f(x)=4-x^2 simplify completly. Use difference quotient.
( x - 2 ) ( x + 2 ) =
x ^ 2 - 2 x + 2 x - 4 =
x ^ 2 - 4 =
- ( x ^ 2 - 4 ) =
4 - x ^ 2
So :
4 - x ^ 2 = - ( x - 2 ) ( x + 2 )
To simplify the function f(x) = 4 - x^2 using the difference quotient, we'll start by finding the value of the difference quotient:
The difference quotient is defined as:
f'(x) = lim(h->0) [f(x + h) - f(x)] / h
Let's calculate this step by step:
1. Start by finding f(x + h), which means replacing x with (x + h) in the function:
f(x + h) = 4 - (x + h)^2
2. Expand (x + h)^2:
f(x + h) = 4 - (x^2 + 2xh + h^2)
3. Simplify the expression inside the parentheses:
f(x + h) = 4 - x^2 - 2xh - h^2
4. Now, calculate f(x):
f(x) = 4 - x^2
5. Next, subtract f(x) from f(x + h):
f(x + h) - f(x) = (4 - x^2 - 2xh - h^2) - (4 - x^2)
= 4 - x^2 - 2xh - h^2 - 4 + x^2
= -2xh - h^2
6. Finally, divide by h and take the limit as h approaches 0:
f'(x) = lim(h->0) (-2xh - h^2) / h
Now, simplify the expression further:
f'(x) = lim(h->0) -2x - h
f'(x) = -2x
Therefore, the derivative of f(x) = 4 - x^2 using the difference quotient is f'(x) = -2x.
To simplify the function f(x) = 4 - x^2 using the difference quotient, we can start by expanding the square term.
First, let's find f(x + h).
f(x + h) = 4 - (x + h)^2
= 4 - (x^2 + 2xh + h^2)
= 4 - x^2 - 2xh - h^2
Next, we can find the difference quotient by subtracting f(x) from f(x + h) and dividing by h.
[f(x + h) - f(x)] / h = [4 - x^2 - 2xh - h^2 - (4 - x^2)] / h
= [-2xh - h^2] / h
= -2x - h
Therefore, the difference quotient of f(x) = 4 - x^2 is -2x - h.