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Find and simplify f(a + h) – f(a) h (h # 0) for the following function. f(x) = 6x2 – 2x + 5

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Answer:To find and simplify the expression \(f(a + h) - f(a) / h\), we need to substitute the given function \(f(x) = 6x^2 - 2x + 5\) into the expression and perform the required operations. Here's how it's done step by step:

1. Start with the function \(f(x) = 6x^2 - 2x + 5\).

2. Replace \(x\) with \(a + h\) to get \(f(a + h) = 6(a + h)^2 - 2(a + h) + 5\).

3. Calculate \(f(a)\) by replacing \(x\) with \(a\) in the function: \(f(a) = 6a^2 - 2a + 5\).

4. Now subtract \(f(a)\) from \(f(a + h)\):

\(f(a + h) - f(a) = [6(a + h)^2 - 2(a + h) + 5] - [6a^2 - 2a + 5]\).

5. Expand the terms inside the brackets:

\(f(a + h) - f(a) = 6(a^2 + 2ah + h^2) - 2a - 2h + 5 - 6a^2 + 2a - 5\).

6. Distribute and combine like terms:

\(f(a + h) - f(a) = 6a^2 + 12ah + 6h^2 - 2a - 2h + 5 - 6a^2 + 2a - 5\).

7. Notice that the \(6a^2\) terms cancel out, and the \(5\) and \(-5\) terms also cancel out:

\(f(a + h) - f(a) = 12ah + 6h^2 - 2h\).

8. Factor out \(h\) from the remaining terms:

\(f(a + h) - f(a) = h(12a + 6h - 2)\).

9. Finally, simplify the expression inside the parentheses:

\(f(a + h) - f(a) = h(12a + 6h - 2) = h(6(2a + h - 1))\).

So, the simplified expression \(f(a + h) - f(a) / h\) for the given function \(f(x) = 6x^2 - 2x + 5\) is \(6(2a + h - 1)\).

Explanation:

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