Given the function $f(x) = 4x^2 - 3$, you are asked to express the difference quotient:

$\frac{f(x+h) - f(x)}{h}$

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

Start by substituting $x+h$ into the function:

$f(x+h) = 4(x+h)^2 - 3$

Expand $(x+h)^2$:

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

Thus,

$f(x+h) = 4(x^2 + 2xh + h^2) - 3$ $f(x+h) = 4x^2 + 8xh + 4h^2 - 3$

Step 2: Compute $f(x+h) - f(x)$

Now subtract $f(x) = 4x^2 - 3$ from $f(x+h)$:

$f(x+h) - f(x) = (4x^2 + 8xh + 4h^2 - 3) - (4x^2 - 3)$

Simplifying:

$f(x+h) - f(x) = 4x^2 + 8xh + 4h^2 - 3 - 4x^2 + 3$ $f(x+h) - f(x) = 8xh + 4h^2$

Step 3: Divide by $h$

Now, divide the result by $h$:

$\frac{f(x+h) - f(x)}{h} = \frac{8xh + 4h^2}{h}$

Simplify the expression:

$\frac{f(x+h) - f(x)}{h} = 8x + 4h$

Final Answer

The simplified form of the difference quotient is:

$\frac{f(x+h) - f(x)}{h} = 8x + 4h$

Would you like further details or have any questions? Here are five related questions for more practice:

1. How would you evaluate the difference quotient when $h = 0$?
2. How does the result $8x + 4h$ change as $h \to 0$?
3. What is the geometric meaning of the difference quotient?
4. How do you compute the derivative of $f(x) = 4x^2 - 3$?
5. How does this process differ when $f(x)$ is a linear function?

Tip: The difference quotient is fundamental for finding the derivative of a function. It measures the rate of change as $h$ approaches zero!