Final answer:
To evaluate the difference quotient of f(x) = 3 - 5x - x² at x = 5 with a change of 5h, we calculate f(5h) - f(5)/h which results in the simplified expression -25 - 25h.
Step-by-step explanation:
We need to evaluate the difference quotient of the function f(x) = 3 - 5x - x². The difference quotient is given by the expression f(a + h) - f(a) / h, where a is a particular value of x, and h is a small change in x.
To find the difference quotient for f(x) when a = 5 and x is changed by h (meaning x = 5 + h), we calculate f(5 + h) - f(5) / h. This simplifies to:
- Step 1: Calculate f(5 + h), which is f(5h) in this context, as h is a factor of the change:
- f(5h) = 3 - 5(5h) - (5h)² = 3 - 25h - 25h²
- Step 2: Calculate f(5):
- f(5) = 3 - 5(5) - 5² = 3 - 25 - 25 = -47
- Step 3: Substitute into the difference quotient and simplify:
- (f(5h) - f(5)) / h = (3 - 25h - 25h² + 47) / h
- Simplify the numerator: -25h - 25h² + 50 = -25h (1 + h) + 50
- Divide each term by h:
- (-25h(1 + h) + 50) / h = -25 - 25h + 50/h
- Since h cannot equal zero, the term 50/h is undefined for h = 0. For any nonzero h, this simplifies to -25 - 25h.
The difference quotient for the function f(x) = 3 - 5x - x² at x = 5 with a change of 5h is -25 - 25h.