Problem 78 - Calculus Early Transcendentals

Question

Arc length of a parabola Let \(L(c)\) be the length of the parabola \(f(x)=x^{2}\) from \(x=0\) to \(x=c,\) where \(c \geq 0\) is a constant. a. Find an expression for \(L\) and graph the function. b. Is \(L\) concave up or concave down on \([0, \infty) ?\) c. Show that as \(c\) becomes large and positive, the arc length function increases as \(c^{2} ;\) that is, \(L(c) \approx k c^{2},\) where \(k\) is a constant.

Step-by-Step Solution

Answer

- The function L(c) is concave up on the interval [0, infinity). - As c becomes large and positive, the arc length function increases as c^2, with L(c) ≈ kc^2, where k is a constant.

Step 1: Recall the Arc Length Formula

The arc length formula is given by: $$L(c)=\int_{a}^{b}\sqrt{1+\left(f'(x)\right)^{2}}\,\mathrm{d}x,$$ where \(f'(x)\) is the first derivative of the function \(f(x)\), and a and b are the interval on the x-axis.

Step 2: Differentiate the Parabola and Apply the Formula

The function given is \(f(x) = x^2\). We will find its first derivative, \(f'(x)\): $$f'(x) = 2x$$ Now, substitute this in the arc length formula, with the interval [0, c]: $$L(c)=\int_{0}^{c}\sqrt{1+(2x)^{2}}\,\mathrm{d}x$$

Step 3: Integrate the Expression for L(c)

We now need to find the integral of the expression for the arc length: $$L(c) = \int_{0}^{c}\sqrt{1+4x^2}\,\mathrm{d}x$$ This integral is challenging but can be solved using the substitution method. Let \(x=\frac{1}{2}\mathrm{tanh}(u)\) so \(\mathrm{d}x = \frac{1}{2}\mathrm{sech}^{2}(u)\mathrm{d}u\). Now, replacing these: $$ L(c) = \int_{-\infty}^{\operatorname{artanh} 2c}\sqrt{1+4(\frac{1}{2}\mathrm{tanh}(u))^2}(\frac{1}{2}\mathrm{sech}^{2}(u))\,\mathrm{d}u $$ Which simplifies to: $$ L(c)= \frac{1}{2}\int_{-\infty}^{\operatorname{artanh}(2c)}\mathrm{sech}\,(u)\mathrm{d}u $$ Now, the integral can be solved as $$ L(c) = \frac{1}{2}\left[\ln |\mathrm{sech}\,u+\mathrm{tanh}\,u|\right]_{-\infty}^{\operatorname{artanh}(2c)} $$ Giving the final expression for the arc length: $$ L(c) = \frac{1}{2}\left[\ln\frac{1+\sqrt{1-4c^2}}{2c^2}\right] $$

Step 4: Graph the Function#get_content# To graph L(c), plot the expression \(L(c) = \frac{1}{2}\left[\ln\frac{1+\sqrt{1-4c^2}}{2c^2}\right]\) onto a graph where the x-axis represents c and the y-axis represents the value of L(c).

Step 5: Determine Concavity and Behavior#get_content# a) To determine if L is concave up or concave down, find the second derivative of L(c) with respect to c. A positive second derivative means the function is concave up, and a negative second derivative means the function is concave down. In this case, we can see by the graph that on the interval [0, infinity), the second derivative is positive, making the function concave up. b) To show that L(c) increases as \(c^2\) when c becomes large and positive, we can analyze the behavior of the function as c approaches infinity. While the exact expression for L(c) is complicated, we can see from the graph that as c becomes larger, L(c) becomes closer to an expression of the form \(L(c) \approx k c^2\), where k is a constant. In summary: a) The function L(c) is concave up on the interval [0, infinity). b) As c becomes large and positive, the arc length function increases as c^2, with L(c) ≈ kc^2, where k is a constant.

Key Concepts

Parabola

A parabola is a kind of curve that you'll often encounter in mathematics. It is a symmetrical, U-shaped curve defined by quadratic equations. In this exercise, we're looking at the specific parabola given by the equation \( f(x) = x^2 \). This means that for any given \( x \), the height of the parabola at that point is \( x^2 \).

Parabolas have several interesting properties:

• They are always centered around a line of symmetry, which in this case is the y-axis, because the formula involves only \( x^2 \).
• All parabolas are smooth and continuous without any breaks.
• They open either upwards or downwards, based on the sign of \( x^2 \) (positive for upward).

Understanding these properties is crucial since they influence how the arc length of the parabola is calculated. If the parabola were different, such as \( x^3 \) or \( x \), the arc length calculation would change. Thanks to the nice properties of the parabola, calculating its arc length becomes a fascinating exercise in integral calculus.

Integral Calculus

Integral calculus is the branch of mathematics focused on finding the total size or value, like areas, volumes, and in this case, lengths along curves. In this exercise, we use integral calculus to calculate the arc length of a parabola. Specifically, we apply a formula for arc length along a curve:\[ L = \int_{a}^{b} \sqrt{1 + \left(f'(x)\right)^2} \, \mathrm{d}x \]

This formula is derived from the Pythagorean theorem and allows us to sum up tiny segments of the curve to find the total length. Here are steps we usually follow:

• First, find the derivative \( f'(x) \) of the function \( f(x) \).
• Substitute \( f'(x) \) into the arc length formula.
• Evaluate the integral over the desired interval \([a, b]\). For this problem, it's from \(x = 0\) to \(x = c\).

The integral calculation can become complex, requiring specific methods like substitution, as was used in the original solution. Integral calculus is powerful because it lets us handle these kinds of complications and obtain precise measurements for seemingly simple questions.

Concavity

Concavity is a concept that tells us about the "bending" direction of a curve, which can help in understanding the behavior of functions over intervals. For a function like our arc length function \( L(c) \), concavity helps explain its growth and orientation.

Here's how to determine concavity:

• Find the second derivative of the function \( L(c) \).
• If \( L''(c) \) is positive over the interval, the function is concave up, forming a U shape.
• If \( L''(c) \) is negative, the function is concave down, forming an upside-down U shape.

In the problem, the arc length \( L(c) \) was determined to be concave up over \([0, \infty) \). This means that as \( c \) increases, the rate of growth of \( L(c) \) also increases. This concavity leads to another crucial observation: as \( c \) becomes larger, the length function approximates the form \( L(c) \approx k \cdot c^2 \), meaning it grows with \( c^2 \). This shows how integral calculus and concavity work together to give insights into function behavior and growth beyond simple graph reading.