Find and simplify the difference quotient for the given function.
f(x)= -x^2+2x-1
I need to know how to set up the equations and simplify everything when dividing by h at the end.
I have seen the difference quotients defined as
(f(x+h) - f(x) / h
in your case:
( -(x+h)^2 + 2(x+h) - 1 - (-x^2 + 2x - 1))/h
= ( -x^2 - 2hx - h^2 + 2x + 2h - 1 + x^2 - 2x + 1)/h
= (-2hx - h^2 + 2h)/h
= -2x - h + 2 , h ≠ 0
Found my error! Thank you! 2xh i wans't carrying that in when distributing
To find the difference quotient for the function f(x) = -x^2 + 2x - 1, we need to find the value of (f(x + h) - f(x))/h.
Step 1: Substitute (x + h) into the original function
f(x + h) = -(x + h)^2 + 2(x + h) - 1
Step 2: Simplify f(x + h)
f(x + h) = -(x^2 + 2xh + h^2) + 2x + 2h - 1
f(x + h) = -x^2 - 2xh - h^2 + 2x + 2h - 1
Step 3: Substitute f(x) into the original function
f(x) = -x^2 + 2x - 1
Step 4: Find the difference, f(x + h) - f(x)
f(x + h) - f(x) = (-x^2 - 2xh - h^2 + 2x + 2h - 1) - (-x^2 + 2x - 1)
f(x + h) - f(x) = -x^2 - 2xh - h^2 + 2x + 2h - 1 + x^2 - 2x + 1
Step 5: Simplify the difference
f(x + h) - f(x) = -2xh - h^2 + 2h
Step 6: Divide by h
(f(x + h) - f(x))/h = (-2xh - h^2 + 2h)/h
Step 7: Simplify the expression
(f(x + h) - f(x))/h = -2x - h + 2
Therefore, the difference quotient for the function f(x) = -x^2 + 2x - 1 is -2x - h + 2.
To find the difference quotient for the given function f(x) = -x^2 + 2x - 1, we need to set up the equation using the definition of the difference quotient and then simplify it.
The difference quotient for a function f(x) is given by the formula:
f'(x) = [f(x + h) - f(x)] / h
Here, h represents a small increment along the x-axis.
Let's substitute the function f(x) into the formula:
f'(x) = [(-x^2 + 2x - 1 + h^2 - 2h + 1) - (-x^2 + 2x - 1)] / h
Simplify the expression inside the brackets:
f'(x) = [(-x^2 + h^2) + (2x - 2h) + 1 - (-x^2 + 2x - 1)] / h
Combine like terms:
f'(x) = [-x^2 + h^2 + 2x - 2h + 1 + x^2 - 2x + 1] / h
Now, simplify the numerator:
f'(x) = (h^2 - 2h + 2) / h
We have obtained the difference quotient for the given function f(x). To simplify it further, we can factor out h from the numerator:
f'(x) = h(h - 2) / h
Cancel out the common factor of h from the numerator and denominator:
f'(x) = (h - 2)
Therefore, the simplified difference quotient is f'(x) = (h - 2).
Note: The purpose of finding the difference quotient is to evaluate a derivative when h approaches zero (as h represents a small increment). In this case, when h approaches zero, f'(x) will approach -2, indicating the slope of the tangent line to the function at a particular point.