How to show non-negative tangents ?

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Homework Help Overview

The discussion revolves around demonstrating that there are no tangents to the graph of the function f(x) = (5x + 2) / (x + 2) with a negative slope, as well as determining the equation of the tangent to the curve y = (x² - 1) / (3x) at x = 2. The subject area includes calculus concepts related to derivatives and tangent lines.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants suggest various methods for finding the tangent, including using the product rule with the chain rule, performing polynomial division, and considering inequalities to analyze the function's behavior. There is also a discussion about the necessary information required to define a tangent.

Discussion Status

The discussion is active, with participants exploring different approaches to the problem. Some have raised questions about the assumptions underlying the methods proposed, particularly regarding the use of inequalities and the focus on the derivative rather than the function itself.

Contextual Notes

There is a constraint mentioned regarding the use of the quotient rule, as it is not part of the curriculum for the participants. This has influenced the suggested methods for solving the problems.

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Homework Statement


1. Show that there are no tangents to the graph of [tex]f(x) = \frac{5x+2}{x+2}[/tex] that have a negative slope.

2. Determine the equation of the tangent to the curve [tex]y = \frac{x^{2}-1}{3x}[/tex] at x = 2

Homework Equations


1. Our teacher told us not to use the quotient rule because it is not in our curriculum so please do not give me a solution using the quotient rule. Thanks.
 
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What about using the product rule with the chain rule then? Use the fact that A/B=AB-1.
 
1. Or Perform Polynomial division on f(x), then its quite simple.

2. You need 3 pieces of information for a tangent - what are they?
 
for 1. don't i use inequality ? set the equation f(x) < 0 and solve for x if it can't be solved then u can conclude there are no points that can give me a negative tangent
 
ghostanime2001 said:
for 1. don't i use inequality ? set the equation f(x) < 0 and solve for x if it can't be solved then u can conclude there are no points that can give me a negative tangent

If you solve f(x)<0, you will be finding all x for which the function itself is negative i.e. below the x axis. But that's not what you want to find, you need to show that the DERIVATIVE is never negative.
 

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