Question

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 \(L\) b. Is \(L\) concave up or concave down on \([0, \infty) ?\) c. Show that as \(c\) becomes large and positive, the are length function increases as \(c^{2}\); that is, \(L(c) \approx k c^{2},\) where \(k\) is a constant.

Short answer

Question: Determine the concavity of the arc length function L(c) for the parabola described by f(x) = x^2 from x = 0 to x = c. Also, analyze the growth of the arc length function as c becomes large and positive. Answer: The arc length function L(c) is concave up on the interval [0, ∞). As c becomes large and positive, the arc length function increases as c^2.

Step-by-step solution

Determine the arc length formula for a function

Recall that the arc length of a function \(f(x)\) on the closed interval \([a, b]\) can be given as: \[L = \int_{a}^{b} \sqrt{1 + [f'(x)]^2}dx\]

Compute the derivative of f(x)

We are given the function \(f(x) = x^2\). Now, we need to compute its derivative, \(f'(x)\). Using the power rule, we get: \[f'(x) = 2x\]

Plug f'(x) into the arc length formula and simplify

Now, we'll substitute f'(x) into the formula and integrate from 0 to \(c\). The arc length \(L(c)\) is: \[L(c) = \int_{0}^{c} \sqrt{1 + (2x)^2}dx = \int_{0}^{c} \sqrt{1 + 4x^2}dx\] #b. Is L concave up or concave down on [0, ∞) ?#

Determine concavity by taking the second derivative of L(c)

To find the concavity of the function \(L(c)\) on the given interval, we need to take the second derivative of \(L(c)\). First, compute the first derivative of \(L(c)\) using the arc length expression: \[L'(c) = \sqrt{1 + 4c^2}\] We have to compute next the second derivative: \[L''(c) = \frac{\mathrm{d} L'(c)}{\mathrm{d} c} = \frac{8c}{\sqrt{1 + 4c^2}}\] Note that for \(c\geq0\), the second derivative is always positive. Therefore, the function is concave up on \([0, \infty)\) #c. Show that as c becomes large and positive, the are length function increases as \(c^2\); that is, \(L(c) \approx k c^2\), where k is a constant.#

Comparing L(c) to a quadratic function

For large values of \(c\), the term \(1\) in the expression \(\sqrt{1 + 4c^2}\) becomes negligible compared to \(4c^2\). Therefore, as \(c\) becomes large, \(L'(c)\) behaves similarly to: \[L'(c) \approx \sqrt{4c^2} = 2c\] Now, integrate \(L'(c)\) with respect to \(c\): \[L(c) \approx \int 2c\, dc = c^2 + C\] Since we have to find the length of the parabola between \(x=0\) and \(x=c\), it is reasonable to consider that the length is \(0\) when \(x=0\). Thus, we can set \(C=0\). So we can write: \[L(c) \approx c^2\] As \(c\) becomes large, the arc length function \(L(c)\) increases as \(c^2\).

Key concepts

Calculus

Calculus is a fundamental branch of mathematics, focused on the study of change and motion. At the core of calculus are two concepts: differentiation and integration. Differentiation is the process of finding the rate at which a function is changing at any given point (known as a derivative), while integration is essentially the opposite, measuring the accumulation of quantities over an interval. These two operations are inverse to each other and are indispensable in calculating the properties of shapes, motion of objects, and the changes in quantities over time. Calculus allows us to understand the behavior of complex systems and solve problems that would otherwise be impossible. For instance, calculating the arc length of a parabola requires the use of integrals because we are accumulating the length over an interval.

Arc Length Formula

The arc length of a curve in calculus is found by integrating along the path of the curve. Specifically, the arc length formula for a smooth function, f(x), from point a to b is expressed as:
\[L = \int_{a}^{b} \sqrt{1 + [f'(x)]^2}dx\]
Where \(f'(x)\) is the first derivative of the function f(x), which represents the slope at any point. The integral sums up the length of infinitesimally small segments of the curve, giving the total length as the result. This process is a perfect illustration of integration – where we add up an infinite number of infinitesimally small pieces to get a whole.

Concavity of Functions

The concavity of a function refers to the direction in which a curve bends. A function is said to be concave up if it bends upwards like a cup (\(\cup\)) and concave down if it bends downwards like an arch (\(\cap\)). The mathematical way to determine the concavity of a function over an interval is by examining its second derivative. If the second derivative is positive, the function is concave up; if it's negative, the function is concave down.
As shown in the problem's solution, when a function's second derivative is consistently positive (or consistently negative), it reveals the function's uniform concavity across the interval in question. In the case of the arc length function L(c), the second derivative being always positive on \([0, \infty)\) indicates that the function is concave up over that range, meaning it increases at an increasing rate.

Quadratic Function

A quadratic function is a second-degree polynomial of the form f(x) = ax^2 + bx + c, where a, b, and c are constants and \(a eq 0\). The graph of a quadratic function is a parabola that opens upwards if \(a > 0\) and downwards if \(a < 0\). Quadratic functions are commonly seen in various applications, including physics, engineering, and economics.
In the context of our problem, the function \(f(x) = x^2\) is a simple quadratic function whose arc length we interest ourselves in. As \(c\) becomes very large, the arc length of the parabola approximates to the function \(L(c) \approx c^2\). This indicates that despite its inherent curvature, over large intervals, the parabola’s arc length grows in a way that's proportional to a quadratic function – showcasing an interesting link between linear dimensions and the nature of second-degree polynomials.