Consider the parabola y = 4x − x^2. Find the slope m of the tangent line to the parabola at the point (1, 3).

The equation of the parabola is given as:

\[
y = 4x - x^2
\]

To find the slope \( m \) of the tangent line to the parabola at a specific point, we first need to compute the derivative of the function \( y \) with respect to \( x \). The derivative \( \frac{dy}{dx} \) represents the slope of the tangent line at any point on the curve.

The derivative of \( y = 4x - x^2 \) is:

\[
\frac{dy}{dx} = 4 - 2x
\]

Now, we evaluate the derivative at the given point \( x = 1 \):

\[
\frac{dy}{dx}\bigg|_{x=1} = 4 - 2(1) = 4 - 2 = 2
\]

Therefore, the slope \( m \) of the tangent line to the parabola at the point \( (1, 3) \) is:

\[
m = 2
\]

Would you like more details on the tangent line or any other aspect of this problem?

Here are 5 related questions to explore further:
1. How do you find the equation of the tangent line given a point and the slope?
2. What is the geometric significance of the derivative in terms of curves?
3. How can you find the vertex of the parabola \( y = 4x - x^2 \)?
4. What is the relationship between the second derivative and the concavity of a curve?
5. How would the slope of the tangent line change if the point was \( (2, 4) \)?

**Tip**: The slope of a tangent line at any point on a curve is always equal to the value of the derivative at that point.

Discover how to calculate the slope of the tangent line to the parabola y = 4x − x^2 at the point (1, 3). This detailed solution involves finding the derivative and evaluating it at the given point. Suitable for high school students learning calculus and derivatives.

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