chap3-summary.xml

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<section xml:id="chap3-summary" xmlns:xi="http://www.w3.org/2001/XInclude">
<title>Chapter Summary and Review</title>
<subsection><title>Key Concepts</title>
<p>
	<ol label="*1*">
		<li><p>
			<assemblage><title>Direct and Inverse Variation</title>
			<p><ul>
				<li><p><term><m>y</m> varies directly with <m>x</m></term> if the ratio <m>\dfrac{y}{x} </m> is constant, that is, if <m>y = kx</m> .</p></li>
				<li><p><term><m>y</m> varies directly with a power of <m>x</m></term> if the ratio <m>\dfrac{y}{x^n} </m> is constant, that is, if <m>y = kx^n</m> .</p></li>
				<li><p><term><m>y</m> varies inversely with <m>x</m></term> if the product <m>xy </m> is constant, that is, if <m>y =\dfrac{k}{x}</m> .</p></li>
				<li><p><term><m>y</m> varies inversely with a power of <m>x</m></term> if the product <m>x^ny </m> is constant, that is, if <m>y =\dfrac{k}{x^n}</m> .</p></li>
			</ul></p>
			</assemblage>
		</p></li>
		<li><p>
			The graph of a direct variation passes through the origin. The graph of an inverse variation has a vertical asymptote at the origin.
		</p></li>
		<li><p>
			If <m>y = kx^n</m>, we say that <m>y</m> <term>scales</term><idx>scales</idx> as <m>x^n</m>.
		</p></li>
		<li><p>
			<m>n</m>th roots: <m>s</m> is called an <m>n</m>th root of <m>b</m> if <m>s^n = b</m>.
		</p></li>
		<li><p>
			<assemblage><title>Exponential Notation</title>
			<p>
			   The absolute value has the following properties:
				<md alignment="alignat">
			        <mrow>\amp a^{-n}=\frac{1}{x^n} \amp\hphantom{00000}  \amp a\ne 0</mrow>
			        <mrow>\amp a^{0}=1 \amp  \amp a\ne 0</mrow>
			        <mrow>\amp a^{1/n}=\sqrt[n]{a} \amp  \amp n \text{ an integer, } n\gt 2</mrow>
			        <mrow>\amp a^{m/n}=(a^{1/n})^m=(a^m)^{1/n}, \amp  \amp a\gt 0, ~n\ne 0</mrow>
				</md>
			</p>
			</assemblage>
		</p></li>
		<li><p>
			In particular, a negative exponent denotes a reciprocal, and a fractional exponent denotes a root.
		</p></li>
		<li><p><m>a^{m/n} = \sqrt[n]{a^m} =\left(\sqrt[n]{a}\right)^m</m></p></li>
		<li><p>
			To compute <m>a^{m/n}</m>, we can compute the <m>n</m>th root first, or the <m>m</m>th power, whichever is easier.
		</p></li>
		<li><p>We cannot write down an exact decimal equivalent for an irrational number, but we can approximate an irrational number to as many decimal places as we like.</p></li>
		<li><p>The laws of exponents are valid for all exponents <m>m</m> and <m>n</m>, and for <m>b\ne 0</m>.</p><p></p>
			<assemblage><title>Laws of Exponents</title><p>
                <ol label="*I*" cols="2">
                	<li><p><m>a^m\cdot a^n=a^{m+n}</m></p></li>
                	<li><p><m>\dfrac{a^m}{a^n}=a^{m-n}</m></p></li>
                	<li><p><m>\left(a^m\right)^n=a^{mn}</m></p></li>
                	<li><p><m>\left(ab\right)^n=a^n b^n</m></p></li>
                	<li><p><m>\left(\dfrac{a}{b}\right)^n=\dfrac{a^n}{b^n} </m></p></li>
                </ol>
			</p></assemblage>
		</li>
		<li><p>A function of the form <m>f (x) = kx^p</m>, where <m>k</m> and <m>p</m> are constants, is called a <term>power function</term>.</p></li>
		<li><p>An <term>allometric equation</term> is a power function of the form <m>\text{variable} = k(\text{mass})^p</m>.</p></li>
		<li><p>We can solve the equation <m>x^n = b</m> by raising both sides to the <m>\dfrac{1}{n}</m> power</p></li>
		<li><p>We can solve the equation <m>x^{1/n} = b</m> by raising both sides to the <m>n</m>th power.</p></li>
		<li><p>To solve the equation <m>x^{m/n} = k</m>, we raise both sides to the power <m>n/m</m>.</p></li>
		<li><p>The graphs of power functions <m>y = x^{m/n}</m>, where <m>m/n</m> is positive are all increasing for <m>x\ge 0</m>. If <m>m/n\gt 1</m>, the graph is concave up. If <m>0\lt m/n \lt 1</m>, the graph is concave down.</p></li>
		<li><p>The notation <m>z = f (x, y)</m> indicates that <m>z</m> is a function of two variables, <m>x</m> and <m>y</m>.</p></li>
		<li><p> We can use a table with rows and columns to display the output values for a function of two variables.</p></li>
		<li><p>
			<assemblage><title>Joint Variation</title><p>
				<ul>
					<li><p>We say that <m>z</m> <term>varies jointly</term> with <m>x</m> and <m>y</m> if
					<me>z = kxy, ~~~k \ne 0</me>
					</p></li>
					<li><p>We say that <m>z</m> varies directly with <m>x</m> and inversely with <m>y</m> if
					<me>z=k\frac{x}{y}, ~~~k\ne 0, ~~ y\ne 0 </me>
					</p></li>
				</ul>
			</p></assemblage>
		</p></li>
		<li><p>We can represent a function of two variables graphically by showing a set of graphs for several fixed values of one of the variables.</p></li>
		<li><p>
			<assemblage><title>Roots of Real Numbers</title><p>
				<ul>
					<li><p>Every positive number has two real-valued roots, one positive and one negative, if the index is even.</p></li>
					<li><p>A negative number has no real-valued root if the index is even.</p></li>
					<li><p>Every real number, positive, negative, or zero, has exactly one real-valued root if the index is odd.</p></li>
				</ul>
			</p></assemblage>
		</p></li>
	</ol>.
</p>
</subsection>
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</section>
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