Given the function f(x)=4x , which of the following best represents the value of the numerical derivative f′(2) , determined using the symmetric difference quotient with h = 0.5? (1 point) Responses −1.333 − 1.333 −1.067 − 1.067 −1.010 − 1.010 −1.003 − 1.003 −1.000
To find the value of the numerical derivative f '(2) using the symmetric difference quotient with h = 0.5, we can use the formula:
f '(x) = (f(x+h) - f(x-h)) / (2h)
Substituting the values into the formula:
f '(2) = (f(2+0.5) - f(2-0.5)) / (2*0.5)
= (f(2.5) - f(1.5)) / 1
= (4(2.5) - 4(1.5)) / 1
= (10 - 6) / 1
= 4 / 1
= 4
Therefore, the correct answer is 4.
To find the value of the numerical derivative f'(2) using the symmetric difference quotient with h = 0.5, we need to use the following formula:
f'(x) ≈ (f(x + h) - f(x - h)) / (2h)
First, let's find f(2 + h) and f(2 - h):
f(2 + 0.5) = 4(2 + 0.5) = 4(2.5) = 10
f(2 - 0.5) = 4(2 - 0.5) = 4(1.5) = 6
Now, we can plug these values into the formula:
f'(2) ≈ (10 - 6) / (2 * 0.5) = 4 / 1 = 4
So the value of the numerical derivative f'(2) is 4. None of the provided responses (-1.333, -1.067, -1.010, -1.003, -1.000) represent the correct value, which is 4.
To find the numerical derivative of the function f(x) = 4x using the symmetric difference quotient, we need to calculate the slope of the function at x = 2.
The symmetric difference quotient formula is:
f'(x) = [f(x+h) - f(x-h)] / 2h
Given h = 0.5, we can plug in the values:
f'(2) = [f(2+0.5) - f(2-0.5)] / (2*0.5)
Simplifying further:
f'(2) = [f(2.5) - f(1.5)] / 1
Now, let's calculate f(2.5):
f(2.5) = 4 * 2.5 = 10
And f(1.5):
f(1.5) = 4 * 1.5 = 6
Plugging in the values:
f'(2) = (10 - 6) / 1 = 4 / 1 = 4
Therefore, the value of the numerical derivative f'(2) using the symmetric difference quotient with h = 0.5 is 4.