let f(x)=1-x^2. Find the difference quotient from x to h+h

Given the function \( f(x) = 1 - x^2 \), we want to find the difference quotient from \( x \) to \( h + h \) (which is essentially \( x + 2h \)).

The difference quotient is given by:

\[
\frac{f(x+2h) - f(x)}{(x+2h) - x}
\]

### Step 1: Compute \( f(x+2h) \)

Substitute \( x + 2h \) into \( f(x) \):

\[
f(x + 2h) = 1 - (x + 2h)^2
\]

Expanding \( (x + 2h)^2 \):

\[
(x + 2h)^2 = x^2 + 4xh + 4h^2
\]

So,

\[
f(x + 2h) = 1 - (x^2 + 4xh + 4h^2) = 1 - x^2 - 4xh - 4h^2
\]

### Step 2: Compute the difference \( f(x+2h) - f(x) \)

Now subtract \( f(x) \) from \( f(x+2h) \):

\[
f(x + 2h) - f(x) = (1 - x^2 - 4xh - 4h^2) - (1 - x^2)
\]

Simplifying:

\[
f(x + 2h) - f(x) = 1 - x^2 - 4xh - 4h^2 - 1 + x^2
\]

\[
f(x + 2h) - f(x) = -4xh - 4h^2
\]

### Step 3: Divide by \( (x + 2h) - x = 2h \)

Now, compute the difference quotient:

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

Simplifying:

\[
\frac{-4xh - 4h^2}{2h} = -2x - 2h
\]

### Final Answer:

The difference quotient from \( x \) to \( x + 2h \) is:

\[
-2x - 2h
\]

Would you like more details or have any questions?

Here are 5 related questions you can explore:
1. What is the general form of the difference quotient for any function \( f(x) \)?
2. How do you apply the difference quotient to polynomial functions?
3. What happens to the difference quotient as \( h \to 0 \)?
4. How is the difference quotient related to the derivative of a function?
5. Can you find the derivative of \( f(x) = 1 - x^2 \) using the difference quotient?

**Tip:** Always simplify algebraic expressions carefully before substituting them into the difference quotient to avoid mistakes.

Discover how to find the difference quotient for the function f(x) = 1 - x^2 from x to x + 2h, with a detailed step-by-step solution. This problem involves expanding a polynomial function and applying the difference quotient formula. Suitable for high school students in Grades 10-12.

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