Let f(x) be the function 11x2−9x+7. Then the quotient
f(6+h)−f(6)h can be simplified to ah+b for:
a=
and
b=
a= 57
b= -85
To simplify the expression f(6+h) - f(6)/h, we need to substitute the given function f(x) = 11x^2 - 9x + 7 into that expression.
Step 1: Evaluate f(6+h)
Replace x with (6+h) in the function f(x) = 11x^2 - 9x + 7:
f(6+h) = 11(6+h)^2 - 9(6+h) + 7
= 11(36 + 12h + h^2) - 54 - 9h + 7
= 396 + 132h + 11h^2 - 47 - 9h
= 11h^2 + 123h + 349
Step 2: Evaluate f(6)
Replace x with 6 in the function f(x) = 11x^2 - 9x + 7:
f(6) = 11(6)^2 - 9(6) + 7
= 11(36) - 54 + 7
= 396 - 54 + 7
= 349
Step 3: Calculate the expression f(6+h) - f(6)
Subtract the expression f(6) from f(6+h):
f(6+h) - f(6) = (11h^2 + 123h + 349) - 349
= 11h^2 + 123h
Step 4: Divide the above expression by h
Divide f(6+h) - f(6) by h:
(11h^2 + 123h) / h = 11h + 123
Therefore, a = 11 and b = 123.
To simplify the quotient f(6+h) - f(6) / h, we need to substitute the given function f(x) = 11x^2 - 9x + 7 and simplify the expression.
Step 1: Evaluate f(6+h)
Substitute x = 6+h into the function f(x):
f(6+h) = 11(6+h)^2 - 9(6+h) + 7
Simplify f(6+h):
f(6+h) = 11(36 + 12h + h^2) - 54 - 9h + 7
f(6+h) = 396 + 132h + 11h^2 - 47 - 9h
Simplify further:
f(6+h) = 11h^2 + 123h + 349
Step 2: Evaluate f(6)
Substitute x = 6 into the function f(x):
f(6) = 11(6)^2 - 9(6) + 7
f(6) = 11(36) - 54 + 7
f(6) = 396 - 54 + 7
f(6) = 349
Step 3: Simplify the quotient
Now substitute the values we found into the quotient:
(f(6+h) - f(6)) / h = (11h^2 + 123h + 349 - 349) / h
(f(6+h) - f(6)) / h = 11h^2 + 123h / h
(f(6+h) - f(6)) / h = 11h + 123
Therefore, a = 11 and b = 123.
I have a sneaking suspicion that you mean
[f(6+h)−f(6)]/ h
f(6+h) = 11(6+h)^2 - 9(6+h) + 7
f(6) = 11(6)^2 - 9(6) + 7
---------------------------------
f(6+h) -f(h) = 11(12h+h^2) -9h
= 132 h+ 11 h^2 - 9h
= 123 h + 11 h^2
now divide by h
= 11 h + 123
a = 11
b = 123
check with calculus
f' = 22 x - 9
when x = 6
f' = 132 - 9 = 123 check