(b)
f(x + h) − f(x) h

Given the function $f(x) = x - 4x^2$, let's solve each part step by step.

(a) Find $f(x + h)$:

We need to substitute $x + h$ into the function $f(x) = x - 4x^2$.

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

Now expand the terms:

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

So,

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

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

We already have $f(x + h)$ and we know that $f(x) = x - 4x^2$. Now, let's compute the difference $f(x + h) - f(x)$:

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

Simplify the terms:

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

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

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

Now, divide by $h$:

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

Simplify the expression by dividing each term by $h$:

$\frac{f(x + h) - f(x)}{h} = -8x - 4h + 1$

Thus, the simplified expression is:

$\frac{f(x + h) - f(x)}{h} = -8x - 4h + 1$

Summary of results:

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

(b) $\frac{f(x + h) - f(x)}{h} = -8x - 4h + 1$

Would you like more details on any of these steps? Here are 5 related questions:

1. How would this change if the function were cubic instead of quadratic?
2. What does the expression $\frac{f(x + h) - f(x)}{h}$ represent in calculus?
3. Can you simplify $f(x + h)$ for a different function, say $f(x) = 3x^2 - 2x$?
4. How would the result change if $h$ approaches zero?
5. How would you apply the same steps to a trigonometric function?

Tip: The difference quotient $\frac{f(x + h) - f(x)}{h}$ is a fundamental concept in calculus and is used to find the derivative of a function.