Problem 45 - Essential Calculus

Question

The function \(f(x)=\left\\{\begin{array}{ll}0, & x=0 \\ 1-x, & 0

Step-by-Step Solution

Answer

No, this does not contradict Rolle's Theorem. Even though the function is differentiable in (0, 1) and f(0) = f(1), the function is not continuous at $x = 0$. Thus, the function doesn't meet all the conditions of Rolle's theorem.

Step 1: Understand the Function

Firstly, understand the given function $f(x) = \begin{cases} 0, & x=0 \\ 1-x, & 0

Step 2: Evaluate the Derivative

Next, calculate the derivative of the function in its domain where it's differentiable. Its derivative, for $0 < x \leq 1$, is $f'(x) = -1$, which is never 0.

Step 3: Analyze Rolle's Theorem

According to Rolle's theorem, there should exist at least one point c in the interval (0,1) where the derivative of the function is zero. However, in this case, it seems there is no such c that makes $f'(c) = 0$. Hence, we have to analyze the apparent contradiction.

Step 4: Resolve the Discrepancy

Upon looking closely into the conditions of Rolle's theorem, we notice that besides differentiability in interval (0,1) and $f(0) = f(1)$, there’s another condition - the function must also be continuous on the closed interval [0, 1]. However, the function $f(x)$ given is not continuous at $x = 0$, as on the immediate right of 0, the function value is 1 and not 0. Thus, the function does not satisfy all the conditions of Rolle's theorem and hence there's no contradiction.

Key Concepts

Piecewise Functions

In calculus, piecewise functions are mathematical expressions defined by multiple sub-functions, each of which applies to a certain interval of the main function's domain. They are like a collage of functions, each piece neatly defined within specific parts of the x-axis.

For example, the function
\( f(x) = \begin{cases} 0, & x=0 \ 1-x, & 0<x \leq 1 \end{cases} \)
is a piecewise function because it has two different expressions: one for \( x=0 \) and another for the interval \( 0<x \leq 1 \). While these functions can sometimes appear complex, they describe situations where a rule changes depending on the input value — similar to how postage fees might change based on the weight of the package.

Differentiability

Differentiability is a cornerstone of calculus, and it refers to the ability to calculate a derivative at a point or over an interval. A function is differentiable at a point if it has a defined slope there — think of it as being able to sketch a perfect, non-breaking tangent line at that spot on the curve.

For the function given in our exercise, the derivative when \( 0<x \leq 1 \) is \( f'(x) = -1 \). However, at \( x=0 \), we don't define a derivative because the rule for the function shifts. The absence of differentiability can sometimes throw a wrench in the application of certain theorems and needs to be considered when analyzing the behavior of a function.

Continuity

Continuity is another essential concept in calculus, which is intimately related to differentiability. A function is continuous at a point when you can draw its graph at that point without lifting your pencil — the function's value flows unbroken through that point.

In our exercise, the function \( f(x) \) shows a peculiarity: at \( x=0 \), the function is defined outright as zero. However, immediately to the right, for any positive value, no matter how small, the function jumps to \( 1-x \). This jump indicates a discontinuity at \( x=0 \). For Rolle's Theorem to apply, a function must be continuous on the closed interval, which is not the case here.

Calculus

Calculus is a branch of mathematics focused on change and motion. Through its fundamental operations — differentiation and integration — calculus provides tools for analyzing the behavior of functions. It is essential for solving real-world problems in physics, engineering, economics, statistics, and beyond.

The problem at hand involves applying one of calculus's famous theorems, Rolle's Theorem, which states that if a function is continuous on a closed interval and differentiable on the open interval, then there's at least one point where the derivative (slope) is zero. Understanding this theorem's requirements, as they relate to continuity and differentiability, underscores the logic and beauty of calculus.

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Problem 45

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Key Concepts

Piecewise Functions

In calculus, piecewise functions are mathematical expressions defined by multiple sub-functions, each of which applies to a certain interval of the main function's domain. They are like a collage of functions, each piece neatly defined within specific parts of the x-axis.

For example, the function
\( f(x) = \begin{cases} 0, & x=0 \ 1-x, & 0<x \leq 1 \end{cases} \)
is a piecewise function because it has two different expressions: one for \( x=0 \) and another for the interval \( 0<x \leq 1 \). While these functions can sometimes appear complex, they describe situations where a rule changes depending on the input value — similar to how postage fees might change based on the weight of the package.

Differentiability

Differentiability is a cornerstone of calculus, and it refers to the ability to calculate a derivative at a point or over an interval. A function is differentiable at a point if it has a defined slope there — think of it as being able to sketch a perfect, non-breaking tangent line at that spot on the curve.

For the function given in our exercise, the derivative when \( 0<x \leq 1 \) is \( f'(x) = -1 \). However, at \( x=0 \), we don't define a derivative because the rule for the function shifts. The absence of differentiability can sometimes throw a wrench in the application of certain theorems and needs to be considered when analyzing the behavior of a function.

Continuity

Continuity is another essential concept in calculus, which is intimately related to differentiability. A function is continuous at a point when you can draw its graph at that point without lifting your pencil — the function's value flows unbroken through that point.

In our exercise, the function \( f(x) \) shows a peculiarity: at \( x=0 \), the function is defined outright as zero. However, immediately to the right, for any positive value, no matter how small, the function jumps to \( 1-x \). This jump indicates a discontinuity at \( x=0 \). For Rolle's Theorem to apply, a function must be continuous on the closed interval, which is not the case here.

Calculus

Calculus is a branch of mathematics focused on change and motion. Through its fundamental operations — differentiation and integration — calculus provides tools for analyzing the behavior of functions. It is essential for solving real-world problems in physics, engineering, economics, statistics, and beyond.

The problem at hand involves applying one of calculus's famous theorems, Rolle's Theorem, which states that if a function is continuous on a closed interval and differentiable on the open interval, then there's at least one point where the derivative (slope) is zero. Understanding this theorem's requirements, as they relate to continuity and differentiability, underscores the logic and beauty of calculus.