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Extra resources for Calculus with analytic geometry
If the degrees of the numerator and denominator case, are equal, then tends to a finite non-zero number, the quotient of the leading coefficients. 8. Polynomials and Rational Functions 39 To prove this assertion, we use· the technique introduced at the beginning of this section. We write the numerator as a,. x'"h(x) and the denominator as b,. x"k(x), where h(x) - 1 and k(x) - 1 as l x l - oo. x"k(x) b,. x" b,. and the conclusion follows. " I x I is large. Examples Assume x5 + 3X X5 r(x) = i � =x x + 12 x2 hence r(x) - oo as x - oo, and r(x) - - oo as x 6x2 + 7x - 3 6x2 3 2.
X". Hence for I x I very large, the graph of y = f(x) is like the graph of y = a,. x". As x ---+ oo or x ---+ - oo, it either zooms up or down, depending on the sign of a,. and (for x ---+ oo ) whether n is even or odd. - 1 . F UN CT I O NS AND G RAPHS 38 Polynomials of the form Factored Polynomials f(x) = (x - r 1 )(x - r2 ) • • • (x - r,,) are particularly easy to graph. Each r1 is a zero off(x), that is,f(r1 ) = 0. )(x - r2 ) • • • (x - r,, ) can equal 0 only if one of the factors equals 0, that is, only if x is one of the numbers r1 , r2 , .
A) The other intersection is P • (m - a,(m - a) 2 ). x (b) It coincides with (a, a2 ) when m • 2a. Fia. 3 Tangent to y = x1 at (a, a1) 62 1 . F U NCT I O N S A N D G RAPHS It meets y = xl where xl - a2 = m(x - a). This quadratic equation has two solutions. One we know in advance is x = a; divide it out: x + a = m. The other is x = m - a. It also is equal to a if and only if m = 2a. So here is our desired slope. Consequently the tangent line (Fig. 3) is • y - al = 2a(x - a), that is, }' = 2ax - a1.