Chapter 6: The Calculus of Exponential and Logarithmic Functions

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fundamental theorem of calculus: derivative of an integral form

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fundamental theorem of calculus: derivative of an integral form

if g(x) = ∫xa f(t)dt where a stands for a constant, and f is continuos in the neighbourhood of a, then g’(x) = f(x)

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2

the natural logarithm function

ln x = ∫x1 1/t dt

where x is a positive number

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derivative of lnx

d/dx (lnx) = 1/x

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integral of the reciprocal function

1/u du = ln |u| + C

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the uniqueness theorem for derivatives

If: 1. f’(x) = g’(x) for all values of x in the domain, and

  1. f(a) = g(a) for one value, x = a, in the domain,

then f(x) = g(x) for all values of x in the domain

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6

the uniqueness theorem verbally

if two functions have the same derivative everywhere and they also have a point in common, then they are the same function

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logarithm properties of ln - product

ln (ab) = lna + lnb

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logarithm properties of ln - quotient

ln(a/b) = lna - lnb

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logarithm properties of ln - power

ln(ar) = r lna

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logarithm properties of ln - intercept

ln 1 = 0

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algebraic definition: logarithm

a = logbc if and only if ba = c where b > 0, b ≠ 1, and c > 0

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change-of-base property for logarithms

logbx = logax/logab

in particular: logbx = logex/logeb = lnx/lnb = 1/lnb * lnx

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13

e (functionally)

e = lim n→0 (1+n)1/n = lim n→∞ (1 + 1/n)n

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e (numerically)

e = 2.7182818284…

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indeterminate forms

0/0, 00, 1^∞, ∞/∞, 0(∞), ∞0, ∞-∞

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determinate forms

∞+∞=∞, -∞-∞=-∞, 0^∞=0, 0^-∞=∞, ∞*∞=∞

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transcendental numbers

like e or π, numbers that are not only irrational, but that are not able to be expressed using only a finite number of the operations of algebra performed on integers, it goes beyond these operations

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inverse relationship between ex and lnx

ln(ex) = x, and elnx = x

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exponential with base b

bx = exlnb

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derivative and integral of a base-b exponential function

for any positive constant b≠1 (to avoid division by zero)

d/dx (bx) = bxlnb ∫budu = bu/lnb + C

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logarithmic differentiation

  1. take ln of both sides

  2. ln of a power

  3. differentiate implicitly with respect to x

  4. solve for y’

  5. substitute for y

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22

l’hopitals rule

if f(x) = g(x)/h(x) and if lim g(x0 as x—>c = lim h(x) as x—>c = 0

then lim f(x) as x—>c = lim g’(x)/h’(x) as x—>c provided the latter limit exists

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limit-function interchange for continuous functions

for the function f(x) = g(h(x)), if h(x) has a limit, L, as x approaches c and if g is continuous at L, then lim g(h(x)) as x—>c = g(lim h(x) as x—>c)

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24

∫sinxdx

= -cosx + c

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25

∫cosxdx

= sinx + c

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∫tanxdx

= -ln|cosx| + C = ln|secx| + C

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∫cotxdx

= ln|sinx| + C = -ln|cscx| + C

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∫secxdx

= ln|secx + tanx| + C

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∫cscxdx

= -ln|cscx + cotx| + C = ln|cscx - cotx| + C

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