By Robert Miller
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Additional resources for Calc I
As we will see, they have no asymptotes at all. Definition A Degree—if a polynomial has one variable, it is the highest exponent. Definition B Leading coefficient—the coefficient of the highest power. Example 7— Degree is 6. Leading coefficient is -7. Horizontal Asymptote Type I (Don't be scared. ) Suppose y = P(x)/Q(x). P and Q are polynomials. If the degree of P (top) is less than the degree of Q (bottom), the horizontal asymptote is y = 0, the x axis. Example 8— As x goes to infinity 3/X2, -7/X3, and 8/x4 all go to 0.
F(c) exists [f(c) is some number]. 3. A. f'(c-) is negative and f'(c+) is positive, and the cusp looks like this: B. f'(c-) is positive and f'(c +) is negative, and the cusp looks like this: As we will see, if f'(c-) and f'(c+) have the same sign, we will get another kind of inflection point. Example 27— We will test for the cusp first or second kind of inflection point. 1. f'(2)= infinity 2. f(2) = 3 (2,3) 3. f'(2-) is negative, f'(2+) is positive. Cusp with the point down. -3 = (x - 2)4/5. (x - 2)1/5 = ±(-3)1/4, which is imaginary.
F(5-) is positive; f(5+) is negative. The curve near x = 5 is... To summarize, if the exponent is odd positive in the denominator, on one side of the asymptote the curve goes to plus infinity and on the other the curve goes to minus infinity. We are now ready to put all the pieces together. With some study and a little practice, you positively will be able to sketch curves with intercepts and asymptotes only in under two minutes!!!!!!! Example 19— First locate the intercepts. x intercept means y = 0 means top of fraction = 0.
Calc I by Robert Miller