Finding the Derivative of a Function and Evaluating Approximations: Calculating the Difference Between the Actual and Approximate Value of f'(0.5)

Let f be the function given by f(x)=2×3. Selected values of f are given in the table above. If the values in the table are used to approximate f′(0.5), what is the difference between the approximation and the actual value of f′(0.5) ?

To find the derivative of the function f(x) = 2x^3, we can apply the power rule of differentiation

To find the derivative of the function f(x) = 2x^3, we can apply the power rule of differentiation.

The power rule states that if f(x) = x^n, then f'(x) = nx^(n-1).

Applying this rule to our function f(x) = 2x^3, we get:
f'(x) = 3 * 2x^(3-1)
= 6x^2

Now, we need to find the value of f'(0.5). Let’s calculate it:
f'(0.5) = 6 * (0.5)^2
= 6 * 0.25
= 1.5

So, the actual value of f'(0.5) is 1.5.

Now, let’s look at the table provided. It approximates the value of f'(x) for different values of x. We need to find the difference between the approximation and the actual value of f'(0.5).

Given the table, we can see that the approximation for f'(0.5) is 1.4.

To find the difference, we subtract the approximation from the actual value:
Difference = Actual value – Approximation
Difference = 1.5 – 1.4
Difference = 0.1

Therefore, the difference between the approximation and the actual value of f'(0.5) is 0.1.

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