/*! 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 1 \(y=|\cos x|+|\sin x|\) \(y=\s... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 1

\(y=|\cos x|+|\sin x|\) \(y=\sin x-\cos x\) is the neighbourhood of \(x=\frac{2 \pi}{3}\) \(y^{\prime}=\cos x+\sin x\) \(y^{\prime}=\frac{-1}{2}+\frac{\sqrt{3}}{2}\)

Short Answer

Expert verified
Answer: The value of the second function at \(x=\frac{2\pi}{3}\) is \(\frac{1}{2}+\frac{\sqrt{3}}{2}\), and the slope of the function at this point is \(\frac{-1}{2}+\frac{\sqrt{3}}{2}\).

Step by step solution

01

Evaluate the functions at \(x=\frac{2\pi}{3}\)

We first need to evaluate both functions at the given point \(x=\frac{2\pi}{3}\): $$y_1=|\cos\left(\frac{2\pi}{3}\right)|+|\sin\left(\frac{2\pi}{3}\right)|$$ $$y_2=\sin\left(\frac{2\pi}{3}\right)-\cos\left(\frac{2\pi}{3}\right)$$
02

Simplify and confirm equality

Simplify both expressions: $$y_1=|-\frac{1}{2}|+|\frac{\sqrt{3}}{2}|=\frac{1}{2}+\frac{\sqrt{3}}{2}$$ $$y_2=\frac{\sqrt{3}}{2}-\left(-\frac{1}{2}\right)=\frac{\sqrt{3}}{2}+\frac{1}{2}$$ Since both \(y_1\) and \(y_2\) evaluate to the same value, we can now proceed to compute the derivative of the second function.
03

Compute derivative of \(y=\sin x-\cos x\)

Differentiate the second function to find the slope of the function. $$y^{\prime}=\frac{d}{dx}\left(\sin x-\cos x\right) = \cos x+\sin x$$
04

Evaluate the derivative at \(x=\frac{2\pi}{3}\)

Substitute the given value of \(x=\frac{2\pi}{3}\) into the expression for the derivative: $$y^{\prime}\left(\frac{2\pi}{3}\right) = \cos\left(\frac{2\pi}{3}\right)+\sin\left(\frac{2\pi}{3}\right)$$ $$y^{\prime}\left(\frac{2\pi}{3}\right) = \frac{-1}{2}+\frac{\sqrt{3}}{2}$$ The slope of the function \(y=\sin x-\cos x\) at \(x=\frac{2\pi}{3}\) is \(\frac{-1}{2}+\frac{\sqrt{3}}{2}\).

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

\((f(x))^{n}=f(n x)\) Differentiating it \((f(x))^{n-1} f^{\prime}(x)=f^{\prime}(n x)\) Multiply by \(f(x)\) $\mathrm{f}(\mathrm{nx}) \mathrm{f}^{\prime}(\mathrm{x})=\mathrm{f}^{\prime}(\mathrm{nx}) \mathrm{f}(\mathrm{x})$

Assume that \(f\) is differentiable for all \(x\). The sign of \(f^{\prime}\) is as follows: \(\mathrm{f}^{\prime}(\mathrm{x})>0\) on \((-\infty,-4)\) \(\mathrm{f}^{\prime}(\mathrm{x})<0\) on \((-4,6)\) \(\mathrm{f}^{\prime}(\mathrm{x})>0\) on \((6, \infty)\) Let \(g(x)=f(10-2 x)\). The value of \(g^{\prime}(L)\) is (A) positive (B) negative (C) zero (D) the function \(g\) is not differentiable at \(x=5\)

Find \(\frac{\mathrm{dy}}{\mathrm{dx}}\) Column-I (A) $\sin ^{-1}\left(2 x \sqrt{1-x^{2}}\right),\left(x<-\frac{1}{\sqrt{2}}\right)$ (B) $2 \sin ^{-1}\left(\sqrt{1-x}+\sin ^{-1}(2 \sqrt{x(1-x)}),\left(01\) Column-II (P) 0 (Q) \(-\frac{2}{1+x^{2}}\) (R) \(\frac{3}{\sqrt{1-x^{2}}}\) (S) \(-\frac{2}{\sqrt{1-x^{2}}}\)

\(f(x)=x^{2} \ln g(x)\) \(f^{\prime}(x)=2 x \ln g(x)+\frac{x^{2} g^{\prime}(x)}{g(x)}\) \(f^{\prime}(2)=4 \ln 3-\frac{16}{3}\)

\(\sqrt{\mathrm{y}+\mathrm{x}}+\sqrt{\mathrm{y}-\mathrm{x}}=\mathrm{c}\) Squaring both sides. \(2 y+2 \sqrt{y^{2}-x^{2}}=c^{2}\) Differentiating $2 y^{\prime}+\frac{2\left(y y^{\prime}-x\right)}{\sqrt{y^{2}-x^{2}}}=0$ \(y^{\prime}=\frac{x}{y+\sqrt{y^{2}-x^{2}}}=\frac{2 x}{c^{2}}\) Rationilising, we get \(y^{\prime}=\frac{y-\sqrt{y^{2}-x^{2}}}{x}\)
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

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.