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Unformatted text preview: Lecture 10 18.01 Fall 2006 Lecture 10: Curve Sketching Goal : To draw the graph of f using the behavior of f and f . We want the graph to be qualitatively correct, but not necessarily to scale. Typical Picture: Here, y is the minimum value, and x is the point where that minimum occurs. x = critical point y Figure 1: The critical point of a function Notice that for x < x , f ( x ) < 0. In other words, f is decreasing to the left of the critical point. For x > x , f ( x ) > 0: f is increasing to the right of the critical point. Another typical picture: Here, y is the critical (maximum) value, and x is the critical point. f is decreasing on the right side of the critical point, and increasing to the left of x . x = critical point y f’(x) < 0 x > x Figure 2: A concave-down graph 1 Lecture 10 18.01 Fall 2006 Rubric for curve-sketching 1. (Precalc skill) Plot the discontinuities of f — especially the infinite ones! 2. Find the critical points. These are the points at which f ( x ) = (usually where the slope changes from positive to negative, or vice versa.) 3. (a) Plot the critical points (and critical values), but only if it’s relatively easy to do so....
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- Fall '08