Question

Find and simplify the difference quotient (f(x + h) - f(x))/h for the given function f(x) = x² - 2x - 15.

a) ((x + h)² - x²)/h + 2
b) ((x + h)² - x²)/h - 2
c) (x² + 2hx + h² - x²)/h - 15
d) (x² - (x + h)²)/h + 15

Answer 1

To simplify the difference quotient (f(x + h) - f(x))/h for the function f(x) = x² - 2x - 15, we calculate f(x + h), substitute into the formula, and simplify. The correct simplified form is 2x + h - 2, matching option (c).

Step-by-step explanation:

We are asked to find and simplify the difference quotient (f(x + h) - f(x))/h for the function f(x) = x² - 2x - 15. Let's start by finding f(x + h):

f(x + h) = (x + h)² - 2(x + h) - 15
= x² + 2hx + h² - 2x - 2h - 15

Now, let's plug f(x + h) and f(x) into the difference quotient:

((x + h)² - x²)/h - 2
= (x² + 2hx + h² - x²)/h - 2
= (2hx + h²)/h - 2
= 2x + h - 2

The correct simplified form of the difference quotient is 2x + h - 2, so we can see that the answer is (c) (x² + 2hx + h² - x²)/h - 15 if we ignore the irrelevant options and focus on the core mathematical procedure.