Evaluate the difference quotient for the given function. Simplify your answer.
f(x) = 1+2x− x^2, (f(4+h)-f(4))/h
what's the trouble, forgot your Algebra I already?
f(4+h) = 1+2(4+h)-(4+h)^2 = -7-6h-h^2
f(4) = 1+2(4)-4^2 = -7
f(4+h)-f(4) = -6h-h^2
Now divide all that by h and you have
-6-h
Oh, I see you're trying to find the difference quotient of a function. Well, let's dive right into the silliness, shall we?
First, let's calculate f(4+h). We substitute '4+h' into the function:
f(4+h) = 1 + 2(4+h) - (4+h)^2
Now, let's compute f(4), which means substituting '4' into the function:
f(4) = 1 + 2(4) - (4)^2
Now, we'll subtract f(4) from f(4+h):
f(4+h) - f(4) = (1 + 2(4+h) - (4+h)^2) - (1 + 2(4) - (4)^2)
Now, let's simplify this expression a bit. I'll distribute those pesky multiplications:
f(4+h) - f(4) = (1 + 8 + 2h - (16 + 8h + h^2)) - (1 + 8 - 16)
Oh, boy! Time to combine like terms:
f(4+h) - f(4) = (9 + 2h - 16 - 8h - h^2) - (-7)
Alright, now let's simplify this a bit more:
f(4+h) - f(4) = -h^2 - 6h - 7
Finally, we divide by 'h' to get the difference quotient:
(f(4+h) - f(4))/h = (-h^2 - 6h - 7) / h
And there you have it! The simplified difference quotient for the function f(x) = 1+2x− x^2 is: (-h^2 - 6h - 7) / h.
Remember, it's important to simplify but never forget to have a good laugh along the way!
To evaluate the difference quotient for the function f(x) = 1+2x− x^2, we need to find (f(4+h) - f(4))/h.
First, let's find f(4+h):
f(x) = 1+2x− x^2
f(4+h) = 1+2(4+h)− (4+h)^2
Expanding (4+h)^2:
f(4+h) = 1+2(4+h)− (16+8h+h^2)
= 1+8+2h− 16−8h−h^2
= -6-6h-h^2
Next, let's find f(4):
f(4) = 1+2(4)− 4^2
= 1+8−16
= -7
Now, we can substitute these values into the difference quotient formula:
(f(4+h) - f(4))/h = (-6-6h-h^2 - (-7))/(h)
Simplifying, we get:
=(-6-6h-h^2+7)/h
=(-h^2-6h+1)/h
Therefore, the simplified difference quotient for the given function is -(h+6)+1/h.
To evaluate the difference quotient for the given function, we need to find the value of (f(4+h) - f(4))/h.
Step 1:
First, let's find the value of f(4+h).
To do this, substitute 4+h into the function f(x) = 1+2x− x^2:
f(4+h) = 1 + 2(4+h) - (4+h)^2.
Step 2:
Next, let's find the value of f(4).
To do this, substitute 4 into the function f(x):
f(4) = 1 + 2(4) - (4)^2.
Step 3:
Now, let's plug the values we found into the difference quotient formula: (f(4+h) - f(4))/h.
(f(4+h) - f(4))/h = (1 + 2(4+h) - (4+h)^2 - (1 + 2(4) - (4)^2))/h.
Step 4:
Simplify the numerator:
(1 + 8 + 2h - (16 + 8h + h^2) - 1 - 8 + 16)/(h).
Step 5:
Combine like terms:
(2h - 8h - h^2 + 8)/(h).
Step 6:
Simplify further:
(-h^2 - 6h + 8)/(h).
This is the simplified expression of the difference quotient for the given function f(x) = 1 + 2x − x^2.