Final answer:
To evaluate the difference quotient, substitute the given values into the formula: (f(3+h) - f(3)) / h. For the function f(x) = 4 + 3x - x², the difference quotient simplifies to 3 - h. For the function f(x) = x³, the difference quotient simplifies to 3a² + 3ah + h².
Step-by-step explanation:
To evaluate the difference quotient, we need to substitute the given values into the formula: (f(3+h) - f(3)) / h. Let's start with the first function, f(x) = 4 + 3x - x²:
f(3+h) = 4 + 3(3+h) - (3+h)²
= 4 + 9 + 3h - 9 - 6h - h²
= 4 + 3h - h²
f(3) = 4 + 3(3) - 3²
= 4 + 9 - 9
= 4
Now, substitute these values into the formula:
(f(3+h) - f(3)) / h = [(4 + 3h - h²) - 4] / h
= (3h - h²) / h
= 3 - h
So, the difference quotient for the first function is 3 - h.
For the second function, f(x) = x³:
f(a + h) = (a + h)³
= a³ + 3a²h + 3ah² + h³
f(a) = a³
Substituting these values into the formula:
(f(a + h) - f(a)) / h = [(a³ + 3a²h + 3ah² + h³) - a³] / h
= (3a²h + 3ah² + h³) / h
= 3a² + 3ah + h²
Therefore, the difference quotient for the second function is 3a² + 3ah + h².