https://file.aiquickdraw.com/mathbot/file/1726500364620gk0kmudq.jpg

Let's break down the problem from the image.

### Given Curve:
\( f(x) = x - x^3 \)

#### (a) Find the slope of the tangent line to the curve at the point \( (1, 0) \).

The slope of the tangent line at any point on the curve is given by the derivative of the function \( f(x) \) evaluated at that point. So, let's start by finding the derivative of \( f(x) \).

1. \( f(x) = x - x^3 \)
2. The derivative is:
\[
f'(x) = \frac{d}{dx}(x - x^3) = 1 - 3x^2
\]

Now, evaluate the derivative at \( x = 1 \):
\[
f'(1) = 1 - 3(1)^2 = 1 - 3 = -2
\]
So, the slope of the tangent line at \( (1, 0) \) is **-2**.

#### (b) Find the equation of the tangent line in part (a).

The equation of the tangent line is given by the point-slope form:
\[
y - y_1 = m(x - x_1)
\]
Where:
- \( m \) is the slope (found in part (a) as -2),
- \( (x_1, y_1) = (1, 0) \) is the point of tangency.

Substitute the values:
\[
y - 0 = -2(x - 1)
\]
Simplifying:
\[
y = -2x + 2
\]
So, the equation of the tangent line is \( y = -2x + 2 \).

#### (c) Graph the curve and the tangent line.

From the image, the blue curve represents the function \( f(x) = x - x^3 \), and the red line represents the tangent line at \( (1, 0) \), which is \( y = -2x + 2 \).

Would you like further details or explanations for any of these parts?

Here are 5 related questions you might explore:
1. How do you find the second derivative of \( f(x) = x - x^3 \)?
2. What is the geometric interpretation of the second derivative at \( (1, 0) \)?
3. How does the behavior of the function change as \( x \to \infty \) and \( x \to -\infty \)?
4. What are the critical points of \( f(x) = x - x^3 \), and what do they represent?
5. How do you find the concavity of the curve at different points?

**Tip:** Always verify the slope at the point of tangency by double-checking your derivative calculations!

The given curve is f(x) = x - x^3. (a) Find the slope of the tangent line to the curve at the point (1, 0). (b) Find the equation of the tangent line in part (a). (c) Graph the curve and the tangent line.

Learn how to find the slope and equation of the tangent line to the curve f(x) = x - x^3 at the point (1, 0). This problem covers key calculus concepts, including finding derivatives and using the point-slope form. Suitable for high school calculus students.

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