SOLUTION: Evaluate [f(x+h)-f(x)]/h and simplify if f(x)=x^2+x+1 Help please!

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Question 1086729: Evaluate [f(x+h)-f(x)]/h and simplify if f(x)=x^2+x+1
Help please!
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
f%28x%29=x%5E2%2Bx%2B1

f%28x%2Bh%29=%28x%2Bh%29%5E2%2B%28x%2Bh%29%2B1 Replace every x with (x+h)

f%28x%2Bh%29=x%5E2%2B2xh%2Bh%5E2%2Bx%2Bh%2B1 Expand out (x+h)^2

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Now that we know what f(x+h) is equal to, we can find the difference quotient

We'll be making two substitutions. The first substitution will have us replace the "f(x+h)" with the expression we got above

So we go from this



to this



Note the red boxes highlighted to show exactly what is replaced and how it looks after the replacement is done

Similarly, we will replace f(x) with x^2+x+1 to go from this



to this



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After the two substitutions, we now have



Let's simplify the right hand side (RHS)



%28f%28x%2Bh%29-f%28x%29%29%2Fh=%28x%5E2%2B2xh%2Bh%5E2%2Bx%2Bh%2B1-x%5E2-x-1%29%2Fh Distribute the negative through

%28f%28x%2Bh%29-f%28x%29%29%2Fh=%282xh%2Bh%5E2%2Bx%2Bh%2B1-x-1%29%2Fh Note how x^2-x^2 = 0x^2 = 0 so the x^2 terms go away

%28f%28x%2Bh%29-f%28x%29%29%2Fh=%282xh%2Bh%5E2%2Bh%2B1-1%29%2Fh Similarly, x-x = 0x = 0

%28f%28x%2Bh%29-f%28x%29%29%2Fh=%282xh%2Bh%5E2%2Bh%29%2Fh And also 1-1 = 0. Each term now has something with an 'h' as a factor

%28f%28x%2Bh%29-f%28x%29%29%2Fh=%28h%282x%2Bh%2B1%29%29%2Fh Factor out that common term h

%28f%28x%2Bh%29-f%28x%29%29%2Fh=%28cross%28h%29%282x%2Bh%2B1%29%29%2F%28cross%28h%29%29 The 'h's cancel on top and bottom (since h/h = 1)

%28f%28x%2Bh%29-f%28x%29%29%2Fh=2x%2Bh%2B1

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The final simplified answer is 2x%2Bh%2B1