Given the function f(x)=4/x , 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, we need to calculate the values of f(x) at x = 2 and x = 2 + h = 2.5 and take their difference over h.
f(x) = 4/x
f(2) = 4/2 = 2
f(2.5) = 4/2.5 = 1.6
f′(2) ≈ (f(2.5) - f(2))/(2.5 - 2)
= (1.6 - 2)/(0.5)
= -0.4/0.5
= -0.8
Therefore, the best representation of the value of the numerical derivative f′(2) is -0.8.
None of the provided responses match the calculated value of the numerical derivative.
To find the value of the numerical derivative f'(2) using the symmetric difference quotient, we need to use the formula:
f'(x) ≈ (f(x + h) - f(x - h))/(2h)
where h is the step size.
Given that f(x) = 4/x, we can substitute x = 2 and h = 0.5 into the formula:
f'(2) ≈ (f(2 + 0.5) - f(2 - 0.5))/(2 * 0.5)
Simplifying, we get:
f'(2) ≈ (f(2 + 0.5) - f(2 - 0.5))/1
Next, let's evaluate the function at the points:
f(2 + 0.5) = f(2.5) = 4/(2.5) = 1.6
f(2 - 0.5) = f(1.5) = 4/(1.5) = 2.6667
Now, substitute these values back into the formula:
f'(2) ≈ (1.6 - 2.6667)/1
Simplifying, we get:
f'(2) ≈ -1.0667
Therefore, the best representation of the value of the numerical derivative f'(2), determined using the symmetric difference quotient with h = 0.5, is -1.067.
To find the numerical derivative, we can use the symmetric difference quotient formula:
f′(2) = (f(2 + h) - f(2 - h))/ (2h)
Given that h = 0.5, we can calculate f(2 + h) and f(2 - h) first:
f(2 + h) = f(2 + 0.5) = 4/(2 + 0.5) = 4/2.5 = 1.6
f(2 - h) = f(2 - 0.5) = 4/(2 - 0.5) = 4/1.5 = 2.667
Now we can plug these values into the symmetric difference quotient formula:
f′(2) = (1.6 - 2.667) / (2 * 0.5) = -1.067
Therefore, the best representation of the value of the numerical derivative f′(2) is -1.067.
The correct response is: -1.067 - 1.067.