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Chapter 4 | Applications of Derivatives
4.27 Sketch a graph of f ( x ) = ( x −1) 3 ( x +2).
Example 4.29 Sketching a Rational Function
2
x
Sketch the graph of f ( x ) =
.
⎛ ⎝ 1− x 2
⎞ ⎠
Solution Step 1. The function f is defined as long as the denominator is not zero. Therefore, the domain is the set of all real numbers x except x =±1. Step 2. Find the intercepts. If x =0, then f ( x ) =0, so 0 is an intercept. If y =0, then x 2 ⎛ ⎝ 1− x 2 ⎞ ⎠ =0, which implies x =0. Therefore, (0, 0) is the only intercept. Step 3. Evaluate the limits at infinity. Since f is a rational function, divide the numerator and denominator by the highest power in the denominator: x 2 . We obtain lim x →±∞ x 2 1− x 2 = lim x →±∞ 1 1 x 2 −1 =−1. Therefore, f has a horizontal asymptote of y =−1 as x →∞ and x →−∞. Step 4. To determine whether f has any vertical asymptotes, first check to see whether the denominator has any zeroes. We find the denominator is zero when x =±1. To determine whether the lines x =1 or x =−1 are vertical asymptotes of f , evaluate lim x →1 f ( x ) and lim x →−1 f ( x ). By looking at each one-sided limit as x →1, we see that
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