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University of North Texas

# Let's consider the problem of designing a railroad track tomake a smooth transition between sections of straighttrack. Existing track along the negative $x$ -axis is to bejoined smoothly to a track along the line $y=1$ for $x \geqslant 1$(a) Find a polynomial $P=P(x)$ of degree 5 such that thefunction $F$ defined by$$F(x)=\left\{\begin{array}{ll}{0} & {\text { if } x \leqslant 0} \\ {P(x)} & {\text { if } 0<x<1} \\ {1} & {\text { if } x \geqslant 1}\end{array}\right.$$is continuous and has continuous slope and continuous curvature.(b) Usc a graphing calculator or computer to draw thegraph of $F .$

## $P(x)=10 x^{3}-15 x^{4}+6 x^{5}$

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University of North Texas

Vectors

Vector Functions

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