Questions: (f(a+h)-f(a))/h represents the slope of the line joining x=a to x=a+h. For f(x)=(x+2)/(x-9) a. Construct and simplify the difference quotient (f(a+h)-f(a))/h b. By identifying a and h, use your answer above, find the slope of the line joining the points from: i. x=10 to x=11 ii. x=15 to x=12

Solution Steps

To solve the given problem, we need to follow these steps:

1. Construct and Simplify the Difference Quotient:

   • Given \( f(x) = \frac{x+2}{x-9} \), we need to find \( \frac{f(a+h) - f(a)}{h} \).
   • Substitute \( f(a+h) \) and \( f(a) \) into the difference quotient formula and simplify.

2. Calculate the Slope for Given Points:

   • Use the simplified difference quotient formula to find the slope for the given points.
   • For \( x=10 \) to \( x=11 \), set \( a=10 \) and \( h=1 \).
   • For \( x=15 \) to \( x=12 \), set \( a=15 \) and \( h=-3 \).

Given the function \( f(x) = \frac{x+2}{x-9} \), we need to construct the difference quotient: \[ \frac{f(a+h) - f(a)}{h} \] Substituting \( f(a+h) \) and \( f(a) \) into the difference quotient formula, we get: \[ \frac{\left( \frac{a+h+2}{a+h-9} \right) - \left( \frac{a+2}{a-9} \right)}{h} \] Simplifying this expression, we obtain: \[ \frac{\left( \frac{a+h+2}{a+h-9} \right) - \left( \frac{a+2}{a-9} \right)}{h} = \frac{(a+h+2)(a-9) - (a+2)(a+h-9)}{h(a+h-9)(a-9)} \] Further simplification yields: \[ \frac{(a+h+2)(a-9) - (a+2)(a+h-9)}{h(a+h-9)(a-9)} = \frac{-11h}{h(a^2 + ah - 18a - 9h + 81)} \] Finally, canceling \( h \) in the numerator and denominator, we get: \[ \frac{-11}{a^2 + ah - 18a - 9h + 81} \]

For \( x=10 \) to \( x=11 \), we set \( a=10 \) and \( h=1 \). Substituting these values into the simplified difference quotient: \[ \frac{-11}{10^2 + 10 \cdot 1 - 18 \cdot 10 - 9 \cdot 1 + 81} = \frac{-11}{100 + 10 - 180 - 9 + 81} = \frac{-11}{2} \] Thus, the slope from \( x=10 \) to \( x=11 \) is: \[ \boxed{-5.500} \]

For \( x=15 \) to \( x=12 \), we set \( a=15 \) and \( h=-3 \). Substituting these values into the simplified difference quotient: \[ \frac{-11}{15^2 + 15 \cdot (-3) - 18 \cdot 15 - 9 \cdot (-3) + 81} = \frac{-11}{225 - 45 - 270 + 27 + 81} = \frac{-11}{18} \] Thus, the slope from \( x=15 \) to \( x=12 \) is: \[ \boxed{-0.6111} \]

Final Answer