Question

Finding a difference quotient for a line Find the difference quotient (f(x+h)-f(x))/(h) f(x)=3x^(2)-2x+3

Answer 1

The difference quotient for the line
`\( f(x) = 3x^2 - 2x + 3 \) is \( (6h - 2)/(h) \).`

Step-by-step explanation:

To find the difference quotient, we use the formula
`\( (f(x+h)-f(x))/(h) \)`. First, let's calculate ( f(x+h)) and ( f(x) ):

`\[ f(x+h) = 3(x+h)^2 - 2(x+h) + 3 \]`

Expanding and simplifying:

`\[ f(x+h) = 3(x^2 + 2xh + h^2) - 2x - 2h + 3 \]`

`\[ f(x+h) = 3x^2 + 6xh + 3h^2 - 2x - 2h + 3 \]`

Now, subtracting
`\( f(x) = 3x^2 - 2x + 3 \) from \( f(x+h) \):`

`\[ f(x+h) - f(x) = (3x^2 + 6xh + 3h^2 - 2x - 2h + 3) - (3x^2 - 2x + 3) \]`

Simplifying:

`\[ f(x+h) - f(x) = 6xh + 3h^2 - 2h \]`

Now, plug this into the difference quotient formula:

`\[ (f(x+h)-f(x))/(h) = (6xh + 3h^2 - 2h)/(h) \]`

Simplify further:

`\[ (6h - 2)/(h) \]`

So, the difference quotient for the given line is
`\( (6h - 2)/(h) \).` This represents the average rate of change of the function ( f(x) ) with respect to ( x ) over a small interval ( h ).