Advertisements
Advertisements
Question
Show that the percentage error in the nth root of a number is approximately `1/"n"` times the percentage error in the number
Advertisements
Solution
Let x be the number
Let y = `x^(1/"n")`
log y = `1/"n" log x`
Taking differentiate on both sides, we have
`1/y "d"y = 1/"n" xx 1/x "d"x`
`(Deltay)/y = 1/y "d"y`
= `1/"n" xx 1/x "d"x`
`(Deltay)/y xx 100 = 1/"n"(("d"x)/x xx 100)`
`1/"n"` times the percentage error in the number.
APPEARS IN
RELATED QUESTIONS
Let f(x) = `root(3)(x)`. Find the linear approximation at x = 27. Use the linear approximation to approximate `root(3)(27.2)`
Use the linear approximation to find approximate values of `root(3)(26)`
Find a linear approximation for the following functions at the indicated points.
f(x) = x3 – 5x + 12, x0 = 2
Find a linear approximation for the following functions at the indicated points.
h(x) = `x/(x + 1), x_0` = 1
Find the differential dy for the following functions:
y = `(1 - 2x)^3/(3 - 4x)`
Find df for f(x) = x2 + 3x and evaluate it for x = 3 and dx = 0.02
Find Δf and df for the function f for the indicated values of x, Δx and compare:
f(x) = x3 – 2x2, x = 2, Δx = dx = 0.5
Assuming log10 e = 0.4343, find an approximate value of Iog10 1003
The trunk of a tree has a diameter of 30 cm. During the following year, the circumference grew 6 cm. Approximately how much did the tree diameter grow?
The trunk of a tree has a diameter of 30 cm. During the following year, the circumference grew 6 cm. What is the percentage increase in the area of the cross-section of the tree?
In a newly developed city, it is estimated that the voting population (in thousands) will increase according to V(t) = 30 + 12t2 – t3, 0 ≤ t ≤ 8 where t is the time in years. Find the approximate change in voters for the time change from 4 to `4 1/6` years
The relation between the number of words y a person learns in x hours is given by y = `sqrt(x), 0 ≤ x ≤ 9`. What is the approximate number of words learned when x changes from 1 to 1.1 hours?
A circular plate expands uniformly under the influence of heat. If its radius increases from 10.5 cm to 10.75 cm, then find an approximate change in the area and the approximate percentage change in the area
Choose the correct alternative:
A circular template has a radius of 10 cm. The measurement of the radius has an approximate error of 0.02 cm. Then the percentage error in the calculating the area of this template is
Choose the correct alternative:
If f(x) = `x/(x + 1)`, then its differential is given by
