/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 48 (a) Show that \(f(x)=x^{4 / 3}\)... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 48

(a) Show that \(f(x)=x^{4 / 3}\) is differentiable at \(0,\) but not twice differentiable at 0 (b) Show that \(f(x)=x^{7 / 3}\) is twice differentiable at \(0,\) but not three times differentiable at 0 (c) Find an exponent \(k\) such that \(f(x)=x^{k}\) is \((n-1)\) times differentiable at \(0,\) but not \(n\) times differentiable at 0

Short Answer

Expert verified
(a) Differentiable once; not twice. (b) Twice differentiable; not three times. (c) \( k = n-1 \).

Step by step solution

01

Find the first derivative

Let's find the derivative of \( f(x) = x^{4/3} \). The derivative is given by \( f'(x) = \frac{d}{dx} x^{4/3} = \frac{4}{3}x^{-2/3} \). At \( x = 0 \), the limit \( \lim_{x \to 0} \frac{4}{3}x^{-2/3} \) approaches infinity, indicating the derivative exists at each point other than zero giving us \( f'(x) \) undefined at \( x = 0 \). This calculation shows that \( f(x) \) is differentiable at zero.
02

Check the second derivative for \(f(x) = x^{4/3}\)

Find the second derivative \( f''(x) = \frac{d}{dx} \left( \frac{4}{3}x^{-2/3} \right) = -\frac{8}{9}x^{-5/3} \). Since \( x^{-5/3} \) is not defined at \( x = 0 \), \( f''(x) \) is not defined at \( x = 0 \). Thus, \( f(x) \) is not twice differentiable at zero.
03

Find the first and second derivatives for \(f(x) = x^{7/3}\)

Calculate the first derivative: \( f'(x) = \frac{7}{3}x^{4/3} \), which is defined at \( x = 0 \). Calculate the second derivative: \( f''(x) = \frac{28}{9}x^{1/3} \), which is also defined at \( x = 0 \).
04

Check the third derivative for \(f(x) = x^{7/3}\)

Find the third derivative \( f'''(x) = \frac{28}{27}x^{-2/3} \). This derivative is not defined at \( x = 0 \) since it approaches infinity. Therefore, \( f(x) \) is not three times differentiable at zero.
05

Generalize for exponent \(k\) to find \(f(x) = x^k\)

If \( f(x) = x^k \) is \((n-1)\) times differentiable but not \(n\) times at \( x = 0 \), it must be that the nth derivative changes from finite to undefined at \( x = 0 \). The \(n\)th derivative is of the form \( x^{k-n} \). Thus, \( n \) must equal \( k \) to change differentiability. Set \( k-n+1 = 0 \), solving gives \( k = n-1 \).

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.

Higher Order Derivatives
In calculus, the concept of higher order derivatives extends the idea of the derivative to multiple iterations. While the first derivative of a function provides information about its slope or rate of change, higher order derivatives can reveal more about the behavior and shape of the function.
  • The second derivative indicates the function's concavity, helping us understand how the graph curves.
  • The third derivative, also known as the "jerk," gives insight into the rate of change of acceleration, which can be crucial in physics and engineering.
  • Higher order derivatives continue this pattern, providing even deeper insights into the function's behavior.
For example, if we look at the function \( f(x) = x^{4/3} \), finding the first derivative \( f'(x) = \frac{4}{3}x^{-2/3} \) raises a crucial point: it is undefined at \( x = 0 \). This fact plays a key role when examining differentiability at a specific point like zero, particularly for higher order derivatives.
Non-differentiability Points
Non-differentiability points occur where a function does not have a well-defined derivative. Such points may arise due to discontinuities, sharp turns or cusps in the function.
  • A common cause of non-differentiability is when the derivative at a point like \( x = 0 \) becomes infinite or undefined, as with \( f(x) = x^{4/3} \) and its second derivative.
  • Each higher order derivative must also be checked for definition at the desired point, as seen with \( f(x) = x^{7/3} \), where the third derivative \( f'''(x) = \frac{28}{27}x^{-2/3} \) is undefined at zero.
  • Recognizing non-differentiable points helps in understanding not just the geometry of functions, but also their applicability in real-world modeling, such as physics or financial markets.
Understanding where a function becomes non-differentiable allows us to better predict and understand the limits and behavior of real processes modeled by such functions.
Power Functions
Power functions are expressions of the form \( f(x) = x^k \), where \( k \) is a real number. They are a fundamental component of calculus because they provide simple models for many natural phenomena.
  • The derivatives of power functions follow a straightforward rule: the derivative of \( x^k \) is \( kx^{k-1} \).
  • These functions reveal how differentiability changes with the exponent's value. For instance, \( f(x) = x^{4/3} \) is differentiable at zero but not twice, while \( f(x) = x^{7/3} \) is twice differentiable but fails the third derivative at zero.
  • Understanding power functions is essential for solving various calculus problems, especially those involving limits and series expansions.
Exploring how derivatives apply to power functions deepens comprehension of calculus, paving the way for tackling more complex mathematical problems in physics, engineering, and economics.
Calculus Problems
Calculus problems often challenge students by asking them to explore differentiability and continuity, especially regarding higher order derivatives. These problems serve as a foundation for understanding deeper mathematical concepts.
  • Such exercises typically involve showing if functions like \( x^k \) are differentiable at certain points, and to what extent (first, second, or higher orders).
  • They highlight the importance of identifying where and why a function may not be differentiable, enhancing comprehension and application of core calculus principles.
  • By solving these problems, learners develop a better grasp of how calculus tools can be applied to analyze and model real scenarios.
Addressing these problems effectively requires a mastery of basic calculus rules and the ability to generalize findings to various situations, providing a comprehensive skill set for future mathematical ventures.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Use implicit differentiation to find the specified derivative. $$\sqrt{u}+\sqrt{v}=5 ; d u / d v$$

Verify that L'H么pital's rule is of no help in finding the limit, then find the limit, if it exists, by some other method. $$\lim _{x \rightarrow+\infty} \frac{x+\sin 2 x}{x}$$

A particle is moving along the curve \(16 x^{2}+9 y^{2}=144\) Find all points \((x, y)\) at which the rates of change of \(x\) and y with respect to time are equal. I Assume that \(d x / d t\) and dy/dt are never both zero at the same point. \(]\)

Find all rational values of \(r\) such that \(y=x^{r}\) satisfies the given equation. $$3 x^{2} y^{\prime \prime}+4 x y^{\prime}-2 y=0$$

Use a calculating utility and the change of base formula (9) to find the values of \(\log _{2} 7.35\) and \(\log _{5} 0.6,\) rounded to four decimal places.
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Mechanics Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.