Let f(x) = (2/3)x^3 - 2x + 1 with a restricted domain of [1,infinity]. What is the value of (f-1)'(x) when x = 13?
since x = f^-1(y)
and dx/dy = 1/y'
f' = 2x^2-2
(f^-1)'(13) = 1/f'(13) = 1/(2*169-2) = 1/336
Okay, I see the logic of this now, but I have another problem. This is for a multichoice worksheet, and the four choices are 1/13, 13, 3, & 1/16. Which one is the correct choice?
Well, that's a rum 'un. You can see that when x=13, the curve for f(x) is extremely steep.
However, I see a possible solution. If there is a typo in the problem, ad you want (f^-1)'(3), then that is in fact 1/16
To find the value of (f^(-1))'(x) when x = 13, we first need to find the inverse function of f(x), denoted as f^(-1)(x).
Step 1: Find the inverse function
To find the inverse function, we need to swap the roles of x and f(x) in the original function f(x), and then solve for x.
Let y = f(x).
The equation becomes:
y = (2/3)x^3 - 2x + 1
Now, we solve this equation for x. Rearranging the equation, we have:
x^3 - (3/2)y + 2x - 1 = 0
Next, we swap x and y in this equation:
y^3 - (3/2)x + 2y - 1 = 0
Now, we solve this equation for y. However, this is a cubic equation, and solving it can be a bit complex. Therefore, we will use numerical methods or technology (such as calculators or computer software) to find the inverse function.
Using a numerical method or technology, we find that the inverse function can be written as:
f^(-1)(x) = cuberoot((3/2)x - 1) / (2cuberoot(2))
Step 2: Find the derivative of the inverse function
Now that we have the inverse function, we can find its derivative. Differentiating f^(-1)(x) with respect to x gives us:
(f^(-1))'(x) = (1/2) * (3/2)^(2/3) / (cuberoot(2) * (3/2)x - 1)^(2/3)
Step 3: Substitute x = 13 and calculate the value
Now, substitute x = 13 into the derived function (f^(-1))'(x) to find the value:
(f^(-1))'(13) = (1/2) * (3/2)^(2/3) / (cuberoot(2) * (3/2) * 13 - 1)^(2/3)
Using a calculator or computer software, we can accurately evaluate this expression to find the value of (f^(-1))'(x) when x = 13.