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?
To find the value of the numerical derivative f'(2) using the symmetric difference quotient with h = 0.5, we need to compute the values of f(x) at x = 1.5 and x = 2.5 and use them to calculate the difference quotient.
f(1.5) = 4 * 1.5 = 6
f(2.5) = 4 * 2.5 = 10
The symmetric difference quotient formula is given by:
f'(x) = (f(x + h) - f(x - h)) / (2 * h)
Substituting the values obtained earlier:
f'(2) = (f(2.5) - f(1.5)) / (2 * 0.5)
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.
Option D. 4
To find the numerical derivative of f(x) at x = 2 using the symmetric difference quotient with h = 0.5, we will use the following formula:
f'(x) = (f(x + h) - f(x - h)) / (2h)
First, let's substitute the given values into the formula:
x = 2 (as specified in the question)
h = 0.5 (given in the question)
Now, let's calculate f(x + h) and f(x - h):
f(x + h) = f(2 + 0.5) = f(2.5)
= 4(2.5)
= 10
f(x - h) = f(2 - 0.5) = f(1.5)
= 4(1.5)
= 6
Now, let's substitute these values back into the formula:
f'(2) = (f(2 + 0.5) - f(2 - 0.5)) / (2 * 0.5)
= (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.
To find the numerical derivative of a function using the symmetric difference quotient, we need to calculate the difference in function values at two points and divide it by the difference in x-values. The formula for the symmetric difference quotient is:
f'(x) = (f(x+h) - f(x-h))/ (2h)
In this case, the function is f(x) = 4x, and we want to find f'(2) using h = 0.5.
1. Let's find f(2+h) and f(2-h) first.
- f(2+h) = 4(2 + 0.5) = 4(2.5) = 10
- f(2-h) = 4(2 - 0.5) = 4(1.5) = 6
2. Now, substitute the values into the formula for the symmetric difference quotient:
f'(2) = (f(2+h) - f(2-h))/(2h)
= (10 - 6)/(2(0.5))
= 4/1
= 4
Therefore, the value of the numerical derivative f'(2), determined using the symmetric difference quotient with h = 0.5, is 4.