Calculus 220 section 2 3 Sketching a Curve Part 1 notes by Tim Pilachowski Section 2 2 examined first and second derivative attributes of a graph Now we get to begin with the equation and work toward its graph In this section we ll only work with f f extrema and concavity Example A Sketch the graph of f x x 2 5 x 6 Using a table of signs we can determine the slope of the curve on the intervals on either side of the critical value and thus determine whether this value represents a maximum minimum or point of inflection interval sign of f Example B Sketch the graph of f x x 3 3 x 2 9 x 1 Using the factors of f we can determine the slope of the curve on the intervals on either side of and in between these two critical values to determine whether these values represent maxima minima or points of inflection The factors of f enable us to determine concavity Example C Sketch the graph of f x 1 2 x 3 Using the factors of f we can determine the slope of the curve on the intervals on either side of the critical value and thus determine whether this value represents a maximum minimum or point of inflection The factors of f enable us to determine concavity Example D Find an example for which f 0 does not imply a point of inflection A table of signs indicates the nature of the curve at the origin interval sign of y sign of sign of dy dx d2y dx 2 behavior of graph f 0 implies increasing function increasing function does not imply f 0 at all points e g y x 3 f 0 implies a graph is concave up a graph concave up does not imply f 0 at all points e g y x 4 Similar statements can be made with regard to decreasing functions and graphs that are concave down
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