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Let's solve Problem 34 step by step:

Given:
\[
f(x) = x^4 + 2x - 1 \quad \text{at the point } (1, 2)
\]

### Part (a): Find the slope of the tangent line
To find the slope of the tangent line, we need the derivative of \( f(x) \), which gives us the slope of the tangent at any point \( x \).

1. Find the derivative of \( f(x) \):
\[
f'(x) = \frac{d}{dx}\left(x^4 + 2x - 1\right)
\]
Using the power rule:
\[
f'(x) = 4x^3 + 2
\]

2. Evaluate \( f'(x) \) at \( x = 1 \):
\[
f'(1) = 4(1)^3 + 2 = 4 + 2 = 6
\]
So, the slope of the tangent line at \( x = 1 \) is **6**.

### Part (b): Find the equation of the tangent line
To find the equation of the tangent line, we use the point-slope form of a line:
\[
y - y_1 = m(x - x_1)
\]
where \( m \) is the slope, and \( (x_1, y_1) \) is the point \( (1, 2) \).

Substitute \( m = 6 \), \( x_1 = 1 \), and \( y_1 = 2 \):
\[
y - 2 = 6(x - 1)
\]
Simplifying:
\[
y - 2 = 6x - 6 \quad \Rightarrow \quad y = 6x - 4
\]
So, the equation of the tangent line is:
\[
y = 6x - 4
\]

### Part (c): Find the equation of the normal line
The normal line is perpendicular to the tangent 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 6, the slope of the normal line is:
\[
m_{\text{normal}} = -\frac{1}{6}
\]

Using the point-slope form again, with \( (1, 2) \) and the slope \( -\frac{1}{6} \):
\[
y - 2 = -\frac{1}{6}(x - 1)
\]
Simplifying:
\[
y - 2 = -\frac{1}{6}x + \frac{1}{6} \quad \Rightarrow \quad y = -\frac{1}{6}x + \frac{13}{6}
\]
So, the equation of the normal line is:
\[
y = -\frac{1}{6}x + \frac{13}{6}
\]

### Part (d): Graph the function, tangent line, and normal line
To graph these, we would plot:
1. The function \( f(x) = x^4 + 2x - 1 \).
2. The tangent line \( y = 6x - 4 \).
3. The normal line \( y = -\frac{1}{6}x + \frac{13}{6} \).

I can generate a plot of these functions if you'd like.

Would you like me to generate this graph or clarify any step further?

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Here are 5 follow-up questions for practice:
1. What is the geometric interpretation of a tangent line?
2. How do we find the slope of a normal line?
3. How does the power rule apply to different polynomial functions?
4. What happens to the tangent line when the slope is 0?
5. Can you find the tangent line to \( f(x) = x^4 + 2x - 1 \) at a different point?

**Tip**: When finding the tangent and normal lines, always ensure you're using the correct form of the slope—it's easy to confuse tangent and normal slopes!

Solve the tangent and normal lines problem for f(x) = x^4 + 2x - 1 at the point (1, 2).

Learn how to solve a calculus problem involving finding the tangent and normal lines to the function f(x) = x^4 + 2x - 1 at the point (1, 2). This problem involves derivative calculations and applying point-slope formulas. Suitable for high school students in calculus courses.

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