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 numerical derivative using the symmetric difference quotient with h = 0.5, we can use the formula:
f′(x) = (f(x+h) - f(x-h)) / (2h)
In this case, f(x) = 4x, so we have:
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 value of the numerical derivative f′(2) is 4. None of the given responses match this value.
To find the numerical derivative of the function f(x) = 4x at x = 2 using the symmetric difference quotient, we can use the formula:
f'(x) = (f(x + h) - f(x - h)) / (2h)
Given that h = 0.5, we can substitute the values into the formula:
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 these values:
f(2.5) = 4 * 2.5 = 10
f(1.5) = 4 * 1.5 = 6
Substituting these values back into the formula:
f'(2) = (10 - 6) / 1
f'(2) = 4
Therefore, the value of the numerical derivative f'(2) using the symmetric difference quotient with h = 0.5 is 4.
To find the numerical derivative of a function, we can use the symmetric difference quotient formula. The symmetric difference quotient is given by:
f'(x) = (f(x+h) - f(x-h)) / (2h)
In this case, we have the function f(x) = 4x and we want to find f'(2) using h = 0.5.
First, let's evaluate f(2+h) and f(2-h).
f(2+h) = 4(2+h) = 8 + 4h
f(2-h) = 4(2-h) = 8 - 4h
Now we can substitute these values into the symmetric difference quotient formula:
f'(2) = (f(2+h) - f(2-h)) / (2h)
= ((8 + 4h) - (8 - 4h)) / (2h)
= (8 + 4h - 8 + 4h) / (2h)
= 8h / (2h)
= 4
So, the numerical derivative f'(2) is equal to 4.
So, none of the provided responses (−1.333, −1.067, −1.010, −1.003, −1.000) represents the correct value for f'(2).