Logarithmic differentials Let f be a differentiable function of one or more variables that is positive on its domain. a. Show that d ( ln f ) = d f f . b. Use part (a) to explain the statement that the absolute change in ln f is approximately equal to the relative change in f . c. Let f ( x, y ) = xy , note that ln f = ln x + ln y , and show that relative changes add: that is, df/f = dx/x + dy/y. d. Let f ( x, y ) = x/ y , note that ln f = ln x = ln y , and show that relative changes subtract; that is df /f = dx / x – dy / y. e. Show that in a product of n numbers, f = x 1 x 2 … x n , the relative change in f is approximately equal to the sum of the relative changes in the variables.
Logarithmic differentials Let f be a differentiable function of one or more variables that is positive on its domain. a. Show that d ( ln f ) = d f f . b. Use part (a) to explain the statement that the absolute change in ln f is approximately equal to the relative change in f . c. Let f ( x, y ) = xy , note that ln f = ln x + ln y , and show that relative changes add: that is, df/f = dx/x + dy/y. d. Let f ( x, y ) = x/ y , note that ln f = ln x = ln y , and show that relative changes subtract; that is df /f = dx / x – dy / y. e. Show that in a product of n numbers, f = x 1 x 2 … x n , the relative change in f is approximately equal to the sum of the relative changes in the variables.
Solution Summary: The author explains the differential equation d(mathrmlnf)=df
Logarithmic differentials Let f be a differentiable function of one or more variables that is positive on its domain.
a. Show that
d
(
ln
f
)
=
d
f
f
.
b. Use part (a) to explain the statement that the absolute change in ln f is approximately equal to the relative change in f.
c. Let f(x, y) = xy, note that ln f = ln x + ln y, and show that relative changes add: that is, df/f = dx/x + dy/y.
d. Let f(x, y) = x/y, note that ln f = ln x = ln y, and show that relative changes subtract; that is df /f = dx/x – dy/y.
e. Show that in a product of n numbers, f = x1x2…xn, the relative change in f is approximately equal to the sum of the relative changes in the variables.
Dialysis The project on page 458 models the removal of
urea from the bloodstream via dialysis. Given that the
initial urea concentration, measured in mg/mL, is c (where
c > 1), the duration of dialysis required for certain condi-
tions is given by the equation
3c + V9c? – 8c
t = In
2
Calculate the derivative of t with respect to c and interpret it.
Find the first derivative ____, where a, b, c, m and n, are constants.
A clear step by step solving of this question would be much appreciated. Thank you!
University Calculus: Early Transcendentals (3rd Edition)
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