Use differentials to approximate the quantity (give answer to 4 decimal places)
4th root of 256.6.
4.00234169288
If y = x^1/4,
dy = 1/4 x^-3/4 dx
setting x = 256
dx = .6, we get
dy = 1/4 (256^-3/4)(.6)
= (1/4)(1/64)(.6)
= 0.00234
so, y ~= y+dy = 4.00234
To approximate the quantity, we can use differentials. The differential of a function f(x) is given by:
df = f'(x) * dx
In this case, we want to find the differential of the 4th root of 256.6. Let's define our function as f(x) = x^(1/4), then the differential of f(x) is:
df = (1/4) * x^(-3/4) * dx
Now, we can plug in the value x = 256.6 and dx = 0.1 into the differential to approximate the change in the quantity:
df ≈ (1/4) * (256.6)^(-3/4) * 0.1
Calculating this expression, we can approximate the change in the 4th root of 256.6.
Let me do the math for you.
To use differentials to approximate the quantity, we can consider the derivative of the function and use it to find the change in the function value (delta y) corresponding to a small change in the input variable (delta x).
Let's denote the function as f(x) = x^(1/4), where x represents our input value. We want to find the approximate value of f(x+dx) - f(x), which is the change in the function value when the input changes by a small amount dx.
Taking the derivative of f(x) with respect to x, we have:
f'(x) = (1/4) * x^(-3/4)
Now, we can use the differential approximation:
delta y = f'(x) * delta x
In our case, x = 256.6 and we want to find the approximate value of the 4th root of 256.6, so dx = 0.1 (a small change in x).
Substituting the values:
delta y = (1/4) * 256.6^(-3/4) * 0.1
Calculating this expression, we get:
delta y ≈ 0.1458
Now, to find the approximate value of f(x+dx), we adjust the input value by adding the small change:
f(x+dx) ≈ f(x) + delta y
Substituting the values, we have:
f(x+dx) ≈ f(256.6) + 0.1458
Calculating the expression, we get:
f(x+dx) ≈ 4.8178
Therefore, the approximate value of the 4th root of 256.6, rounded to four decimal places, is 4.8178.