) Let f(x) = 2x
2 − 4.
(a) Define the “Newton quotient” for this function at a point a.
(b) Simplify this by cancelling “h” from the numerator and the denominator, where possible.
(c) Find the limit of the “Newton quotient” as “h” approaches zero.

Let's break down the problem step by step.

### (a) Define the “Newton quotient” for the function \( f(x) = 2x^2 - 4 \) at a point \( a \).

The **Newton quotient** is essentially the difference quotient used to compute the derivative of a function. It is given by:

\[
N(a, h) = \frac{f(a+h) - f(a)}{h}
\]

For the function \( f(x) = 2x^2 - 4 \), we need to substitute \( f(a+h) \) and \( f(a) \) into this formula.

- \( f(a+h) = 2(a+h)^2 - 4 \)
- \( f(a) = 2a^2 - 4 \)

Thus, the Newton quotient becomes:

\[
N(a, h) = \frac{[2(a+h)^2 - 4] - [2a^2 - 4]}{h}
\]

### (b) Simplify by cancelling “h” from the numerator and denominator.

First, we need to expand \( (a+h)^2 \):

\[
(a+h)^2 = a^2 + 2ah + h^2
\]

So:

\[
f(a+h) = 2(a^2 + 2ah + h^2) - 4 = 2a^2 + 4ah + 2h^2 - 4
\]

Now, subtract \( f(a) = 2a^2 - 4 \) from \( f(a+h) \):

\[
f(a+h) - f(a) = (2a^2 + 4ah + 2h^2 - 4) - (2a^2 - 4)
\]

Simplifying:

\[
f(a+h) - f(a) = 2a^2 + 4ah + 2h^2 - 4 - 2a^2 + 4 = 4ah + 2h^2
\]

Now, divide by \( h \) to get the Newton quotient:

\[
N(a, h) = \frac{4ah + 2h^2}{h}
\]

Factor out \( h \) from the numerator:

\[
N(a, h) = \frac{h(4a + 2h)}{h}
\]

Cancel \( h \) from the numerator and denominator:

\[
N(a, h) = 4a + 2h
\]

### (c) Find the limit of the “Newton quotient” as \( h \) approaches zero.

To find the derivative of the function at \( x = a \), take the limit of the Newton quotient as \( h \to 0 \):

\[
\lim_{h \to 0} N(a, h) = \lim_{h \to 0} (4a + 2h)
\]

Since \( 2h \to 0 \) as \( h \to 0 \), the limit is:

\[
\lim_{h \to 0} (4a + 2h) = 4a
\]

Thus, the limit of the Newton quotient as \( h \to 0 \) is \( 4a \), which is the derivative of \( f(x) = 2x^2 - 4 \) at \( x = a \).

---

### Summary:

- **(a)** The Newton quotient for \( f(x) = 2x^2 - 4 \) at \( a \) is:
  \[
  N(a, h) = \frac{[2(a+h)^2 - 4] - [2a^2 - 4]}{h}
  \]
- **(b)** After simplifying and cancelling \( h \), the Newton quotient becomes:
  \[
  N(a, h) = 4a + 2h
  \]
- **(c)** The limit of the Newton quotient as \( h \to 0 \) is:
  \[
  \lim_{h \to 0} N(a, h) = 4a
  \]

Would you like more details on any part of the solution?

### 5 Related Questions:
1. How is the derivative of a quadratic function generally calculated?
2. What is the geometric interpretation of the Newton quotient?
3. Can the Newton quotient be used for non-differentiable functions?
4. How does the result \( 4a \) relate to the slope of the tangent line at \( x = a \)?
5. How would this process change for a higher-degree polynomial?

### Tip:
When simplifying expressions with limits, always factor out terms that can cancel to avoid indeterminate forms (like \( 0/0 \)).

Let f(x) = 2x^2 − 4.
(a) Define the “Newton quotient” for this function at a point a.
(b) Simplify this by cancelling “h” from the numerator and the denominator, where possible.
(c) Find the limit of the “Newton quotient” as “h” approaches zero.

Explore how to calculate the Newton quotient for the quadratic function f(x) = 2x^2 − 4, simplify the expression by canceling terms, and find the derivative by taking the limit as h approaches zero. This problem introduces key concepts of calculus, including the difference quotient and limits, suitable for Grades 11-12 and early college students.

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