Detailed Example of Curve Sketching Example Sketch the graph of f (x) = for x > 0) x . (Note: this function is only deﬁned ln x 1. Plot a The function is discontinuous at x = 1, because ln 1 = 0. 1 1 = + =∞ + ln 1 0 f (1+ ) = f (1− ) = 1 1 = − = −∞ − ln 1 0 b endpoints (or x → ±∞) f (0+ ) = 0+ 0+ = =0 ln 0+ −∞ The situation is a little more complicated at the other end; we’ll get a feel for what happens by plugging in x = 1010 . f (1010 ) = 1010 1010 109 = = >> 1 ln 1010 10 ln 10 ln 10 We conclude that f (∞) = ∞. We can now start sketching our graph. The point (0, 0) is one endpoint of the graph. There’s a vertical asymptote at x = 1, and the graph is descending before and after the asymptote. Finally, we know that f (x) increases to positive inﬁnity as x does. We already have a pretty good idea of what to expect from this graph! 2. Find the critical points f � (x) 1 · ln x − x (ln x)2 (ln x) − 1 (ln x)2 = = �1� x a f � (x) = 0 when ln x = 1, so when x = e. This is our only critical point. b f (e) = lnee = e is our critical value. The point (e, e) is a critical point on our graph; we can label it with the letter c. (It’s ok if our graph is not to scale; we’ll do the best we can.) 1 (e, e) C x=1 Figure 1: Sketch using starting point, asymptote, critical point and endpoints. We now know the qualitative behavior of the graph. We know exactly where f is increasing and decreasing because the graph can only change direction at critical points and discontinuities; we’ve identiﬁed all of those. The rest is more or less decoration. 3. Double check using the sign of f � . We already know: f f is decreasing on is decreasing on 0 e2 . The denominator is positive when x > 1 and negative when x < 1. Combining these, we get: 0 0 (concave up) =⇒ f �� (x) < 0 (concave down) This means that there’s a “wiggle” at the point � 2 e2 , e2 � on the graph. The value of f (x) is still increasing and the graph continues to rise, but the graph is rising less and less steeply as the values of f � (x) decrease. 5. Combine this information to draw the graph. We’ve been doing this as we go. If you’re working a homework problem, at this point you might copy your graph to a clean sheet of paper. This is probably as detailed a graph as we’ll ever draw. In fact, one advantage of our next topic is that it will reduce the need to be this detailed. 3 2 (e2 , e2 ) I (e, e) C x=1 Figure 2: Final sketch. 4