SOLUTION: Find the difference quotient of the following function f(x)=(3x+4)^2

Algebra ->  Functions -> SOLUTION: Find the difference quotient of the following function f(x)=(3x+4)^2      Log On


   

Question 1008200: Find the difference quotient of the following function
f(x)=(3x+4)^2
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Let's expand out f(x)

f%28x%29+=+%283x%2B4%29%5E2

f%28x%29+=+%283x%2B4%29%283x%2B4%29

f%28x%29+=+3x%283x%2B4%29%2B4%283x%2B4%29

f%28x%29+=+9x%5E2%2B12x%2B12x%2B16

f%28x%29+=+9x%5E2+%2B+24x+%2B+16

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Now compute f(x+h)

f%28x%29+=+9x%5E2+%2B+24x+%2B+16

f%28x%2Bh%29+=+9%28x%2Bh%29%5E2+%2B+24%28x%2Bh%29+%2B+16

f%28x%2Bh%29+=+9%28x%5E2%2B2xh%2Bh%5E2%29+%2B+24%28x%2Bh%29+%2B+16

f%28x%2Bh%29+=+9x%5E2%2B18xh%2B9h%5E2+%2B+24x%2B24h+%2B+16
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With that info, we can finally find the difference quotient





%28f%28x%2Bh%29-f%28x%29%29%2Fh=%2818xh%2B9h%5E2%2B24h%29%2Fh

%28f%28x%2Bh%29-f%28x%29%29%2Fh=%28h%2818x%2B9h%2B24%29%29%2Fh

%28f%28x%2Bh%29-f%28x%29%29%2Fh=18x%2B9h%2B24


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So the difference quotient of f(x) is 18x%2B9h%2B24

Side Note: if you let h approach 0, then 18x%2B9h%2B24 turns into 18x%2B24 which is exactly the derivative of f(x).