5.3 Using Derivatives for Curve Sketching AP Calculus AB 5.3 Using Derivatives for Curve Sketching
In the past, one of the important uses of derivatives was as an aid in curve sketching. We usually use a calculator of computer to draw complicated graphs, it is still important to understand the relationships between derivatives and graphs.
First derivative: is positive Curve is rising. is negative Curve is falling. is zero Possible local maximum or minimum. Second derivative: is positive Curve is concave up. is negative Curve is concave down. is zero Possible inflection point (where concavity changes).
Example: Graph There are roots at and . Possible extreme at . We can use a chart to organize our thoughts. Set First derivative test: negative positive positive
Example: Graph There are roots at and . Possible extreme at . Set First derivative test: maximum at minimum at
Example: Graph NOTE: On the AP Exam, it is not sufficient to simply draw the chart and write the answer. You must give a written explanation! First derivative test: There is a local maximum at (0,4) because for all x in and for all x in (0,2) . There is a local minimum at (2,0) because for all x in (0,2) and for all x in .
Example: Graph There are roots at and . Possible extreme at . Or you could use the second derivative test: Because the second derivative at x = 0 is negative, the graph is concave down and therefore (0,4) is a local maximum. Because the second derivative at x = 2 is positive, the graph is concave up and therefore (2,0) is a local minimum.
Example: Graph We then look for inflection points by setting the second derivative equal to zero. Possible inflection point at . negative positive inflection point at There is an inflection point at x = 1 because the second derivative changes from negative to positive.
Make a summary table: rising, concave down local max falling, inflection point local min rising, concave up
Example: Use the concavity test to determine intervals on which the graph of the function is (a) concave up, and (b) concave down. First, find the first and second derivatives. Notice that at x = 0, y '' is undefined. That means something is happening at x = 0. So, check values on both sides of x = 0.
Example: Find all points of inflection of the function. First, find the first and second derivatives. x = -2 is a point of inflection because the graph changes from concave down to concave up at x = -2.
Example: Use the graph of the function f to estimate where (a) f ' and (b) f '' are 0, positive, and negative. (a) f ' = 0 at any maximum or minimum values; or x = -1 and x = 1. f ' > 0 where f is increasing; (-∞, -1) and (1, ∞) f ' < 0 where f is decreasing; (-1, 1)
Example: Use the graph of the function f to estimate where (a) f ' and (b) f '' are 0, positive, and negative. (b) f '' = 0 at any points of inflection; or x = 0. f '' > 0 where f is concave up; (0, ∞) f '' < 0 where f is concave down; (- ∞, 0) p