Question

Hello! I need some assistance with this homework question, pleaseQ8

Answer 1

The difference quotient can be found by the formula:

`(f(x+h)-f(x))/(h)`

Where h is not 0, because it would be a division by 0.

In this case, we have the function:

`f(x)=x^2-7x+1`

Then using the forumla:

`(f(x+h)-f(x))/(h)=((x+h)^2-7(x+h)+1-(x^2-7x+1))/(h)`

The next step is to solve the binomial squared, and do a distributive property.

We can solve the squared binomial using:

`(a+b)^2=a^2+2ab+b^2`

Then:

`((x+h)^2-7(x+h)+1-(x^2-7x+1))/(h)=(x^2+2xh+h^2-7x-7h+1-x^2+7x-1)/(h)`

Here we can see that there are several terms equal but with different sign. We can cancel out the terms x², 7x and 1

`(x^2+2xh+h^2-7x-7h+1-x^2+7x-1)/(h)=(x^2-x^2-7x+7x+1-1+2hx+7h+h^2)/(h)=(h^2+2hx-7h)/(h)`

Now we can factor out the h in the numerator:

`(h(h+2x-7))/(h)`

Which cancels out with the denominator, and we get:

`(f(x+h)-f(x))/(h)=2x-7+h`

And that's the result of the problem.