SOLUTION: Find and simplify the difference quotient for the given function. f(x) = 6x2

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Question 1176449: Find and simplify the difference quotient for the given function.
f(x) = 6x2

Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

6x^2 is the same as writing 6x%5E2

f%28x%29+=+6x%5E2

f%28x%2Bh%29+=+6%28x%2Bh%29%5E2 We replace every x with x+h

f%28x%2Bh%29+=+6%28x%5E2%2B2xh%2Bh%5E2%29 FOIL

f%28x%2Bh%29-f%28x%29+=+6%28x%5E2%2B2xh%2Bh%5E2%29-6x%5E2 Subtract f(x+h) and f(x). This is the "difference" portion of "difference quotient".

f%28x%2Bh%29-f%28x%29+=+6x%5E2%2B12xh%2B6h%5E2-6x%5E2 Distribute

f%28x%2Bh%29-f%28x%29+=+12xh%2B6h%5E2 The 6x^2 terms subtract and cancel.

%28f%28x%2Bh%29-f%28x%29%29%2Fh+=+%2812xh%2B6h%5E2%29%2Fh Divide both sides by 'h'. This is where the "quotient" part comes in of "difference quotient".

%28f%28x%2Bh%29-f%28x%29%29%2Fh+=+%28h%2812x%2B6h%29%29%2Fh Factor out h from the numerator

%28f%28x%2Bh%29-f%28x%29%29%2Fh+=+12x%2B6h Cancel the pair of 'h' terms from the numerator and denominator


Answer: %28f%28x%2Bh%29-f%28x%29%29%2Fh+=+12x%2B6h

Side note: Later in calculus you'll learn that as h approaches 0, the expression 12x+6h will approach 12x. This is the derivative of f(x) = 6x^2. Keep in mind that h = 0 is not allowed, since division by zero is not allowed, but we can get closer and closer to it.