Questions: Find the difference quotient (f(x+h)-f(x))/h, where h ≠ 0, for the function below. f(x)=4x^2-2x+1 Simplify your answer as much as possible. (f(x+h)-f(x))/h=

Find the difference quotient (f(x+h)-f(x))/h, where h ≠ 0, for the function below.
f(x)=4x^2-2x+1

Simplify your answer as much as possible.
(f(x+h)-f(x))/h=
Transcript text: Find the difference quotient $\frac{f(x+h)-f(x)}{h}$, where $h \neq 0$, for the function below. \[ f(x)=4 x^{2}-2 x+1 \] Simplify your answer as much as possible. \[ \frac{f(x+h)-f(x)}{h}= \]
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Solution

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Solution Steps

To find the difference quotient \(\frac{f(x+h)-f(x)}{h}\) for the function \(f(x) = 4x^2 - 2x + 1\), follow these steps:

  1. Substitute \(x + h\) into the function to find \(f(x + h)\).
  2. Calculate \(f(x + h) - f(x)\).
  3. Divide the result by \(h\).
  4. Simplify the expression as much as possible.
Step 1: Define the Function

We start with the function given by \[ f(x) = 4x^2 - 2x + 1. \]

Step 2: Calculate \(f(x + h)\)

Next, we substitute \(x + h\) into the function: \[ f(x + h) = 4(x + h)^2 - 2(x + h) + 1. \] Expanding this, we have: \[ f(x + h) = 4(x^2 + 2xh + h^2) - 2x - 2h + 1 = 4x^2 + 8xh + 4h^2 - 2x - 2h + 1. \]

Step 3: Find the Difference Quotient

Now, we calculate the difference quotient: \[ \frac{f(x + h) - f(x)}{h} = \frac{(4x^2 + 8xh + 4h^2 - 2x - 2h + 1) - (4x^2 - 2x + 1)}{h}. \] This simplifies to: \[ \frac{8xh + 4h^2 - 2h}{h} = \frac{h(8x + 4h - 2)}{h}. \] Assuming \(h \neq 0\), we can cancel \(h\): \[ 8x + 4h - 2. \]

Step 4: Simplify the Expression

Thus, the simplified form of the difference quotient is: \[ 4h + 8x - 2. \]

Final Answer

The difference quotient is \[ \boxed{4h + 8x - 2}. \]

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