Estimate for x = 2, 4, 6, using the given values of f'(x) and the fact that f(0)=50 .
To estimate the values of f(x) for x = 2, 4, and 6, using the given values of f'(x) and the fact that f(0) = 50, we need to use the concept of the derivative.
The derivative of a function f(x) represents the rate of change of the function at any given point. In this case, we have the values of f'(x), which are the derivatives of the function f(x) evaluated at x = 0, 2, 4, and 6.
To estimate f(x), we can use the derivative values to approximate the change in f(x) from one point to another. This can be done using the following formula:
f(x) ≈ f(a) + (x - a) * f'(a)
where a is the initial point at which we know the function value (in this case, a = 0).
Let's calculate the estimates for x = 2, 4, and 6:
For x = 2:
f(2) ≈ f(0) + (2 - 0) * f'(0)
Using the fact that f(0) = 50, let's substitute the given value of f'(0):
f(2) ≈ 50 + (2 - 0) * f'(0)
Now substitute the value of f'(0), if provided, to get the estimate for f(2).
Similarly, you can calculate the estimates for f(4) and f(6) using the formula above and the given values of f'(4) and f'(6).