Find and Simplify the difference quotient
f(3+h)-f(3)/h for the function f(x)=-4x^2-7
thanks again
Someone sure wants you to learn what derivatives are about
f(3+h) = -4(3+h)^2 -7
f(3+h) = -4(9 + 6h + h^2 ) - 7
f(3+h) = -36 - 24 h - 4 h^2 - 7
f(3) = -36 - 7
f(3+h) - f(3) = -24 h - 4 h^2
divide by h
-24 - 4 h
NOTE AGAIN
lrt h--->0
you get slope = -24
which is the value of the derivative (which is -8 x) at x = 3 !!!!
Well, well, well! It looks like we've got a fun little math problem here. Let's dive right in!
To find the difference quotient for the function f(x) = -4x^2 - 7, we'll need to calculate f(3+h) and f(3).
First, let's find f(3+h):
f(3+h) = -4(3+h)^2 - 7
= -4(9 + 6h + h^2) - 7
= -36 - 24h - 4h^2 - 7
= -4h^2 - 24h - 43
Next, let's find f(3):
f(3) = -4(3)^2 - 7
= -4(9) - 7
= -36 - 7
= -43
Now that we have f(3+h) = -4h^2 - 24h - 43 and f(3) = -43, we can plug these values into the difference quotient formula:
[f(3+h) - f(3)] / h
[(-4h^2 - 24h - 43) - (-43)] / h
[-4h^2 - 24h] / h
-4h - 24
Ta-da! We've got our simplified difference quotient: -4h - 24.
Put on your clown shoes and juggle those variables with a laugh! Remember, if you have any more math questions or just want to have some fun, I'm here for you!
To find and simplify the difference quotient for the function f(x) = -4x^2 - 7, we'll follow these steps:
Step 1: Begin by substituting (3 + h) for x in the function f(x):
f(3 + h) = -4(3 + h)^2 - 7
Step 2: Expand and simplify the expression inside the brackets:
f(3 + h) = -4(9 + 6h + h^2) - 7
= -36 - 24h - 4h^2 - 7
= -43 - 24h - 4h^2
Step 3: Next, substitute 3 for x in the function f(x):
f(3) = -4(3)^2 - 7
= -4(9) - 7
= -36 - 7
= -43
Step 4: Now, we can evaluate the difference quotient by subtracting f(3) from f(3 + h) and dividing the result by h:
[f(3 + h) - f(3)] / h = [(-43 - 24h - 4h^2) - (-43)] / h
= (-43 - 24h - 4h^2 + 43) / h
= (-24h - 4h^2) / h
Step 5: Simplify the expression by factoring out h:
(-24h - 4h^2) / h = h(-24 - 4h) / h
= -24 - 4h
So, the simplified difference quotient for the function f(x) = -4x^2 - 7 is -24 - 4h.