Question

Determine whether the following statements are true and give an explanation or counterexample. a. \(\int_{a}^{b} \sqrt{1+f^{\prime}(x)^{2}} d x=\int_{a}^{b}\left(1+f^{\prime}(x)\right) d x.\) b. Assuming \(f^{\prime}\) is continuous on the interval \([a, b]\), the length of the curve \(y=f(x)\) on \([a, b]\) is the area under the curve \(y=\sqrt{1+f^{\prime}(x)^{2}}\) on \([a, b].\) c. Arc length may be negative if \(f(x)<0\) on part of the interval in question.

Short answer

Provide a brief explanation. Answer: False. Their derivatives with respect to x are different, so these two expressions are not the same. 2. True or false: "The arc length of the curve y = f(x) on the interval [a, b] is equal to the area under the curve y = √(1 + f'(x)²) on the same interval." Provide a brief explanation. Answer: True. The formula for finding the arc length of a curve y=f(x) is given by L = ∫[a, b] √(1 + f'(x)²) dx, which is the same as the given statement. 3. Is the statement "Arc length can be negative if f(x) < 0" true or false? Provide a brief explanation. Answer: False. Arc length represents a distance which is always non-negative, and it depends on the behavior of the first derivative (f'(x)), not the actual function values.

Step-by-step solution

a. Validate the Equality

Note that the left side of the equation represents the integral of \(\sqrt{1+f'(x)^2}\) while the right side represents the integral of \(1+f'(x)\). To see if these two expressions are the same, let us consider their derivatives with respect to \(x\). The left side would yield \(\frac{d}{dx}\sqrt{1+f'(x)^2} = \frac{f'(x)f''(x)}{\sqrt{1 + f'(x)^2}}\). For the right side, the derivative would be \(\frac{d}{dx}(1 + f'(x)) = f''(x)\). These two expressions are different, which means the given statement is incorrect. Therefore, the correct answer is "false".

b. Arc Length as Area under the Curve

The statement says that the arc length of the curve \(y = f(x)\) on the interval \([a, b]\) is equal to the area under the curve \(y = \sqrt{1 + f'(x)^2}\) on the same interval. The formula for finding the arc length of a curve \(y=f(x)\) is given by: $$L = \int_a^b \sqrt{1 + f'(x)^2} dx$$ Comparing this with the given statement, we see that they indeed have the same form. Since the function \(f'(x)\) is continuous, this integral exists. Therefore, the statement is "true".

c. Negative Arc Length

Arc length is defined as the length along a curve and represents a distance. Distances are always positive (or zero, if the length is degenerate) quantities and can never be negative. The statement suggests that arc length may be negative if \(f(x) < 0\). However, arc length does not depend on the actual function values, but rather on the first derivative \((f'(x))\) and its behavior. Therefore, the statement is "false".

Key concepts

Integration

Integration is a fundamental concept in calculus, allowing us to find areas under curves, among other things. In this exercise, one focus is on comparing the formula \[ \int_{a}^{b} \sqrt{1+f^{\prime}(x)^{2}} \, dx \] with another integral of the form \[ \int_{a}^{b} (1+f^{\prime}(x)) \, dx. \]Understanding integration helps clarify why these two expressions aren't equivalent. While both involve calculating definite integrals over the interval \([a, b]\), the distinct integrands mean they solve different problems. The square root in the first integral captures changes due to both horizontal and vertical displacements along the curve. On the other hand, the second integral, with its linear term, simply evaluates the sum of 1 with the derivative.

When we integrate, we're essentially finding accumulated values, but it's crucial to recognize what quantity. Integration can give insights into physical properties such as distance, area, and volume, depending on the function you're integrating. Thus, by integrating \( \sqrt{1+f^{\prime}(x)^{2}} \), we actually measure arc length, a geometric property, rather than just sum the changes in the function.

It's important to interpret these integrals correctly based on their structures to understand the distinct information each conveys.

Derivatives

Derivatives are at the heart of calculus. They express how functions change and are crucial for understanding motion, growth, and optimization. In this exercise, \( f'(x) \) is the derivative of \( f(x) \), describing its slope at any point. When you differentiate a function, you're finding the instantaneous rate of change or slope of the tangent line to the curve. This concept applies directly when determining if two integrals are the same in the exercise.

The exercise highlights the misunderstanding of how derivatives behave. For part (a), both derivatives of \( \sqrt{1+f'(x)^2} \) and \( 1+f'(x) \)are distinct:* The derivative of \( \sqrt{1+f'(x)^2} \) is \( \frac{f'(x)f''(x)}{\sqrt{1 + f'(x)^2}} \), depending on both the first and second derivatives.* Whereas, the derivative of \( 1 + f'(x) \) is simply \( f''(x) \), indicating a linear rate of change.

This difference shows how derivatives affect arc length calculation by connecting to motion along a curve. Recognizing these links between derivations, integrals, and geometry is crucial for correctly solving calculus problems.

Curve Analysis

Curve analysis involves understanding the geometric properties of curves beyond mere function plotting. In calculus, we often study characteristics like slope, curvature, and notably in this case, arc length. Arc length measures the "distance" along a curve from one point to another, crucial for understanding shapes and trajectories. The arc length formula \( L = \int_a^b \sqrt{1+f'(x)^2} \, dx \)quantifies this distance by incorporating both changes in height and width as we move along the curve.

* **Arc Length Calculation**: It's calculated using the derivative \( f'(x) \) to understand both vertical and horizontal changes. Because of the square root in the formula, arc length considers diagonal "steps," unlike a simple measure of height or width.

* **Misconceptions Cleared**: Part (c) of the original exercise illustrates common misconceptions, such as arc length being negative. Arc length, like any distance, is always non-negative, regardless of whether a function \( f(x) \) is below the \( x \)-axis at certain intervals.

Through curve analysis, we gain deeper insight into the fundamental spatial nature of calculus and how derivatives and integrals enable us to describe such properties precisely. This understanding is essential for applying calculus to real-world problems like engineering, physics, and architecture.