Question

Evaluate the difference quotient for the given function. Simplify your answer.

f(x) = 4 + 3x - x², (f(3+h) - f(3)) / h
f(x) = x³, (f(a+h) - f(a)) / h

Answer 1

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.