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Chapter 4: Problem 12

This problem focuses on the difference between being piecewise linear (made up of straight lines) and being locally linear (being approximately linear when magni ed enough). Consider the functions \(f, g\), and \(h\) below. \(f(x)=|x+2|-3\) \(g(x)=\left\\{\begin{array}{ll}x & \text { for } x \leq 0 \\ x^{2} & \text { for } x>0\end{array}\right.\) \(h(x)=(x-1)^{10}+1\) (a) Graph \(f, g\), and \(h\). (b) Specify all intervals for which the given function is linear (a straight line.) i. \(f\) ii. \(g\) iii. \(h\) (c) Specify the point(s) at which the given function is not locally linear (that is, where it does not look like a straight line, no matter how much you zoom in). i. \(\bar{f}\) ii. \(g\) iii. \(h\)

Short Answer

Expert verified
The functions f, g, and h are linear over the intervals (-infinity,-2), [2,infinity); -infinity,0] and none, respectively. The points at which they are not locally linear are -2, 0, and for h, it is always not locally linear.

Step by step solution

01

Graphing the functions

One can graph \(f(x) = |x+2| - 3\), \(g(x) = x \text{ for } x \leq 0 \text{ and } x^2 \text{ for } x > 0\), and \(h(x) = (x-1)^{10} + 1\) using a graphing calculator or plotting tool. This will provide a visual representation of their behaviours.
02

Finding the intervals of linearity

To find the intervals of linearity, examine where each function forms a straight line. For \(f\), it is linear everywhere except at \(x = -2\); for \(g\), it is linear when \(x \leq 0\), and for \(h\), there's no interval where it is linear as \(h\) is a degree 10 polynomial function.
03

Identifying non-locally linear points

The points at which a function fails to be locally linear are those where, no matter how closely one examines the point, it does not appear to lay on a straight line. For \(f\), this point is \(x = -2\); for \(g\), it is \(x = 0\); and for \(h\) since it's not linear on any open interval, it's not locally linear at any point.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Local Linearity
Local linearity is an important concept in understanding how a function behaves at a small scale. Imagine zooming in on a curve until it looks straight; this is the essence of local linearity. Not all functions have this property. For example, the function \(f(x) = |x+2| - 3\) is not locally linear at \(x = -2\). This is because, no matter how much you zoom in around this point, the sharp corner persists, and it does not resemble a straight line.

Similarly, for the function \(g(x) = x^2\) when \(x > 0\), the smooth curve does not appear linear even under strong magnification, therefore \(x = 0\) is a point where it is not locally linear. On the other hand, \(h(x) = (x-1)^{10} + 1\) does not exhibit local linearity at any point due to the nature of polynomial shapes inherently possessing curves.

When examining local linearity, keep in mind:
  • Sharp corners or edges typically indicate non-local linearity.
  • Smaller zoom on smooth curves can still show non-linear characteristics.
Graphing Functions
Graphing functions helps us visualize how they behave over different intervals. When graphing the function \(f(x) = |x+2| - 3\), you will see a V shape, which breaks at the point \(x = -2\).

For \(g(x)\), the graph consists of two parts: a line for \(x \leq 0\) and a parabola for \(x > 0\). This piecewise nature can create interesting visual features.

The graph of \(h(x) = (x-1)^{10} + 1\) shows a steep curve due to the high degree of the polynomial function. When graphing functions, remember to:
  • Identify key points like intercepts and breaks.
  • Observe the continuity and smoothness of the graph.
  • Use graphing tools for accuracy and a clear visual understanding.
Intervals of Linearity
Intervals of linearity are the sections of a graph where the function behaves like a straight line. The function \(f(x) = |x+2| - 3\) is linear on intervals except for the point \(x = -2\). This means at every other point, the graph appears as a straight line.

In the case of \(g(x)\), it remains linear for \(x \leq 0\) as the graph is a straight line in this section. However, for \(x > 0\), \(g(x)\) describes a curve and thus is not linear.

For \(h(x) = (x-1)^{10} + 1\), there is no interval of linearity due to its polynomial nature. To understand intervals of linearity better:
  • Look for sections where graphs don鈥檛 curve.
  • Identify discontinuities which might end linear intervals.
  • Recognize how multiplication or powers affect straightness.

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Most popular questions from this chapter

A horseman has some ponies of his own and boards horses for other people. For his own ponies, he orders 9 bales of hay from the supplier. The total number of bales he orders increases linearly with the number of horses he boards. When he boards 6 horses, he orders a total of 36 bales of hay (for these horses and his ponies). Express the number of bales of hay he orders as a function of the number of horses he boards.

$$ \text { For Problems } \text { find the equation of the line with the given characteristics. } $$ $$ \text { Passing through points }(\pi, 3) \text { and }(-\pi, 5) $$

$$ \text { For Problems } \text { find the equation of the line with the given characteristics. } $$$$ \text { Passing through point }(\sqrt{3}, \sqrt{2}) \text { and parallel to } 3 x-4 y=7 $$

On the same set of axes, sketch lines through point \((0,1)\) with the slopes indicated. Label the lines. (a) slope \(=0\) (b) slope \(=\frac{1}{2}\) (c) slope \(=1\) (d) slope \(=2\) (e) slope \(=-\frac{1}{2}\) (f) slope \(=-1\) (g) slope \(=-2\)

You ve been presented with two different pay plans for the same job. Plan A offers $$\$ 12$$ per hour with overtime (hours above 40 per week) paying time and a half. Plan B offers $$\$ 14$$ per hour with no overtime. Let \(x\) denote the number of hours you work each week. Let \(P_{\mathrm{A}}(x)\) give the weekly pay under plan \(\mathrm{A}\) and \(P_{\mathrm{B}}(x)\) give the weekly pay under plan B. (a) What is the algebraic formula for \(P_{\mathrm{B}}(x) ?\) (b) What is the algebraic formula for \(P_{\mathrm{A}}(x) ?\) Note that you must de ne this function differently for \(x \leq 40\) and for \(x>40\). Check your answer and make sure that the pay for a 50 -hour work week is $$\$ 660 .$$ (c) i. For what value(s) of \(x\) are the two plans equivalent? ii. For what values of \(x\) is plan B better? (Hint: A good problem-solving strategy is to draw a graph so you can really see what is going on.) (d) True or False: i. \(P_{\mathrm{B}}(x+y)=P_{\mathrm{B}}(x)+P_{\mathrm{B}}(y)\) ii. \(P_{\mathrm{A}}(x+y)=P_{\mathrm{A}}(x)+P_{\mathrm{A}}(y)\) (Hint: If you are not sure how to approach a problem, a good strategy frequently used by mathematicians everywhere! is to try a concrete case. If the statement is false for this special case, then you know the statement is de nitely false. If the statement holds for this special case, then the process of working through the special case may help you determine whether the statement holds in general.) Caution: Since the rule for \(P_{\mathrm{A}}(x)\) changes for \(x>40\), you need to check several cases. If any case doesn \(\mathrm{t}\) hold, then the statement is false.
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