Question

a)

x	k(x)
1.1	1
1.2	-1
1.3	-2

\\(\frac{1 - 4}{1.1 - 1} =\\)
\\(\frac{-1 - 1}{1.2 - 1.1} =\\)
\\(\frac{-2 - 1}{1.3 - 1.2} =\\)

Explanation:

Step1: Calculate the first difference quotient

We have the points \((1, 4)\) and \((1.1, 1)\). The difference quotient formula is \(\frac{k(x_2)-k(x_1)}{x_2 - x_1}\). Substituting \(x_1 = 1\), \(k(x_1)=4\), \(x_2 = 1.1\), \(k(x_2)=1\), we get \(\frac{1 - 4}{1.1-1}=\frac{- 3}{0.1}=- 30\).

Step2: Calculate the second difference quotient

We have the points \((1.1, 1)\) and \((1.2, - 1)\). Using the difference quotient formula \(\frac{k(x_2)-k(x_1)}{x_2 - x_1}\), substituting \(x_1 = 1.1\), \(k(x_1)=1\), \(x_2 = 1.2\), \(k(x_2)=-1\), we get \(\frac{-1 - 1}{1.2 - 1.1}=\frac{-2}{0.1}=-20\).

Step3: Calculate the third difference quotient

We have the points \((1.2, - 1)\) and \((1.3, - 2)\). Using the difference quotient formula \(\frac{k(x_2)-k(x_1)}{x_2 - x_1}\), substituting \(x_1 = 1.2\), \(k(x_1)=-1\), \(x_2 = 1.3\), \(k(x_2)=-2\), we get \(\frac{-2-(-1)}{1.3 - 1.2}=\frac{-2 + 1}{0.1}=\frac{-1}{0.1}=-10\).

Answer:

The values of the difference quotients are \(-30\), \(-20\), \(-10\) respectively.