To estimate the value of (27.07)^(2/3) using linear approximation, we'll start by finding the equation of the tangent line to the curve at a nearby point.
Let's start by finding a nearby point that we can use as a starting point for our linear approximation. One possibility is to use 27, which is the closest perfect cube to 27.07. The cube root of 27 is 3, so we can use this point (27, 3) as our starting point.
Next, we need to find the slope of the tangent line at this point. We can do this by finding the derivative of the function f(x) = x^(2/3) and evaluating it at x = 27.
The derivative of f(x) = x^(2/3) can be found using the power rule. We have:
f'(x) = (2/3)x^(-1/3)
Evaluating this at x = 27:
f'(27) = (2/3)(27)^(-1/3) = (2/3)(1/3) = 2/9
Now we have the slope of the tangent line at the point (27, 3), which is 2/9.
Finally, we can use the point-slope form of a linear equation to write the equation of the tangent line:
y - 3 = (2/9)(x - 27)
Now, we can use this tangent line equation to estimate the value of (27.07)^(2/3).
Plug in x = 27.07 into the equation and solve for y:
y - 3 = (2/9)(27.07 - 27)
y - 3 = (2/9)(0.07)
y - 3 = 0.0156
Adding 3 to both sides, we get:
y = 3.0156
Therefore, using linear approximation, we can estimate that (27.07)^(2/3) is approximately 3.0156.