Using an appropriate linear approximation approximate 26.9^(4/3).
27^(4/3) = 81
y = x^(4/3)
dy = (4/3)x^(1/3) dx
so, if dx = -0.1
dy = (4/3)(3)(-0.1) = -0.4
So, 26.9^(4/3) ≈ 80.6
In fact 26.9^(4/3) = 80.6002
pretty close, eh?
To approximate 26.9^(4/3) using linear approximation, we start by choosing a function that is easy to compute and differentiate. In this case, we can choose the function f(x) = x^(4/3), which is similar to the function we want to approximate.
Next, we need to choose a value near which we will approximate. Let's choose x = 27, which is close to 26.9.
Then, we find the equation of the tangent line to the function at x = 27. To do this, we compute the derivative of f(x) with respect to x and evaluate it at x = 27. The derivative of f(x) = x^(4/3) can be found using the power rule:
f'(x) = (4/3)x^(1/3)
Evaluating at x = 27:
f'(27) = (4/3)(27)^(1/3) = (4/3)(3) = 4
The tangent line to the function f(x) at x = 27 is given by:
y = f(27) + f'(27)(x - 27)
Substituting the values:
y = 27^(4/3) + 4(x - 27)
Simplifying:
y ≈ 27^(4/3) + 4x - 4*27
y ≈ 27^(4/3) + 4x - 108
Finally, we substitute x = 26.9 into the equation to obtain an approximation for 26.9^(4/3):
y ≈ 27^(4/3) + 4(26.9) - 108
Calculating this expression will give us the approximate value.
To approximate 26.9^(4/3) using linear approximation, we can start by finding a linear approximation equation. The general form of linear approximation is given as:
f(a + Δx) ≈ f(a) + f'(a) * Δx
In this case, let's define the function f(x) = x^(4/3), which we want to approximate at a = 26.9. To use the linear approximation, we need the first derivative f'(x).
Taking the derivative of f(x) = x^(4/3) using the power rule, we get:
f'(x) = (4/3) * x^(1/3)
Now, we can plug in the values into the linear approximation equation:
f(26.9 + Δx) ≈ f(26.9) + f'(26.9) * Δx
Calculating f(26.9):
f(26.9) = 26.9^(4/3)
Calculating f'(26.9):
f'(26.9) = (4/3) * 26.9^(1/3)
Now, we can approximate the value by choosing a small value for Δx. Let's say Δx = 0.1.
Plugging in the values:
f(26.9 + 0.1) ≈ f(26.9) + f'(26.9) * 0.1
Calculating f(26.9 + 0.1):
f(26.9 + 0.1) = (26.9 + 0.1)^(4/3)
Substituting the values and calculating the approximation, we get:
(26.9 + 0.1)^(4/3) ≈ f(26.9) + f'(26.9) * 0.1
Now, you can evaluate the expression to get the linear approximation of 26.9^(4/3).