Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 3: Problem 75
Show that if \(f(x)=|x|\), then \(\left(f^{2}\right)^{\prime}(0)\) exists even though \(f^{\prime}(0)\) does not.
Short Answer
Expert verified
\((f^2)'(0) = 0\) exists even though \(f'(0)\) does not.
Step by step solution
01
Understand the function
The function given is \(f(x) = |x|\), which is the absolute value function. For \(x > 0\), \(f(x) = x\), and for \(x < 0\), \(f(x) = -x\). At \(x = 0\), \(f(x) = 0\). This function is continuous at \(x = 0\) but is not differentiable at \(x = 0\) because the left-hand derivative and right-hand derivative at \(x = 0\) are \(-1\) and \(1\), respectively.
02
Define the square of the function
We need to square the function: \(f^2(x) = (f(x))^2 = |x|^2 = x^2\). The squared function is \(x^2\) for all \(x\), which is a polynomial.
03
Differentiate the squared function
Now, differentiate \(f^2(x) = x^2\) with respect to \(x\). The derivative of \(x^2\) is \(2x\).
04
Evaluate the derivative at zero
Substitute \(x = 0\) into the derivative \(2x\): \(2 \cdot 0 = 0\). Thus, \((f^2)'(0) = 0\).
05
Conclude differentiability
Although \(f'(0)\) does not exist, \((f^2)'(x)\) is \(2x\), which is a polynomial and differentiable everywhere, hence \((f^2)'(0) = 0\) exists.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Differentiability
Differentiability is a key concept in calculus that relates to how a function behaves at specific points. If a function is differentiable at a point, it means the function has a well-defined tangent at that point, indicating a smooth transition or change in its value. For a function to be differentiable at a point, the left-hand and right-hand derivatives at that point must be equal.
However, a function may not always be differentiable even if it is continuous at a point. A classic example is the absolute value function, which is continuous everywhere but not differentiable at the origin. At any point where a function is not differentiable, the graph can have a corner or cusp.
- **Key Characteristics of Differentiability:** - The function must be continuous at the point. - The left-hand limit and right-hand limit of the derivative must exist and be equal.
In the context of the exercise, the function \( f(x) = |x| \) is not differentiable at \( x = 0 \) because the left and right-hand derivatives at this point are \( -1 \) and \( 1 \) respectively. This difference indicates a sharp corner at \( x = 0 \).
However, a function may not always be differentiable even if it is continuous at a point. A classic example is the absolute value function, which is continuous everywhere but not differentiable at the origin. At any point where a function is not differentiable, the graph can have a corner or cusp.
- **Key Characteristics of Differentiability:** - The function must be continuous at the point. - The left-hand limit and right-hand limit of the derivative must exist and be equal.
In the context of the exercise, the function \( f(x) = |x| \) is not differentiable at \( x = 0 \) because the left and right-hand derivatives at this point are \( -1 \) and \( 1 \) respectively. This difference indicates a sharp corner at \( x = 0 \).
Absolute Value Function
The absolute value function, represented as \( f(x) = |x| \), takes any real number and outputs its non-negative value. It plays an essential role in mathematical analysis as it provides the distance of a number from zero on a number line, effectively measuring magnitude.
The function can be split into two linear parts:- If \( x > 0 \), then \( f(x) = x \)- If \( x < 0 \), then \( f(x) = -x \)- At \( x = 0 \), \( f(x) = 0 \)
Understanding these pieces is crucial when analyzing its derivative.
- **Graphical Appearance:** - The graph consists of two linear parts meeting at a point (the origin), forming a 'V'-shape. - Each side of the 'V' corresponds to the linear parts \( x \) and \( -x \), illustrating why the derivative is undefined at \( x = 0 \).
Despite its piecewise nature, when squared, the absolute value function transforms into a simple quadratic form \( f^2(x) = x^2 \), where differentiability concerns are simplified.
The function can be split into two linear parts:- If \( x > 0 \), then \( f(x) = x \)- If \( x < 0 \), then \( f(x) = -x \)- At \( x = 0 \), \( f(x) = 0 \)
Understanding these pieces is crucial when analyzing its derivative.
- **Graphical Appearance:** - The graph consists of two linear parts meeting at a point (the origin), forming a 'V'-shape. - Each side of the 'V' corresponds to the linear parts \( x \) and \( -x \), illustrating why the derivative is undefined at \( x = 0 \).
Despite its piecewise nature, when squared, the absolute value function transforms into a simple quadratic form \( f^2(x) = x^2 \), where differentiability concerns are simplified.
Polynomials
Polynomials are crucial in calculus due to their simplicity and well-defined behaviors, which make them highly tractable for differentiation and integration. A polynomial is an expression consisting of variables and coefficients, constructed using operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
- **Basic Properties of Polynomials:** - They are continuous everywhere. - They are differentiable everywhere. - The derivative of a polynomial is another polynomial.
In the exercise, squaring the absolute value function \( f(x) = |x| \) simplifies it to \( f^2(x) = x^2 \), which is a polynomial of degree 2. This transformation is key because \( x^2 \) assumes differentiability everywhere, including at \( x = 0 \).
- **Why Polynomials Matter:** - The polynomial \( x^2 \) illustrates how operations on non-differentiable functions can result in differentiable ones. - It shows that polynomial derivatives offer smooth and continuous rate-of-change information across their domains.
Thus, while the original function \( f(x) = |x| \) is not differentiable at its peak, its square \( f^2(x) = x^2 \) provides clarity and smoothness, highlighted by the derivative \( (f^2)'(x) = 2x \), which clearly exists even at \( x = 0 \).
- **Basic Properties of Polynomials:** - They are continuous everywhere. - They are differentiable everywhere. - The derivative of a polynomial is another polynomial.
In the exercise, squaring the absolute value function \( f(x) = |x| \) simplifies it to \( f^2(x) = x^2 \), which is a polynomial of degree 2. This transformation is key because \( x^2 \) assumes differentiability everywhere, including at \( x = 0 \).
- **Why Polynomials Matter:** - The polynomial \( x^2 \) illustrates how operations on non-differentiable functions can result in differentiable ones. - It shows that polynomial derivatives offer smooth and continuous rate-of-change information across their domains.
Thus, while the original function \( f(x) = |x| \) is not differentiable at its peak, its square \( f^2(x) = x^2 \) provides clarity and smoothness, highlighted by the derivative \( (f^2)'(x) = 2x \), which clearly exists even at \( x = 0 \).
