Let f(x) = \sqrt[3] x. The equation of the tangent line to f(x) at x = 125 is y = . Using this, we find our approximation for \sqrt[3] {125.4} is
df = √3 dx
so, plug in dx=.4 to see how much f has increased
it says that it's wrong
so, what did you plug in? What was your final answer?
1.2
To find the approximation for the cube root of 125.4 using the given tangent line, let's start by finding the equation of the tangent line to f(x) = \sqrt[3] x at x = 125.
To find the equation of the tangent line, we need to calculate the slope of the tangent line at x = 125. This can be done using the derivative of the function f(x).
The derivative of f(x) = \sqrt[3] x can be found using the power rule for differentiation. Since the cube root function can also be written as x^(1/3), its derivative is (1/3)x^(-2/3).
Now, let's find the slope of the tangent line at x = 125. Plugging x = 125 into the derivative, we get:
f'(125) = (1/3)(125)^(-2/3)
Simplifying this expression, we get:
f'(125) = (1/3)(1/125^(2/3)) = 1/3 * 1/(5^2) = 1/3 * 1/25 = 1/75
So, the slope of the tangent line at x = 125 is 1/75.
Now that we have the slope of the tangent line, we can find the equation of the tangent line using the point-slope form. The point-slope form of a line is given by:
y - y1 = m(x - x1)
where (x1, y1) is a point on the line and m is the slope.
Since the tangent line passes through the point (125, f(125)), we can substitute these values into the point-slope equation:
y - f(125) = (1/75)(x - 125)
Since f(x) = \sqrt[3] x, we can substitute in this value as well:
y - (\sqrt[3] 125) = (1/75)(x - 125)
Simplifying the equation, we have:
y - 5 = (1/75)(x - 125)
To approximate the value of \sqrt[3] {125.4} using the tangent line, we substitute x = 125.4 into the equation of the tangent line:
y - 5 = (1/75)(125.4 - 125)
Simplifying this expression, we find:
y - 5 = (1/75)(0.4) = 0.4/75 = 4/750 = 2/375
Adding 5 to both sides of the equation, we have:
y = 5 + 2/375 = 1875/375 + 2/375 = 1877/375
Therefore, our approximation for \sqrt[3] {125.4} using the tangent line is y = 1877/375.