find the numerical deriviative of the given function at the indicated point. use h=0.001. is the function differentiable at the indicated point?
4x- x^2, x=0
Looks like you are starting the concept of a derivative and are finding it by First Principles
f(0) = 0
f(0+.001) = .003999
so the slope of the secant
= (.00399 - 0)/(.001 - 0)
= 3.999
as you make your value of h get closer to zero, the slope of the secant will approach the slope of the tangent (the derivative) and it will get closer and closer to 4.
To find the numerical derivative of a function at a given point, we can use the formula for the numerical derivative:
f'(x) ≈ (f(x + h) - f(x)) / h
where h is the step size or the interval at which we are taking the derivative. In this case, it is given that h = 0.001.
The function given is f(x) = 4x - x^2, and we need to find the derivative at x = 0, which is denoted as f'(0).
To calculate f'(0), we substitute the values into our formula:
f'(0) ≈ (f(0 + 0.001) - f(0)) / 0.001
Next, we evaluate f(0.001) and f(0) by substituting x = 0.001 and x = 0, respectively, into the function:
f(0.001) = 4(0.001) - (0.001)^2 = 0.004 - 0.000001 = 0.003999
f(0) = 4(0) - (0)^2 = 0 - 0 = 0
Substituting these values back into our formula:
f'(0) ≈ (0.003999 - 0) / 0.001
Simplifying:
f'(0) ≈ 0.003999 / 0.001
f'(0) ≈ 3.999
So, the numerical derivative of the function f(x) = 4x - x^2 at x = 0 is approximately 3.999.
Now that we have the derivative, we can determine if the function is differentiable at the indicated point (x=0). In order for a function to be differentiable at a point, the derivative must exist at that point. Since we have obtained a derivative value of 3.999, it means that the function is indeed differentiable at x=0.