Final answer:
The difference quotient for the function f(x) = (x + 7) / (x + 1) is calculated by finding f(5), substituting x into the function, and then computing and simplifying the expression (f(x) - f(5)) / (x - 5).
Step-by-step explanation:
The student asks to evaluate the difference quotient for the function f(x) = (x + 7) / (x + 1), specifically the expression (f(x) - f(5)) / (x - 5). The difference quotient is a formula used in calculus to find the slope of the secant line between two points on a graph of a function, and it's also the basis for the definition of the derivative. As the value of x approaches 5, the difference quotient approaches the derivative of f at x = 5.
To evaluate the quotient, you need to first calculate f(x) and f(5), then subtract and divide as indicated:
- First, find f(5) by substituting 5 into the function: f(5) = (5 + 7) / (5 + 1) = 12 / 6 = 2.
- Next, calculate the difference quotient: (f(x) - f(5)) / (x - 5) = ((x + 7) / (x + 1) - 2) / (x - 5).
- Simplify the expression as needed to evaluate or further analyze.
If you are given specific values for x (like x = 0.0216 or x = -0.0224), substitute them into the simplified expression to find the corresponding values of the difference quotient.