We are tasked with finding the tangent line and the normal line to the curve $y = 3x^2 - 2x + 1$ at the point $(1, 2)$.

Step 1: Find the derivative of $y = 3x^2 - 2x + 1$

The derivative of a function represents the slope of the tangent line at any point on the curve.

$y' = \frac{d}{dx} \left( 3x^2 - 2x + 1 \right)$ Using the power rule: $y' = 6x - 2$

Step 2: Find the slope of the tangent line at $x = 1$

To find the slope of the tangent line at $(1, 2)$, we substitute $x = 1$ into the derivative $y'$.

$y'(1) = 6(1) - 2 = 6 - 2 = 4$ So, the slope of the tangent line is 4.

Step 3: Equation of the tangent line

The equation of a line can be written as $y - y_1 = m(x - x_1)$, where $m$ is the slope and $(x_1, y_1)$ is a point on the line.

For the tangent line, we have:

• Slope $m = 4$
• Point $(x_1, y_1) = (1, 2)$

Substitute these values into the point-slope form:

$y - 2 = 4(x - 1)$ Simplifying: $y - 2 = 4x - 4$ $y = 4x - 2$ Thus, the equation of the tangent line is $y = 4x - 2$.

Step 4: Find the slope of the normal line

The slope of the normal line is the negative reciprocal of the slope of the tangent line. Since the slope of the tangent line is 4, the slope of the normal line is:

$m_{\text{normal}} = -\frac{1}{4}$

Step 5: Equation of the normal line

Using the same point $(1, 2)$, we apply the point-slope form for the normal line:

$y - 2 = -\frac{1}{4}(x - 1)$ Simplifying: $y - 2 = -\frac{1}{4}x + \frac{1}{4}$ $y = -\frac{1}{4}x + \frac{9}{4}$ Thus, the equation of the normal line is $y = -\frac{1}{4}x + \frac{9}{4}$.

Summary:

• The equation of the tangent line is: $y = 4x - 2$
• The equation of the normal line is: $y = -\frac{1}{4}x + \frac{9}{4}$

Would you like more details on any of the steps?

Here are 5 related questions to explore:

1. How do you find the tangent line for a trigonometric function?
2. What is the geometric significance of the normal line?
3. How does the second derivative help in analyzing the concavity of curves?
4. How would you find the tangent and normal lines if the curve was parametric?
5. What changes when finding tangents to implicit curves?

Tip: The tangent line is always parallel to the curve at a point, while the normal line is perpendicular to the tangent at that same point.