Problem 55 - Calculus Single Variable

Question

Suppose that \(f\) is a function with a continuous derivative on an open interval containing \([a, b] .\) Let \(\mathcal{C}\) be the graph of \(f\) over the interval \([a, b],\) and let \(\mathcal{C}^{\prime}\) be the curve with endpoints \((a+h, f(a)+v)\) and \((b+h, f(b)+v), \quad\) is obtained by translating \(\mathcal{C}\) horizontally by an amount \(h\) and vertically by an amount \(v\). Prove that the lengths of \(\mathcal{C}\) and \(\mathcal{C}^{\prime}\) are equal.

Step-by-Step Solution

Answer

The lengths of \(\mathcal{C}\) and \(\mathcal{C}^{\prime}\) are equal as translations do not affect curve length.

Step 1: Understand the Length of a Curve Formula

The length of a curve \(y = f(x)\) over the interval \([a, b]\) is given by the integral \( L = \int_{a}^{b} \sqrt{1 + [f'(x)]^2} \, dx \). This comes from the Pythagorean theorem considering infinitesimally small line segments of the curve.

Step 2: Define the Curves \(\mathcal{C}\) and \(\mathcal{C}^{\prime}\)

The curve \(\mathcal{C}\) is defined by \((x, f(x))\) over \([a, b]\). The translated curve \(\mathcal{C}^{\prime}\) has points \((x+h, f(x)+v)\) defined over \([a, b]\) after translation.

Step 3: Adjust Curve Equations for Translation

For \(\mathcal{C}^{\prime}\), let's consider \( u = x + h \), which implies \( x = u - h \). Then the function becomes \( g(u) = f(u-h) + v \) and covers the interval \([a+h, b+h]\).

Step 4: Compute the Derivative of the Translated Function

Compute the derivative of the translated function: \(g'(u) = \frac{d}{du}[f(u-h) + v] = f'(u-h)\). The derivative doesn't change magnitude, preserving the modification factor of the original curve.

Step 5: Establish Length Expression for Translated Curve

The length of \(\mathcal{C}^{\prime}\) over \([a+h, b+h]\) is given by \( L_{\mathcal{C}^{\prime}} = \int_{a+h}^{b+h} \sqrt{1 + [g'(u)]^2} \, du \). Substitute \(g'(u) = f'(u-h)\), leading to \(\sqrt{1 + [f'(u-h)]^2}\).

Step 6: Perform Variable Substitution to Compare Lengths

By substituting \(x = u-h\), \( dx = du \), change the limits of integration from \( [a, b] \) to \([a+h, b+h]\). This shows that \( L_{\mathcal{C}^{\prime}} = \int_{a}^{b} \sqrt{1 + [f'(x)]^2} \, dx = L \).

Step 7: Conclude the Equality of Lengths

Since \(\int_{a}^{b} \sqrt{1 + [f'(x)]^2} \, dx\) is identical both with and without translation, the lengths of \(\mathcal{C}\) and \(\mathcal{C}^{\prime}\) remain equal.

Key Concepts

Length of a Curve

The length of a curve is an important concept in integral calculus, often used to determine the distance along a path defined by a function. When we have a function represented by the curve \(y = f(x)\) over an interval \([a, b]\), the length of this curve can be calculated using an integral. This is done by considering the curve as a series of infinitesimally small straight lines. Using the Pythagorean theorem, these small lines' contributions are summed up as an integral of the form:
\[L = \int_{a}^{b} \sqrt{1 + [f'(x)]^2} \, dx\]where \(f'(x)\) denotes the derivative of the function \(f(x)\).
This formula essentially provides the total length of the curve from \(x=a\) to \(x=b\) by accounting for changes in both the horizontal and vertical directions. When a curve is translated, either horizontally or vertically, this length remains unchanged, as shown through the example.

Integral Calculus

Integral calculus is a branch of mathematics focused on the process of finding integrals, which are used to determine quantities like areas under curves, accumulated quantities, and in this context, the total length of a curve. At its core, integral calculus revolves around the fundamental theorem of calculus, which links differentiation and integration. Integration can be considered as the reverse process of differentiation.

• It evaluates the area under or over a curve through summation of infinitesimally small quantities.
• This approach can calculate the total length of a curve by summarizing small line segments' contributions throughout its domain.
• Integral calculus allows us to perform variable substitution, which is a technique used to simplify the computation of integrals by changing variables in a reversible manner.

By utilizing substitution in the curve translation exercise, we confirmed that integral constants or formulas do not change, ensuring that translated curves have identical lengths.

Continuous Derivative

A function having a continuous derivative on an interval means that the derivative of the function exists and does not jump or have gaps throughout that interval. This smoothness is crucial when calculating the length of a curve, as it ensures the function behaves predictably and has a well-defined slope at every point. A continuous derivative, \(f'(x)\), ensures that the function maintains a steady rate of change without abrupt shifts.

• A continuous derivative guarantees that the curve's behavior is predictable and smooth.
• This property is essential for applying integral calculus to compute curve lengths accurately.
• In our specific exercise, the continuity of the derivative allows the length formula \(\sqrt{1 + [f'(x)]^2}\) to remain valid throughout the entire interval.

When translating a curve, this property ensures that any smooth transformations preserve the characteristics required for calculating the curve's length, demonstrating why translations do not affect the total curve length.