Problem 51 - A Graphical Approach to Precalculus with Limits

Question

Find the equation of the tangent line to the function \(f\) at the given point. Then graph the function and the tangent line together. $$f(x)=x-x^{2} \text { at }(-1,-2)$$

Step-by-Step Solution

Answer

The equation of the tangent line is \(y = 3x + 1\).

Step 1: Verify the Point on the Curve

Check if the point \((-1, -2)\) lies on the curve defined by \(f(x) = x - x^2\). Substitute \(x = -1\) into the function to ensure the point is valid: \(f(-1) = -1 - (-1)^2 = -1 - 1 = -2\). Thus, the point \((-1, -2)\) is indeed on the curve.

Step 2: Find the Derivative of the Function

Find the derivative of the function \(f(x) = x - x^2\) to determine the slope of the tangent line. The derivative \(f'(x)\) is calculated as follows: \\[ f'(x) = \frac{d}{dx}(x - x^2) = 1 - 2x.\]

Step 3: Calculate the Slope at the Given Point

Substitute \(x = -1\) into the derivative to calculate the slope \(m\) of the tangent line at the point \((-1, -2)\): \\[ m = f'(-1) = 1 - 2(-1) = 1 + 2 = 3.\] Therefore, the slope of the tangent line is 3.

Step 4: Use the Point-Slope Form to Find the Equation

Use the point-slope form of a line, \(y - y_1 = m(x - x_1)\), with the slope \(m = 3\) and point \((-1, -2)\) to find the equation: \\[ y - (-2) = 3(x - (-1)), \]which simplifies to \\[ y + 2 = 3(x + 1).\] Further simplification gives \\[ y = 3x + 3 - 2 \Rightarrow y = 3x + 1.\] Thus, the equation of the tangent line is \(y = 3x + 1\).

Step 5: Graph the Function and the Tangent Line

Graph the function \(f(x) = x - x^2\), a downward-opening parabola, along with the tangent line \(y = 3x + 1\). Ensure the tangent line intersects the function precisely at \((-1, -2)\). This visual confirmation ensures the tangent line is correctly plotted.

Key Concepts

Derivative Calculation

When you want to find the equation of a tangent line to a function, one of the first steps is to calculate the derivative. The derivative is a powerful tool in calculus that gives us the slope of the tangent line at any given point on the curve. For our function, which is \(f(x) = x - x^2\), the derivative is found by applying the power rule. This rule states that for any term like \(ax^n\), the derivative is \(anx^{n-1}\).
So, for our function, the derivative calculation proceeds as follows:

• For the term \(x\), the derivative is simply 1, since the power of \(x\) is 1.
• For \(-x^2\), using the power rule gives \(-2x\), because you bring down the 2 and reduce the power by one.

Thus, the derivative of the function is \(f'(x) = 1 - 2x\).
This derivative will help to find the slope of the tangent line when evaluated at a specific point.

Point-Slope Form

The point-slope form of a line is extremely useful when you know a point on the line and the slope. It looks like this: \(y - y_1 = m(x - x_1)\). Here, \((x_1, y_1)\) is a point on the line, and \(m\) is the slope.
In the given exercise, we've determined the slope \(m\) to be 3 (after evaluating the derivative at \(x = -1\)). When using

• \((-1, -2)\) as the point \((x_1, y_1)\).
• the slope \(m\) as 3,

the point-slope form becomes \(y - (-2) = 3(x - (-1))\).
This simplifies further to \(y + 2 = 3(x + 1)\). Simplifying, you'll find the final equation of the tangent line: \(y = 3x + 1\).
Point-slope form is versatile and allows for quick transition to the line's equation with minimal fuss.

Function Graphing

Graphing functions and their tangent lines gives concrete insight into how these mathematical elements behave together. For the function \(f(x) = x - x^2\), the graph forms a parabola. More specifically, it opens downward due to the negative sign in front of the \(x^2\) term.
When you're graphing:

• Plot points to get the shape of the parabola. You might select points such as \(-1\), \(0\), \(1\) to see how they plot.
• Graph the tangent line \(y = 3x + 1\) alongside the parabola.

Ensure the tangent line makes contact at \((-1, -2)\), just brushing the curve. This means that it should not intersect again. Both components in the graph help verify the math visually.

Parabola

The function \(f(x) = x - x^2\) is very typical of a parabolic shape, specifically a downward-facing one due to the negative \(x^2\). Parabolas are a significant topic in mathematics, modeling many natural phenomena.

Key features of parabolas:

• They have a symmetric axis, often a vertical line through a vertex point.
• The vertex for \(f(x) = x - x^2\) can be found using vertex formula or completing the square.
• In our case, the vertex is at \((0.5, -0.25)\).

Understanding the vertex is important because it is the highest or lowest point, depending on the parabola's orientation. For the exercise, using these properties assists in sketching and comprehending the full picture of the function's behavior.