/*! 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 100 Recall that \(f\) is even if \(f... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 100

Recall that \(f\) is even if \(f(-x)=f(x),\) for all \(x\) in the domain of \(f,\) and \(f\) is odd if \(f(-x)=-f(x),\) for all \(x\) in the domain of \(f\) a. If \(f\) is a differentiable, even function on its domain, determine whether \(f^{\prime}\) is even, odd, or neither. b. If \(f\) is a differentiable, odd function on its domain, determine whether \(f^{\prime}\) is even, odd, or neither.

Short Answer

Expert verified
Answer: If f is a differentiable, even function, its derivative f鈥� is an odd function. If f is a differentiable, odd function, its derivative f鈥� is an even function.

Step by step solution

01

Recall the definition of an even function

An even function is defined as \(f(-x) = f(x)\) for all \(x\) in its domain.
02

Apply the definition of a differentiable function

A differentiable function has a derivative at every point in its domain. For our even function \(f\), let's consider its derivative \(f^{\prime}(x)\).
03

Investigate the derivative of the even function

We want to determine if \(f^{\prime}(-x) = f^{\prime}(x)\) (even) or \(f^{\prime}(-x) = -f^{\prime}(x)\) (odd) by analyzing the even function, \(f(-x) = f(x)\), and taking its derivative. Differentiate both sides with respect to \(x\): \(\frac{d}{dx}(f(-x)) = \frac{d}{dx}(f(x))\) Using the chain rule, the left side becomes: \(-f^{\prime}(-x) = f^{\prime}(x)\) Since \(-f^{\prime}(-x) = f^{\prime}(x)\), we can conclude that \(f^{\prime}(x)\) is an odd function. #b. If f is a differentiable, odd function on its domain, determine whether f鈥� is even, odd, or neither.#
04

Recall the definition of an odd function

An odd function is defined as \(f(-x) = -f(x)\) for all \(x\) in its domain.
05

Apply the definition of a differentiable function

A differentiable function has a derivative at every point in its domain. For our odd function \(f\), let's consider its derivative \(f^{\prime}(x)\).
06

Investigate the derivative of the odd function

We want to determine if \(f^{\prime}(-x) = f^{\prime}(x)\) (even) or \(f^{\prime}(-x) = -f^{\prime}(x)\) (odd) by analyzing the odd function, \(f(-x) = -f(x)\), and taking its derivative. Differentiate both sides with respect to \(x\): \(\frac{d}{dx}(f(-x)) = \frac{d}{dx}(-f(x))\) Using the chain rule, the left side becomes: \(-f^{\prime}(-x) = -f^{\prime}(x)\) Since \(-f^{\prime}(-x) = -f^{\prime}(x)\), we can conclude that \(f^{\prime}(x)\) is an even function.

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!

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

Suppose \(y=L(x)=a x+b\) (with \(a \neq 0\) ) is the equation of the line tangent to the graph of a one-to-one function \(f\) at \(\left(x_{0}, y_{0}\right) .\) Also, suppose that \(y=M(x)=c x+d\) is the equation of the line tangent to the graph of \(f^{-1}\) at \(\left(y_{0}, x_{0}\right)\) a. Express \(a\) and \(b\) in terms of \(x_{0}\) and \(y_{0}\) b. Express \(c\) in terms of \(a,\) and \(d\) in terms of \(a, x_{0},\) and \(y_{0}\) c. Prove that \(L^{-1}(x)=M(x)\)

An observer stands \(20 \mathrm{m}\) from the bottom of a 10 -m-tall Ferris wheel on a line that is perpendicular to the face of the Ferris wheel. The wheel revolves at a rate of \(\pi\) rad/min and the observer's line of sight with a specific seat on the wheel makes an angle \(\theta\) with the ground (see figure). Forty seconds after that seat leaves the lowest point on the wheel, what is the rate of change of \(\theta ?\) Assume the observer's eyes are level with the bottom of the wheel.

Use the following table to find the given derivatives. $$\begin{array}{llllll} x & 1 & 2 & 3 & 4 & 5 \\ \hline f(x) & 5 & 4 & 3 & 2 & 1 \\ f^{\prime}(x) & 3 & 5 & 2 & 1 & 4 \\ g(x) & 4 & 2 & 5 & 3 & 1 \\ g^{\prime}(x) & 2 & 4 & 3 & 1 & 5 \end{array}$$ $$\left.\frac{d}{d x}\left[\frac{f(x)}{(x+2)}\right]\right|_{x=4}$$

The hands of the clock in the tower of the Houses of Parliament in London are approximately \(3 \mathrm{m}\) and \(2.5 \mathrm{m}\) in length. How fast is the distance between the tips of the hands changing at 9:00? (Hint: Use the Law of cosines.)

One of the Leibniz Rules One of several Leibniz Rules in calculus deals with higher-order derivatives of products. Let \((f g)^{(n)}\) denote the \(n\) th derivative of the product \(f g,\) for \(n \geq 1\) a. Prove that \((f g)^{(2)}=f^{\prime \prime} g+2 f^{\prime} g^{\prime}+f g^{\prime \prime}\) b. Prove that, in general,$$(f g)^{(n)}=\sum_{k=0}^{n}\left(\begin{array}{l} n \\\k\end{array}\right) f^{(k)} g^{(n-k)}$$ where \(\left(\begin{array}{l}n \\\ k\end{array}\right)=\frac{n !}{k !(n-k) !}\) are the binomial coefficients. c. Compare the result of (b) to the expansion of \((a+b)^{n}\).
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Pure 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.