Chapter 12: Q 32. (page 916)

In exercise, let
$g_{1} (t) = \sin t, g_{2} (t) = \cos t, g_{3} (t) = 1 - t, f_{1} (x, y) = x^{2} + y^{2}, f_{2} (x, y) = \frac{x^{2}}{y^{2}}, f_{3} (x, y, z) = \frac{x + y}{y + z}, r_{1} (t) = ⟨ 1 + t, t - 1 ⟩, r_{2} (t) = <t, t^{2}, t^{3}>$
Either simplify the specified composition or explain why the composition cannot be formed.
$f_{1} (r_{1} (t))$

Short Answer

Expert verified

Answer is $2 t^{2} + 2$

Step by step solution

Step 1. Given information

$g_{1} (t) = \sin t, g_{2} (t) = \cos t, g_{3} (t) = 1 - t, f_{1} (x, y) = x^{2} + y^{2}, f_{2} (x, y) = \frac{x^{2}}{y^{2}}, f_{3} (x, y, z) = \frac{x + y}{y + z}, r_{1} (t) = ⟨ 1 + t, t - 1 ⟩, r_{2} (t) = <t, t^{2}, t^{3}>$

Step 2. Explanation

$\begin{array}{rcl} f_{1} (x, y) & = & x^{2} + y^{2} \\ r_{1} (t) & = & < 1 + t, t - 1 > \\ f_{1} (r_{1} (t)) & = & f_{1} (< 1 + t, t - 1 >) \\ = & {(1 + t)}^{2} + {(t - 1)}^{2} \\ = & 1 + 2 t + t^{2} + t^{2} - 2 t + 1 \\ = & 2 t^{2} + 2 \end{array}$

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In exercise, letg1(t)=sint,g2(t)=cost,g3(t)=1-t,f1(x,y)=x2+y2,f2(x,y)=x2y2,f3(x,y,z)=x+yy+z,r1(t)=⟨1+t,t-1⟩,r2(t)=t,t2,t3Either simplify the specified composition or explain why the composition cannot be formed.f1r1t

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