Advertisements
Advertisements
Question
Use the linear approximation to find approximate values of `root(4)(15)`
Advertisements
Solution
`root(4)(15) = (15)^(1/4)`
f(x) = `x^(1/4), f(x0) = `(16)^(1/4)` = 2
We know that
f(x0 + Δx) = f(x0) + f’(x0) Δx
`(15)^(1/4) = 2 + 1/(4x^(1/4)) (- 1)`
= `2 + 1/(4(16)^(3/4)) (- 1)`
= `2 + 1/(4 xx 8) (- 1)`
= `2 - 1/32`
= 2 – 0.03125
`(15)^(1/4)` = 1.96875
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)`
Find a linear approximation for the following functions at the indicated points.
g(x) = `sqrt(x^2 + 9)`, x0 = – 4
The radius of a circular plate is measured as 12.65 cm instead of the actual length 12.5 cm. find the following in calculating the area of the circular plate:
Relative error
The radius of a circular plate is measured as 12.65 cm instead of the actual length 12.5 cm. find the following in calculating the area of the circular plate:
Percentage error
A sphere is made of ice having radius 10 cm. Its radius decreases from 10 cm to 9.8 cm. Find approximations for the following:
Change in the surface area
The time T, taken for a complete oscillation of a single pendulum with length l, is given by the equation T = `2pi sqrt(l/g)` where g is a constant. Find the approximate percentage error in the calculated value of T corresponding to an error of 2 percent in the value of l
Find the differential dy for the following functions:
y = `(1 - 2x)^3/(3 - 4x)`
Find the differential dy for the following functions:
y = `(3 + sin(2x))^(2/3)`
Find df for f(x) = x2 + 3x and evaluate it for x = 2 and dx = 0.1
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
Find Δf and df for the function f for the indicated values of x, Δx and compare:
f(x) = x2 + 2x + 3, x = – 0.5, Δx = dx = 0.1
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?
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
A coat of paint of thickness 0.2 cm is applied to the faces of cube whose edge is 10 cm. Use the differentials to find approximately how many cubic centimeters of paint is used to paint this cube. Also calculate the exact amount of paint used to paint this cube
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 u(x, y) = `"e"^(x^2 + y^2)`, then `(delu)/(delx)` is equal to
Choose the correct alternative:
If f(x) = `x/(x + 1)`, then its differential is given by
Choose the correct alternative:
Linear approximation for g(x) = cos x at x = `pi/2` is
