13-1 Vector-Valued Function

13-1 Vector-Valued Function - x = 1+t y = 3t z = -t t is...

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13.1 Vector-valued function - A vector valued function is a function whose domain is all real numbers but its range is a set of vectors. r(t) = <f(t) , g(t), h(t)> r(t) = f(t)i -> + g(t) j -> + h(t) k -> r(t) = <1+t, 3t, -t> r(2) = <3,6,-2> ^ ^ real numbers vector In 2.D, the graph of r -> (t) = <f(t), g(t)> is a curve on the xy axes. Graph: r -> = <Sint, cost> , - <=t < 2pi r -> = <sint, cost> let's eliminate t x = sint 0 <= t < 2pi y = cost x 2 = sin 2 t y 2 = cos 2 t ----------- x 2 +y 2 = 1 t x y 0 0 1 pi/2 1 0 pi 0 -1 3pi/2 -1 0 2pi 0 1 The graph of r(t) = <f(t), g(t), h(t)> is called a curve but we need to show the surface which the curve lies on. Graph r(t) = <1+t, 3t, -t> and show the surface which the curve lies on
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Unformatted text preview: x = 1+t y = 3t z = -t t is any real number x = x + at y = y +bt z = z + ct t x y z -2 (-1-6 2) 1-1 (0-3 1) 2 (1 0) 3 1 (2 3-1) 4 2 (3 6-2) 5 Ex: r -&gt; (t) =r -&gt; (1-t) + t r 1 -&gt; vector notation of a vector-valued function r-&gt; = r-&gt; + t(r-&gt;- r-&gt; ) r-&gt; = r-&gt; + tr 1-&gt;- tr-&gt; r-&gt; = r-&gt; (1-t) + tr 1-&gt; 0 &lt;= t &lt;=1 r-&gt; = r-&gt; + t p-&gt; p r-&gt; = r-&gt; + t p-&gt; p t=scalar I) r-&gt; = r-&gt; (1-t) + tr 1 t is any real number II) r-&gt; = r-&gt; (1-t) + tr 1 0&lt;=t&lt;=1 line-segment Evaluate: &lt;lim t-&gt;inf tan-1 t, lim t-&gt;inf e-2t , lim t-&gt;inf lnt/t &gt; = &lt;pi/2, 0 , 0&gt; vector...
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13-1 Vector-Valued Function - x = 1+t y = 3t z = -t t is...

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