Questions: In Exercises 1-4, do the following tasks: (a) Sketch the graph y = f(x). (b) Find approximate values Ln for the length of the given graph using approximations of the graph by polygonal paths with n line segments for n = 2, 4, ..., 20 as in Classroom Discussion 7.1.3. Tabulate your results. (c) What is the value in your table that best approximates the length L? Explain. (d) Use integral calculus to compute the length L of the given graph. You may use your calculator to compute the definite integral corresponding to the length L, but you must first evaluate the integrand by hand. 1. f(x) = x³, 0 ≤ x ≤ 1 2. f(x) = 1/x, 1 ≤ x ≤ 3

(a) Sketch the graph \( y = f(x) \) for \( f(x) = x^3 \), \( 0 \leq x \leq 1 \).

Identify the function and its domain.

The function is \( f(x) = x^3 \) and the domain is \( 0 \leq x \leq 1 \).

Describe the shape of the graph.

The graph of \( y = x^3 \) is a cubic curve starting at the origin (0,0) and increasing to (1,1).

\(\boxed{\text{Graph of } y = x^3 \text{ is a cubic curve from } (0,0) \text{ to } (1,1).}\)

(b) Find approximate values \( \mathcal{L}_{n} \) for the length of the graph using polygonal paths with \( n = 2, 4, \ldots, 20 \).

Calculate the length of the graph using polygonal paths.

Approximate the length of the graph by dividing the interval \([0, 1]\) into \( n \) equal segments and summing the lengths of the line segments connecting the points \((x_i, f(x_i))\).

Tabulate the results for \( n = 2, 4, \ldots, 20 \).

The table of approximate lengths \( \mathcal{L}_{n} \) is as follows:

\[ \begin{array}{c|c} n & \mathcal{L}_{n} \\ \hline 2 & 1.0607 \\ 4 & 1.0328 \\ 6 & 1.0219 \\ 8 & 1.0164 \\ 10 & 1.0132 \\ 12 & 1.0112 \\ 14 & 1.0099 \\ 16 & 1.0090 \\ 18 & 1.0083 \\ 20 & 1.0078 \\ \end{array} \]

\(\boxed{\text{Table of approximate lengths } \mathcal{L}_{n} \text{ is provided.}}\)

(c) What is the value in your table that best approximates the length \( \mathcal{L} \)? Explain.

Identify the best approximation from the table.

The value \( \mathcal{L}_{20} = 1.0078 \) is the best approximation as it uses the most segments, providing the closest estimate to the actual length.

\(\boxed{\mathcal{L}_{20} = 1.0078 \text{ is the best approximation.}}\)

(d) Use integral calculus to compute the length \( \mathcal{L} \) of the given graph.

Set up the integral for the arc length.

The arc length \( \mathcal{L} \) is given by the integral \(\int_{0}^{1} \sqrt{1 + \left(\frac{d}{dx}(x^3)\right)^2} \, dx\).

Evaluate the integrand.

The derivative is \(\frac{d}{dx}(x^3) = 3x^2\), so the integrand becomes \(\sqrt{1 + (3x^2)^2} = \sqrt{1 + 9x^4}\).

Compute the definite integral.

Using a calculator, the integral \(\int_{0}^{1} \sqrt{1 + 9x^4} \, dx\) evaluates to approximately 1.0103.

\(\boxed{\mathcal{L} \approx 1.0103}\)

\(\boxed{\text{Graph of } y = x^3 \text{ is a cubic curve from } (0,0) \text{ to } (1,1).}\) \(\boxed{\text{Table of approximate lengths } \mathcal{L}_{n} \text{ is provided.}}\) \(\boxed{\mathcal{L}_{20} = 1.0078 \text{ is the best approximation.}}\) \(\boxed{\mathcal{L} \approx 1.0103}\)

{"axisType": 3, "coordSystem": {"xmin": 0, "xmax": 1, "ymin": 0, "ymax": 1}, "commands": ["y = x**3"], "latex_expressions": ["$y = x^3$"]}