{"id":10707,"date":"2017-08-18t17:53:43","date_gmt":"2017-08-19t00:53:43","guid":{"rendered":"\/\/www.catharsisit.com\/hs\/?p=10707"},"modified":"2017-08-18t17:54:01","modified_gmt":"2017-08-19t00:54:01","slug":"ap-calculus-review-inflection-points","status":"publish","type":"post","link":"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-review-inflection-points\/","title":{"rendered":"ap calculus review: inflection points"},"content":{"rendered":"

what are inflection points, and how do you find them? this article explains what you need to know about inflection points for the ap calculus exams.<\/p>\n

inflection points and concavity<\/h2>\n

an inflection point<\/strong> is a point in a graph at which the concavity<\/em> changes. <\/p>\n

\"graph
this graph shows a change in concavity, from concave down to concave up. the inflection point is where the transition occurs.<\/figcaption><\/figure>\n

so let’s talk a little about concavity first.<\/p>\n

concavity<\/h3>\n

concavity is about curving.<\/p>\n

we say that a function y<\/em> = f<\/em>(x<\/em>) is concave up (cu)<\/strong> on a given interval if the graph of the function always lies above<\/em> its tangent lines on that interval. in other words, if you draw a tangent line at any given point, then the graph seems to curve upwards, away from the line.<\/p>\n

\"parabola,
this graph is concave up. at each point, the graph curves upward, away from its tangent line.<\/figcaption><\/figure>\n

conversely, a function is concave down (cd)<\/strong> on a given interval if the graph of the function always lies below<\/em> its tangent lines on that interval. that is the graph seems to curve downwards, away from its tangent line at any given point.<\/p>\n

\"parabola,
this graph is concave down. at each point, the graph curves downward, away from its tangent line.<\/figcaption><\/figure>\n

this saying may help you to remember which one is which: concave up looks like a cup; concave down looks like a frown.<\/em><\/p>\n

inflection<\/h3>\n

some functions, like the parabolas shown above, only have one kind of concavity. but more interesting functions may have a mixture of the two. the graph might be concave down in one interval and concave up in the next. some functions even switch back and forth between concave up and concave down infinitely many times!<\/p>\n

any point at which concavity changes (from cu to cd or from cd to cu) is call an inflection point <\/strong>for the function.<\/p>\n

for example, a parabola f<\/em>(x<\/em>) = ax<\/em>2<\/sup> + bx<\/em> + c<\/em> has no inflection points, because its graph is always concave up or concave down.<\/p>\n

inflection points and derivatives<\/h2>\n

inflection points may be difficult to spot on the graph itself. so we must rely on calculus to find them.<\/p>\n

basically, it boils down to the second derivative<\/em>.<\/p>\n