Achilles would have to move 10 meters to catch up to the tortoise, but the tortoise would already have moved another five meters by then. The tortoise, with a 10-meter advantage, Zeno argued, would win. Achilles could run at 10 m/s, while the tortoise only 5. For example, in the paradox of Achilles and the Tortoise, the warrior Achilles was to race against a tortoise. This series was used as a representation of many of Zeno's paradoxes. 2.3 In a myriad ages it will not be exhaustedġ 2 + 1 4 + 1 8 + 1 16 + ⋯ approaches 0 and so s n tends to 1.This shows that the line x = -2 is a vertical asymptote for the graph of f. (The numerator approaches 36 and the denominator is a positive number approaching 0. (The numerator approaches 36 and the denominator is a negative number approaching 0. Remember, if EITHER of these one-sided limits isĪ vertical asymptote exists. This shows that the line x = 2 is a vertical asymptote for the graph of f. (The numerator approaches 4 and the denominator is a negative number approaching 0. (The numerator approaches 4 and the denominator is a positive number approaching 0. Now check for vertical asymptotes by computing one-sided limits at the zeroes of the denominator, i.e., at x=2 and at x=-2. Thus, the line y=1 is a horizontal asymptote for the graph of f. If x=0, then y=-4 so that y=-4 is the y-intercept. See the adjoining sign chart for the second derivative, f''. However, note that f'' is NOT DEFINED at x=2. Now determine a sign chart for the second derivative, f''. See the adjoining sign chart for the first derivative, f'. In addition, note that f' is NOT DEFINED at x=2. Using the quotient rule, we getįor x= 1, and x=3. Now determine a sign chart for the first derivative, f'. SOLUTION 5 : The domain of f is all x-values EXCEPT x=2, because of division by zero. Ĭlick HERE to return to the list of problems. Thus, the line y = 0 is a a horizontal asymptote for the graph of f. So that 4 x=0 and x=0 is the x-intercept. If x=0, then y=0 so that y=0 is the y-intercept. Now summarize the information from each sign chart.į has an absolute minimum at x=-1, y=-2. Now determine a sign chart for the second derivative, f'' :įor x=1. Now determine a sign chart for the first derivative, f' :įor x=0 and x=2. SOLUTION 1 : The domain of f is all x-values. SOLUTIONS TO GRAPHINGOF FUNCTIONS USING THE FIRST AND SECOND DERIVATIVES Solutions to Graphing Using the First and Second Derivatives

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