/*! 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 6 \(f^{\prime}(x)= \begin{cases}-2... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 6

\(f^{\prime}(x)= \begin{cases}-2 x, & x<0 \\ 2 x, & x \geq 0\end{cases}\) eqn of tangent $\rightarrow \quad y-y_{1}=2 x_{1}\left(x-x_{1}\right), x \geq 0$ \(y-y_{2}=-2 x_{2}\left(x-x_{2}\right) x<0\)

Short Answer

Expert verified
Answer: For x < 0: $$ y - y_{2} = -2 x_{2}(x - x_{2}) $$ For x 鈮� 0: $$ y - y_{1} = 2 x_{1}(x - x_{1}) $$

Step by step solution

01

Case 1: x < 0

In this case, the derivative function, f'(x), is given as -2x. Now we need to find the equation of the tangent line. To do this, consider an arbitrary point (x1, y1) with x1 < 0, since we are working with the case x < 0.
02

Equation of Tangent line for x < 0

Following the standard formula for the equation of a line, the equation of the tangent line when x < 0 will be: $$ y - y_{2} = -2 x_{2}(x - x_{2}), (x_{2} < 0) $$ This equation represents the tangent line for the given derivative function when x < 0.
03

Case 2: x 鈮� 0

In this case, the derivative function, f'(x), is given as 2x. Now we need to find the equation of the tangent line. To do this, we consider an arbitrary point (x1, y1) with x1 鈮� 0.
04

Equation of Tangent line for x 鈮� 0

Following the standard formula for the equation of a line, the equation of the tangent line when x 鈮� 0 will be: $$ y - y_{1} = 2 x_{1}(x - x_{1}), (x_{1} \geq 0) $$ This equation represents the tangent line for the given derivative function when x 鈮� 0. So, the equations of the tangent lines for the given piecewise derivative function are as follows: For x < 0: $$ y - y_{2} = -2 x_{2}(x - x_{2}) $$ For x 鈮� 0: $$ y - y_{1} = 2 x_{1}(x - x_{1}) $$

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!

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

For \(y=x(c-x)\) \(\frac{d y}{d x}=c-2 x\) For \(y=x^{2}+a x+b\) \(\frac{d y}{d x}=2 x+a\) At \((1,0)\) \(0=(C-1) \Rightarrow C=1\) \(0=1+a+b, 1-2=2+a\) \(\Rightarrow \quad a=-3\) \(\Rightarrow b=2\)

\(y=\frac{a x}{1+x}=a-\frac{a}{1+x}\) \(\frac{d y}{d x}=\frac{a}{(1+x)^{2}}=-1\) \(a=-\left(1+x_{1}\right)^{2}\) eqn of tangent \(y-\frac{a x_{1}}{1+x_{1}}=-1\left(x-x_{1}\right)\) \(\mathrm{y}+\mathrm{x}=\mathrm{x}_{1}+\frac{\mathrm{ax}_{1}}{1+\mathrm{x}_{1}}\) \& \(y-3=-x \Rightarrow y+x=3\) \(\Rightarrow \mathrm{x}_{1}+\frac{\mathrm{ax}_{1}}{1+\mathrm{x}_{1}}=3\) \(\Rightarrow \mathrm{x}_{1}-\mathrm{x}_{1}\left(1+\mathrm{x}_{1}\right)=3\)

\(f_{1}^{\prime}(x)=2 x-1 \quad \& \quad f_{2}^{\prime}(x)=3 x^{2}-2 x-2\) \(\Rightarrow 2 x_{1}-1=3 x_{2}^{2}-2 x_{2}-2\) \(\Rightarrow 3 x_{2}^{2}-2 x_{2}-2 x_{1}-1=0\) For \(\mathrm{x}_{2}\) be real, \(\mathrm{D} \geq 0\) \(4-4\left(2 x_{1}+1\right)(3)\) \(4-24 x_{1}-12\) \(\Rightarrow-\left(24 x_{1}+8\right)\) There can be infinite such values of \(\mathrm{x}_{\mathrm{l}}\)

\(f(0)<0\) \(\Rightarrow \mathrm{a}^{2}+\mathrm{a}-2<0\) \(\Rightarrow a \in(-2,1)\) Integral values of a are \(-1,0\).

Let \(y=m x+\frac{1}{m}\) be tangent to \(y^{2}=4 x\) eqn of normal at \(\left(x_{1}, y_{1}\right)\) to \(x^{2}=4\) by is \(y-y_{1}=-\frac{2 b}{x_{1}}\left(x-x_{1}\right)\) \(\Rightarrow y=-\frac{2 b}{x_{1}} x+\frac{x_{1}^{2}}{4 b}+2 b\) Comparing two eqn \(\mathrm{m}=-\frac{2 \mathrm{~b}}{\mathrm{x}_{1}}\) \(\frac{\mathrm{x}_{1}^{2}}{4 \mathrm{~b}}+2 \mathrm{~b}=\frac{1}{\mathrm{~m}}\) Using (I) \& (II)
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

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.