To simplify the difference quotient $\frac{f(x+h) - f(x)}{h}$ for the given function $f(x) = 3x^2 - 8x + 1$, follow these steps:

Step 1: Compute $f(x+h)$

Substitute $x + h$ into the function $f(x)$:

$f(x+h) = 3(x+h)^2 - 8(x+h) + 1$

Step 2: Expand and Simplify $f(x+h)$

First, expand $(x+h)^2$:

$(x+h)^2 = x^2 + 2xh + h^2$

Now, substitute this back into the expression for $f(x+h)$:

$f(x+h) = 3(x^2 + 2xh + h^2) - 8(x + h) + 1$

Distribute the constants:

$f(x+h) = 3x^2 + 6xh + 3h^2 - 8x - 8h + 1$

Step 3: Compute the Difference $f(x+h) - f(x)$

Subtract $f(x)$ from $f(x+h)$:

$f(x+h) - f(x) = \left(3x^2 + 6xh + 3h^2 - 8x - 8h + 1\right) - \left(3x^2 - 8x + 1\right)$

Distribute and combine like terms:

$f(x+h) - f(x) = 3x^2 + 6xh + 3h^2 - 8x - 8h + 1 - 3x^2 + 8x - 1$

Simplify by canceling out the common terms $3x^2$, $-8x$, and $1$:

$f(x+h) - f(x) = 6xh + 3h^2 - 8h$

Step 4: Divide by $h$

Now, divide the simplified difference by $h$:

$\frac{f(x+h) - f(x)}{h} = \frac{6xh + 3h^2 - 8h}{h}$

Factor out $h$ from the numerator:

$\frac{f(x+h) - f(x)}{h} = \frac{h(6x + 3h - 8)}{h}$

Cancel out $h$ from the numerator and denominator:

$\frac{f(x+h) - f(x)}{h} = 6x + 3h - 8$

Final Answer

The simplified difference quotient is:

$\frac{f(x+h) - f(x)}{h} = 6x + 3h - 8$

Would you like any further explanation or have any questions?

Related Questions

1. What happens to the difference quotient as $h$ approaches 0?
2. How do you use the difference quotient to find the derivative of a function?
3. What is the geometric interpretation of the difference quotient?
4. Can the difference quotient be used for non-polynomial functions?
5. How does the difference quotient relate to the concept of a secant line?

Tip

Remember, the difference quotient is a fundamental tool in calculus for finding the derivative, which represents the rate of change or the slope of the tangent line at a specific point on the curve.