We are given the curve $y = f(x) = x^3 - x + 3$ and need to:

(a) Find the equation of the tangent line at the point $(1, 3)$

To find the equation of the tangent line at a given point, we need the slope of the tangent, which is the derivative of $f(x)$ evaluated at the point $x = 1$.

1. Find the derivative $f'(x)$: $f'(x) = \frac{d}{dx}(x^3 - x + 3)$ Using basic differentiation rules: $f'(x) = 3x^2 - 1$

2. Evaluate the derivative at $x = 1$ to find the slope of the tangent line: $f'(1) = 3(1)^2 - 1 = 3 - 1 = 2$ So, the slope of the tangent line at $x = 1$ is $2$.

3. Use the point-slope form of the line equation: The point-slope form of the equation of a line is: $y - y_1 = m(x - x_1)$ where $m$ is the slope and $(x_1, y_1)$ is the point on the line. Here, $m = 2$ and the point is $(1, 3)$, so: $y - 3 = 2(x - 1)$

   Simplifying this: $y - 3 = 2x - 2$ $y = 2x + 1$ Therefore, the equation of the tangent line is $y = 2x + 1$.

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(b) Find the equation of the normal line at the point $(1, 3)$

The normal line is perpendicular to the tangent line. If the slope of the tangent line is $m_t$, then the slope of the normal line, $m_n$, is the negative reciprocal: $m_n = -\frac{1}{m_t}$

Since the slope of the tangent line is 2, the slope of the normal line is: $m_n = -\frac{1}{2}$

Now, we use the point-slope form of the line equation again, with slope $m_n = -\frac{1}{2}$ and the point $(1, 3)$: $y - 3 = -\frac{1}{2}(x - 1)$

Simplifying this: $y - 3 = -\frac{1}{2}x + \frac{1}{2}$ $y = -\frac{1}{2}x + \frac{7}{2}$ Therefore, the equation of the normal line is $y = -\frac{1}{2}x + \frac{7}{2}$.

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Summary of Results:

• The tangent line at $(1, 3)$ is $y = 2x + 1$.
• The normal line at $(1, 3)$ is $y = -\frac{1}{2}x + \frac{7}{2}$.

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Would you like further details or clarification on any of the steps? Here's a list of related questions for you to explore:

1. How do you find the slope of the tangent to a curve?
2. What is the geometric meaning of the normal line?
3. How do you determine the derivative of more complex polynomial functions?
4. How can you apply these concepts in real-world problems, like velocity or optimization?
5. What are other forms of the equation of a line besides point-slope form?

Tip: Always double-check your differentiation and arithmetic when working with derivatives, as small errors in these steps can lead to incorrect results in both tangent and normal line equations!