)*cos(x)
  • 10 14 Integral of (cos(x)^n)*sin(x)
  • 18 00 Integral method to find series
  • 06 02 Inverse hyperbolic Function : d/x(arcsinh(x))
  • 06 03 Inverse hyperbolic Function : d/x(arccosh(x))
  • 06 04 Inverse hyperbolic Function : d/x(arctanh(x))
  • 06 05 Inverse hyperbolic Function : d/x(arccsch(x))
  • 06 06 Inverse hyperbolic Function : d/x(arcsech(x))
  • 06 07 Inverse hyperbolic Function : d/x(arccoth(x))
  • 04 01 Inverse trigonometric Function : d/x(arcsin(x))
  • 04 02 Inverse trigonometric Function : d/x(arccos(x))
  • 04 03 Inverse trigonometric Function : d/x(arctan(x))
  • 04 04 Inverse trigonometric Function : d/x(arccsc(x))
  • 04 05 Inverse trigonometric Function : d/x(arcsec(x))
  • 04 06 Inverse trigonometric Function : d/x(arccot(x))

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C10. J


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C11. K


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C12. L

  • 09 01 Length of arc : Formula
  • 09 02 Length of arc : Example
  • 01 07 Lim[cos(x)/x] = Infinite as x goes to zero
  • 01 02 Lim[sin(x)/x] = 1 as x goes to zero
  • 01 06 Lim[tan(x)/x] = 1 as x goes to zero
  • 01 04 Lim[(1 + 1/x)^(x)] = e as x goes to infinite
  • 01 03 Lim[(1 + x)^(1/x)] = e as x goes to zero
  • 01 01 Lim[(f(x+h)-F(x))/h] = derivative when h goes to zero
  • 06 01 Logarithm : Derivative of ln(x) = 1/x
  • 06 01 Logarithm : Derivative of log10(x) = 1/(x*ln(10))
  • 11 02 Logarithm : Integral ∫ln(x)dx

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C13. M

  • 02 04 Maximum point on curve : y'(x) = 0 and y"(x) = positive
  • 02 04 Minimum point on curve : y'(x) = 0 and y"(x) = negative

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C14. N


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C15. O


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Q16. P

  • 07 01 Parametric equation : signs of y' and y" in x = sec(t) and y = tan(t)
  • 07 02 Parametric equation : signs of y' and y" in x = tan(t) and y = sec(t)
  • 15 00 Partial fractions : Integrals
  • 02 02 Power rule : d/dx(x^n) = n*x^(n-1)
  • 02 05 Product rule : d/dx(F(x)*G(x)) = F'(x)*G(x) + F(x)*G'(x)

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C17. Q

  • 02 06 Quotation rule : d/dx(F(x)/G(x)) = (F'(x)*G(x) - F(x)*G'(x))/(G(x)^2)

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C18. R

  • 16 00 Rational function : Integrals
  • 27 02 Rational function : y = (x - 1)/(x + 1)
  • 27 03 Rational function : y = (x^2 - 1)/(x^2 + 1)
  • 27 04 Rational function : y = (x^3 - 1)/(x^3 + 1)
  • 27 05 Rational function : y = (x^3 + 1)/(x^3 - 1)
  • 27 06 Rational function : y = x + 1/x
  • 27 07 Rational function : y = x^2 + 1/x
  • 27 08 Rational function : y = x^3 + 1/x
  • 27 09 Rational function : y = ((x - 1)^2)/(2*x)
  • 27 10 Rational function : y = ((x - 1)^3)/(2*x)
  • 27 11 Rational function : y = ((x - 1)^4)/(2*x)
  • 27 17 Rational function : y = (x^3)/(x^2 - 1)
  • 27 18 Rational function : y = (x^4)/(x^2 - 1)
  • 27 19 Rational function : y = (2*x^2)/(x^2 - 1)
  • 07 10 R = cos(A) : Slope of tangent to curve

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C19. S

  • 02 04 Second derivative and curve : Critical points and concavity
  • 18 10 Series of arcsin(x)
  • 18 01 Series of arctan(x)
  • 18 04 Series of e^x
  • 18 02 Series of ln(1+x)
  • 18 02 Series of ln(1-x)
  • 18 08 Series of sin(x)
  • 18 06 Series of sinh(x)
  • 18 00 Series : Binomial theory
  • 18 00 Series : Taylor's formula
  • 07 09 Spiral : Property of R = e^A
  • 02 07 Square root of x : d/dx(Sqr(x)) = 1/(2*Sqr(x))
  • 08 05 Substitution rule : Integral
  • 01 08 Summation

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C20. T

  • 18 00 Taylor method to find series of functions
  • 01 09 Trapzoid method find area bounded by y = F(x) and x-axis
  • 03 00 Trigonometry
    • 03 02 d/x(sin(x))
    • 03 03 d/x(cos(x))
    • 03 04 d/x(tan(x))
    • 05 05 d/x(csc(x))
    • 03 06 d/x(sec(x))
    • 03 07 d/x(cot(x))
  • 03 01 Trigonometry : Derivative of inverse functions
  • 10 00 Trigonometry : integral
    • 10 01 ∫sin(x)dx
    • 10 02 ∫cos(x)dx
    • 10 03 ∫tan(x)dx
    • 10 04 ∫csc(x)dx
    • 10 05 ∫sec(x)dx
    • 10 06 ∫cot(x)dx
  • 12 00 Trigonometry : integral of inverse functions
    • 12 01 ∫arcsin(x)dx
    • 12 02 ∫arccos(x)dx
    • 12 03 ∫arctan(x)dx
    • 12 04 ∫arccsc(x)dx
    • 12 05 ∫arcsec(x)dx
    • 12 06 ∫arccot(x)dx

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C21. U


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C22. V

  • 17 00 Volume : Integral method

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C23. W


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C24. X


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C25. Y

  • 27 06 y = x + 1/x
  • 27 07 y = x^2 + 1/x
  • 27 08 y = x^3 + 1/x
  • 27 02 y = (x - 1)/(x + 1)
  • 27 03 y = (x^2 - 1)/(x^2 + 1)
  • 27 04 y = (x^3 - 1)/(x^3 + 1)
  • 27 05 y = (x^3 + 1)/(x^3 - 1)
  • 27 17 y = (x^3)/(x^2 - 1)
  • 27 18 y = (x^4)/(x^2 - 1)
  • 27 19 y = (2*x^2)/(x^2 - 1)
  • 27 09 y = ((x - 1)^2)/(2*x)
  • 27 10 y = ((x - 1)^3)/(2*x)
  • 27 11 y = ((x - 1)^4)/(2*x)

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C26. Z


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