Math 220Calculus INotes for Exam 1It was the best of times, it was the worst of times.Things was real good, things was turribel.It was the best of times, it was the worst of times.Things was real good, things was turribel.Just as in any other class, you have to use the vocabulary correctly and make the meaning of your answers clear. Mathematics has its own “grammar”.You need to know and use the vocabulary of calculus correctly. ⇒′f⇒′′f⇒f(x, y) points on the curve, graphrate of change,slope of the curve, maximum/minimumconcavity of the curve, points of inflectionYou need to know and use the vocabulary of calculus correctly. is increasingff′>′,0f′is positiveis decreasingff′<′,0f′is negativeYou need to know and use the vocabulary of calculus correctly. 00<′′=′fff has a maximumf is concave down00>′′=′fff has a minimumf is concave upYou need to know and use the vocabulary of calculus correctly. is concave upff′′>′′,0f′′is positiveis concave downff′′<′′,0f′′is negative is a point of inflectionff′′=′′,0f′′is zeroYou need to know and use the vocabulary of calculus correctly. Both of these are points of inflection—concavity changes.00=′′>′ff00=′′=′ffAt a point of inflection the first derivative could be anything.A B f (x) ()xF()xfA B f (x) “Explain how you know.”“…because it is…”What is “it” ?the function?the graph?the slope?the first derivative?the second derivative?A B f (x) at f′atat point Aat point Bf′′A B f (x) A horizontal tangent has a slope = 0.A horizontal tangent has no slope.at point A0=′fA B f (x) Point A is a point of inflection.The curve changes from concave up to concave down.at point A0=′′fA B f (x) Point B is a relative minimum point.The curve has a horizontal tangent.at point B0=′fA B f (x) is positive.The curve is concave up.at point B0>′′ff′′In answering word problems (applications), it will be important to distinguish betweenamounts (the function itself) anda rate of change (the derivative of the
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