Kategóriák: Minden - exponential - differentiation - functions - integration

a David Kedrowski 14 éve

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MAT.126 5.5

The text explains the differentiation and integration of functions with bases other than e, including the formulas and rules for handling such cases. It covers the differentiation of exponential functions where the base is a positive real number and not equal to one, as well as the integration of expressions involving these bases.

MAT.126 5.5

MAT.126 5.5 Bases Other Than e and Applications

Differentiation and Integration

A comparison of 4 rules for differentiation

Constant Rule

d/dx[e^e] = 0

Power Rule

d/dx[x^e] = e x^{e-1}

Exponential Rule

d/dx[e^x] = e^x

Logarithmic differentiation

d/dx[x^x] = x^x (1 + ln x)

Integration

When confronted with an integral of the form

S a^x dx

there are two choices.

One is to convert the exponential expression with base a to an equivalent exponential expression with base e. That is, consider

S e^{(ln a)x} dx

remembering that ln a is a constant.

The second option is to use the following integration formula,

S a^x dx = (1/ln a) a^x + C

Derivatives for Bases Other Than e

Let a be a positive real number (a<>1) and let u be a differentiable function of x.

d


  • ---[a^x] = (ln a) a^x
  • dx

    d du


  • ---[a^u] = (ln a) a^u ----
  • dx dx

    d 1

  • ---[log_a x] = --------
  • dx (ln a) x

    d 1 du

  • ---[log_a u] = -------- ---
  • dx (ln a) u dx

    Applications of Exponential Functions

    Logisitic Growth
    Continuously Compounded Interest
    Compound Interest

    Bases Other than e

    Common Logarithmic Function

    The logarithm with base 10.

    Inverse Function Properties

  • y = a^x if and only if x = log_a y

  • a^{log_a x} = x, for x > 0

  • log_a a^x = x, for all x
  • Logarithmic Properties

  • log_a 1 = 0

  • log_a xy = log_a x + log_a y

  • log_a x/y = log_a x - log_a y

  • log_a x^n = n log_a x
  • Definition of Logarithmic Function to Base a

    If a is a positive real number (a<>1) and x is any positive real number, then the logarithmic function to the base a is denoted by log_a x and is defined as

    log_a x = (1/ln a) ln x

    Laws of Exponents

  • a^0 = 1

  • a^x a^y = a^{x+y}

  • a^x / a^y = a^{x-y}

  • (a^x)^y = a^{xy}
  • Definition of Exponential Function to Base a

    If a is a positive real number (a<>1) and x is any real number, then the exponential function to the base a is denoted by a^x and is defined by

    a^x = e^{(ln a)x}

    If a=1, then y=1^x=1 is a constant function.