Question

1) f(x) = 2x - 5
Determine the difference quotient

Answer 1

Difference quotient:
`(f(x+h)-f(x))/(h)` = 2

Explanation:

The difference quotient is the word that we use when trying to calculate the average rate of change of the function over a specific interval, which in this case, is h.

The formula for calculating the difference quotient is:

• 
  `(f(x+h)-f(x))/(h)`

In order to determine the difference quotient of the function
`f(x)=2x-5` we need to first evaluate
`f(x+h)` of this function.

We can do so by substituting x + h for x into the function.

• 
  `f(x+h)=2(x+h)-5`

Distribute 2 inside the parentheses.

• 
  `f(x+h)=2x+2h-5`

Now that we have found f(x + h), we can use the formula for the difference quotient and substitute f(x + h) and f(x) into it:

• 
  `((2x+2h-5)-(2x-5))/(h)`

Distribute the negative sign inside the parentheses.

• 
  `(2x+2h-5-2x+5)/(h)`

Combine like terms in the numerator. The 2x and 5's cancel out. We are left with:

• 
  `(2h)/(h)`

The h's cancel out, and this simplifies to:

• 
  `2`

The difference quotient of f(x) is 2.